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arXiv:2003.11951v2 [math.OC] 10 Jul 2020
On the Complexity and Approximability of Optimal Sensor Selection
and Attack for Kalman Filtering
Lintao Ye, Nathaniel Woodford, Sandip Roy and Shreyas Sundaram
Abstract—Given a linear dynamical system affected by stochas-
tic noise, we consider the problem of selecting an optimal
set of sensors (at design-time) to minimize the trace of the
steady state a priori or a posteriori error covariance of the
Kalman filter, subject to certain selection budget constraints.
We show the fundamental result that there is no polynomial-
time constant-factor approximation algorithm for this problem.
This contrasts with other classes of sensor selection problems
studied in the literature, which typically pursue constant-factor
approximations by leveraging greedy algorithms and submodu-
larity (or supermodularity) of the cost function. Here, we provide
a specific example showing that greedy algorithms can perform
arbitrarily poorly for the problem of design-time sensor selection
for Kalman filtering. We then study the problem of attacking (i.e.,
removing) a set of installed sensors, under predefined attack
budget constraints, to maximize the trace of the steady state
a priori or a posteriori error covariance of the Kalman filter.
Again, we show that there is no polynomial-time constant-factor
approximation algorithm for this problem, and show specifically
that greedy algorithms can perform arbitrarily poorly.
I. INT RO DUC TI ON
In large-scale control system design, the number of sensors
or actuators that can be installed is typically limited by a
design budget constraint. Moreover, system designers often
need to select among a set of possible sensors and actuators,
with varying qualities and costs. Consequently, a key problem
is to select an appropriate set of sensors or actuators in order to
achieve certain objectives. This problem has recently received
much attention from researchers (e.g., [1]–[9]). One specific
instance of this problem arises in the context of linear Gauss-
Markov systems, where the corresponding Kalman filter (with
the chosen sensors) is used to estimate the states of the systems
(e.g., [10], [11]). The problem then becomes how to select
sensors dynamically (at run-time) or statically (at design-
time) to minimize certain metrics of the corresponding Kalman
filter. The former scenario is known as the sensor scheduling
problem, where different sets of sensors can be chosen at
different time steps (e.g., [12]–[14]). The latter scenario is
known as the design-time sensor selection problem, where the
set of sensors is chosen a priori and is not allowed to change
over time (e.g., [15]–[17]).
Since these problems are NP-hard in general (e.g., [18]),
approximation algorithms that provide solutions within a cer-
tain factor of the optimal are then proposed to tackle them.
This research was supported by NSF grants CMMI-1635014 and
CMMI-1635184. The work of the second author was also supported by
the Purdue Military Research Initiative. Lintao Ye, Nathaniel Woodford
and Shreyas Sundaram are with the School of Electrical and Computer
Engineering at Purdue University, West Lafayette, IN 47906 USA. Email:
{ye159,nwoodfor,sundara2}@purdue.edu. Sandip Roy is with the School of
Electrical Engineering and Computer Science at Washington State University,
Pullman, WA 99164 USA. Email: sroy@eecs.wsu.edu.
Among these approximation algorithms, greedy algorithms
have been widely used (e.g, [19], [20]), since such algorithms
have provable performance guarantees if the cost function is
submodular or supermodular (e.g., [21], [22]).
Additionally, in many applications, the sensors that have
been selected and installed on the system are susceptible
to a variety of potential attacks. For instance, an adversary
(attacker) can inject false data to corrupt the state estimation,
which is known as the false data injection attack (e.g., [23]–
[25]). Another type of attack is the Denial-of-Service (DoS)
attack, where an attacker tries to diminish or eliminate a
network’s capacity to achieve its expected objective [26],
including, for example, wireless jamming (e.g., [27], [28]) and
memory exhaustion through flooding (e.g., [29]). One class of
DoS attacks corresponds to removing a set of installed sensors
from the system, i.e., the measurements of the attacked sensors
are not used. This was also studied in [30] and [31], and will
be the type of attack we consider in this work.
In this paper, we consider the sensor selection problem and
the sensor attack problem for Kalman filtering of discrete-
time linear dynamical systems. First, we study the problem of
choosing a subset of sensors to install (under given selection
budget constraints) to minimize the trace of either the steady
state a priori or a posteriori error covariance of the Kalman
filter. We refer to these problems as the priori and posteriori
Kalman Filtering Sensor Selection (KFSS) problems, respec-
tively. Second, we study the problem of attacking the installed
sensors (by removing a subset of them, under given attack
budget constraints) to maximize the trace of either the steady
state a priori or a posteriori error covariance of the Kalman
filter associated with the surviving sensors. These problems are
denoted as the priori and posteriori Kalman Filtering Sensor
Attack (KFSA) problems, respectively.
Contributions
Our contributions are as follows. First, we show that for the
priori and posteriori KFSS problems, there are no polynomial-
time constant-factor approximation algorithms for these prob-
lems (unless P =NP) even for the special case when the
system is stable and all sensors have the same cost. In other
words, there are no polynomial-time algorithms that can find
a sensor selection that is always guaranteed to yield a mean
square estimation error (MSEE) that is within any constant
finite factor of the MSEE for the optimal selection. More
importantly, our result stands in stark contrast to other sensor
selection problems studied in the literature, which leveraged
submodularity of their associated cost functions to provide
greedy algorithms with constant-factor approximation ratios
(e.g., [19], [32]). Second, we show that the same results hold
for the priori and posteriori KFSA problems. Our inapprox-
imability results directly imply that greedy algorithms cannot
provide constant-factor guarantees for our problems. Our third
contribution is to explicitly show how greedy algorithms can
provide arbitrarily poor performance even for very small
instances (with only three states) of the priori and posteriori
KFSS (resp., KFSA) problems.
A portion of the results pertaining to only the priori KFSS
problem appears in the conference paper [33].
Related work
The authors in [11] and [32] studied the design-time sensor
selection problem for discrete-time linear time-varying sys-
tems over a finite time horizon. The objective is to minimize
the estimation error with a cardinality constraint on the chosen
sensors (or alternatively, minimize the number of chosen
sensors while guaranteeing a certain level of performance
in terms of the estimation error). The authors analyzed the
performance of greedy algorithms for this problem. However,
their results cannot be directly applied to the problems that
we consider here, since we aim to optimize the steady state
estimation error.
The papers [16] and [18] considered the same design-
time sensor selection as the one we consider here. In [16],
the authors expressed the problem as a semidefinite program
(SDP). However, they did not provide theoretical guarantees
on the performance of the proposed algorithm. The paper
[18] showed that the problem is NP-hard and provided upper
bounds on the performance of any algorithm for the problem;
these upper bounds were functions of the system matrices.
Although [18] showed via simulations that greedy algorithms
performed well for several randomly generated systems, the
question of whether such algorithms (or other polynomial-time
algorithms) could provide constant-factor approximation ratios
for the problem was left open. We resolve this question in this
paper by showing that there does not exist any polynomial-
time constant-factor approximation algorithm for this problem.
In [30], the authors studied the problem of attacking a given
observation selection in Gaussian process regression [34] to
maximize the posteriori variance of the predictor variable. It
was shown that this problem is NP-hard. Moreover, they also
gave an instance of this problem such that a greedy algorithm
for finding an optimal attack will perform arbitrarily poorly.
In [35], the authors considered the scenario where the attacker
can target a different set of sensors at each time step to
maximize certain metrics of the error covariance of the Kalman
filter at the final time step. Some suboptimal algorithms were
provided with simulation results. Different from [30] and [35],
we study the problem where the attacker removes a set of
installed sensors to maximize the trace of the steady state error
covariance of the Kalman filter associated with the surviving
sensors, and provide fundamental limitations on achievable
performance by any possible algorithm for this problem.
Notation and Terminology
The sets of integers and real numbers are denoted as Z
and R, respectively. The set of integers that are greater than
(resp., greater than or equal to) a∈Ris denoted as Z>a
(resp., Z≥a). Similarly, we use the notations R>a and R≥a.
For any x∈R, let ⌈x⌉be the least integer greater than or
equal to x. For a square matrix P∈Rn×n, let PT, rank(P),
rowspace(P)and trace(P)be its transpose, rank, rowspace
and trace, respectively. We use Pij (or (P)ij ) to denote the
element in the ith row and jth column of P. A diagonal
matrix P∈Rn×nis denoted as diag(P11 ,...,Pnn). The
set of nby npositive definite (resp., positive semi-definite)
matrices is denoted as Sn
++ (resp., Sn
+). The identity matrix
with dimension n×nis denoted as In. The zero matrix with
dimension m×nis denoted as 0m×n. In a matrix, let ∗denote
elements of the matrix that are of no interest. For a vector v,
let vibe the ith element of vand define the support of vto be
supp(v) = {i:vi6= 0}. The Euclidean norm of vis denoted
by kvk2. Define eito be a row vector where the ith element
is 1and all the other elements are zero; the dimension of the
vector can be inferred from the context. Define 1nto be a
column vector of dimension nwith all the elements equal to
1. The set of 0-1indicator vectors of dimension nis denoted as
{0,1}n. For a random variable ω, let E[ω]be its expectation.
For a set A, let |A| be its cardinality.
II. PROB LE M FO RM ULATI ON
Consider the discrete-time linear system
x[k+ 1] = Ax[k] + w[k],(1)
where x[k]∈Rnis the system state, w[k]∈Rnis a zero-mean
white Gaussian noise process with E[w[k](w[k])T] = W, and
A∈Rn×nis the system dynamics matrix. The initial state
x[0] is assumed to be a Gaussian random vector with mean
¯x0∈Rnand covariance Π0∈Sn
+.
Consider a set Qthat contains qsensors. Each sensor i∈ Q
provides a measurement of the system of the form
yi[k] = Cix[k] + vi[k],(2)
where Ci∈Rsi×nis the measurement matrix for sensor i,
and vi[k]∈Rsiis a zero-mean white Gaussian noise process.
We further define y[k],(y1[k])T··· (yq[k])TT,C,
CT
1··· CT
qTand v[k],(v1[k])T··· (vq[k])TT. Thus,
the output provided by all sensors together is given by
y[k] = Cx[k] + v[k],(3)
where C∈Rs×nand s=Pq
i=1 si. Denote E[v[k](v[k])T] =
V. We assume that the system noise and the measurement
noise are uncorrelated, i.e., E[v[k](w[j])T] = 0s×n,∀k , j ∈
Z≥0, and x[0] is independent of w[k]and v[k],∀k∈Z≥0.
A. The Sensor Selection Problem
Consider the scenario where there are no sensors initially
deployed on the system. Instead, the system designer must
select a subset of sensors from Qto install. Each sensor
i∈ Q has a cost bi∈R≥0; define the cost vector b,
b1··· bqT. The designer has a budget B∈R≥0that
can be spent on choosing sensors from Q.
After a set of sensors is selected and installed, the Kalman
filter is applied to provide an optimal estimate of the states
using the measurements from the installed sensors (in the
sense of minimizing the MSEE). Define µ∈ {0,1}qto be
the indicator vector of the selected sensors, where µi= 1 if
and only if sensor i∈ Q is installed. Let C(µ)denote the mea-
surement matrix of the installed sensors indicated by µ, i.e.,
C(µ),CT
i1··· CT
ipT, where supp(µ) = {i1,...,ip}.
Similarly, let V(µ)be the measurement noise covariance
matrix of the installed sensors, i.e., V(µ) = E[˜v[k](˜v[k])T],
where ˜v[k] = (vi1[k])T··· (vip[k])TT. Let Σk|k−1(µ)
(resp., Σk|k(µ)) denote the a priori (resp., a posteriori) error
covariance matrix of the Kalman filter at time step k, when the
sensors indicated by µare installed. Take the initial covariance
Σ0|−1(µ) = Π0,∀µ. We will use the following result [36].
Lemma 1: Suppose that the pair (A, W 1
2)is stabilizable.
For a given indicator vector µ,Σk|k−1(µ)(resp., Σk|k(µ)) will
converge to a finite limit Σ(µ)(resp., Σ∗(µ)), which does not
depend on the initial covariance Π0, as k→ ∞ if and only if
the pair (A, C(µ)) is detectable.
The limit Σ(µ)satisfies the discrete algebraic Riccati equa-
tion (DARE) [36]:
Σ(µ) = AΣ(µ)AT+W−
AΣ(µ)C(µ)TC(µ)Σ(µ)C(µ)T+V(µ)−1C(µ)Σ(µ)AT.
(4)
The limits Σ(µ)and Σ∗(µ)are coupled as
Σ(µ) = AΣ∗(µ)AT+W. (5)
The limit Σ∗(µ)of the a posteriori error covariance matrix
satisfies the following equation [37]:
Σ∗(µ) = Σ(µ)−
Σ(µ)C(µ)T(C(µ)Σ(µ)C(µ)T+V(µ))−1C(µ)Σ(µ).(6)
Note that we can either obtain Σ∗(µ)from Σ(µ)using Eq.
(6) or by substituting Eq. (5) into Eq. (6) and solving for
Σ∗(µ). The inverses in Eq. (4) and Eq. (6) are interpreted as
pseudo-inverses if the arguments are not invertible.
For the case when the pair (A, C(µ)) is not detectable,
we define Σ(µ) = +∞and Σ∗(µ) = +∞. Moreover, for
any sensor selection µ, we note from Lemma 1that the limit
Σ(µ)(resp., Σ∗(µ)), if it exists, does not depend on ¯x0or Π0.
Thus, we can assume without loss of generality that ¯x0=0
and Π0=Inin the sequel. The priori and posteriori Kalman
Filtering Sensor Selection (KFSS) problems are then defined
as follows.
Problem 1: (Priori and Posteriori KFSS Problems). Given
a system dynamics matrix A∈Rn×n, a measurement matrix
C∈Rs×ncontaining all of the individual sensor measurement
matrices, a system noise covariance matrix W∈Sn
+, a sensor
noise covariance matrix V∈Ss
+, a cost vector b∈Rq
≥0and a
budget B∈R≥0, the priori Kalman filtering sensor selection
problem is to find the sensor selection µ, i.e., the indicator
vector µof the selected sensors, that solves
min
µ∈{0,1}qtrace(Σ(µ))
s.t. bTµ≤B
where Σ(µ)is given by Eq. (4) if the pair (A, C (µ)) is
detectable, and Σ(µ) = +∞if otherwise. Similarly, the
posteriori Kalman filtering sensor selection problem is to find
the sensor selection µthat solves
min
µ∈{0,1}qtrace(Σ∗(µ))
s.t. bTµ≤B
where Σ∗(µ)is given by Eq. (6) if the pair (A, C(µ)) is
detectable, and Σ∗(µ) = +∞if otherwise.
B. The Sensor Attack Problem
Now consider the scenario where the set Qof sensors has
already been installed on the system. An adversary desires to
attack a subset of sensors (i.e., remove a subset of sensors
from the system), where each sensor i∈ Q has an attack cost
ωi∈R≥0; define the cost vector ω,ω1··· ωqT. We
assume that the adversary has a budget Ω∈R≥0, which is
the total cost that can be spent on removing sensors from Q.
After a subset of sensors are attacked (i.e., removed), the
Kalman filter is then applied to estimate the states using the
measurements from the surviving sensors. We define a vector
ν∈ {0,1}qto be the indicator vector of the attacked sensors,
where νi= 1 if and only if sensor i∈ Q is attacked.
Hence, the set of sensors that survive is Q \ supp(ν). Define
vc∈ {0,1}qto be the vector such that supp(νc) = Q\supp(ν),
i.e., νc
i= 1 if and only if sensor i∈ Q survives. Similarly
to the sensor selection problem, we let C(νc)and V(νc)
denote the measurement matrix and the measurement noise
covariance matrix, respectively, corresponding to νc. Further-
more, let Σk|k−1(νc)and Σk|k(νc)denote the a priori error
covariance matrix and the a posteriori error covariance matrix
of the Kalman filter at time step k, respectively. Denote
lim
k→∞ Σk|k−1(νc) = Σ(νc)and lim
k→∞ Σk|k(νc) = Σ∗(νc)if
the limits exist, according to Lemma 1. Note that Eq. (4)-(6)
also hold if we substitute µwith νc.
For the case when the pair (A, C(νc)) is not detectable, we
define Σ(νc) = +∞and Σ∗(νc) = +∞. Recall that we have
assumed without loss of generality that ¯x0=0and Π0=
In. The priori and posteriori Kalman Filtering Sensor Attack
(KFSA) problems are defined as follows.
Problem 2: (Priori and Posteriori KFSA Problems). Given
a system dynamics matrix A∈Rn×n, a measurement matrix
C∈Rs×n, a system noise covariance matrix W∈Sn
+, a
sensor noise covariance matrix V∈Ss
+, a cost vector ω∈Rq
≥0
and a budget Ω∈R≥0, the priori Kalman filtering sensor
attack problem is to find the sensor attack ν, i.e., the indicator
vector νof the attacked sensors, that solves
max
ν∈{0,1}qtrace(Σ(νc))
s.t. ωTν≤Ω
where Σ(νc)is given by Eq. (4) if the pair (A, C (νc)) is
detectable, and Σ(νc) = +∞if otherwise. Similarly, the
posteriori Kalman filtering sensor attack problem is to find
the sensor attack νthat solves
max
ν∈{0,1}qtrace(Σ∗(νc))
s.t. ωTν≤Ω
where Σ∗(νc)is given by Eq. (6) if the pair (A, C (νc)) is
detectable, and Σ∗(νc) = +∞if otherwise.
Note that although we focus on the optimal sensor selection
and attack problems for Kalman filtering, due to the duality
between the Kalman filter and the Linear Quadratic Regulator
(LQR) [38], all of the analysis in this paper will also apply if
the priori KFSS and KFSA problems are rephrased as optimal
actuator selection and attack problems for LQR, respectively.
We omit the details of the rephrasing in the interest of space.
Remark 1: Our goal in this paper is to show that for
the priori and posteriori KFSS problems and the priori and
posteriori KFSA problems, the optimal solutions cannot be
approximated within any constant factor in polynomial time.
To do this, it is sufficient for us to consider the special case
when Ci∈R1×n,∀i∈ {1,...,q}, i.e., each sensor provides
a scalar measurement. Moreover, the sensor selection cost
vector and the sensor attack cost vector are considered to be
b=1··· 1Tand ω=1··· 1T, respectively, i.e.,
the selection cost and the attack cost of each sensor are both
equal to 1. By showing that the problems are inapproximable
even for these special subclasses, we obtain that the general
versions of the problems are inapproximable as well.
III. INAP P ROXIM AB ILI TY OF TH E KFSS AND KFSA
PRO BL E MS
In this section, we analyze the approximability of the KFSS
and KFSA problems. We will start with a brief overview
of some relevant concepts from the field of computational
complexity, and then provide some preliminary lemmas that
we will use in proving our results. That will lead into our
characterizations of the complexity of KFSS and KFSA.
A. Review of Complexity Theory
We first review the following fundamental concepts from
complexity theory [39].
Definition 1: Apolynomial-time algorithm for a problem is
an algorithm that returns a solution to the problem in a poly-
nomial (in the size of the problem) number of computations.
Definition 2: Adecision problem is a problem whose answer
is “yes” or “no”. The set P contains those decision problems
that can be solved by a polynomial-time algorithm. The set NP
contains those decision problems whose “yes” answers can be
verified using a polynomial-time algorithm.
Definition 3: An optimization problem is a problem whose
objective is to maximize or minimize a certain quantity,
possibly subject to constraints.
Definition 4: A problem P1is NP-complete if (a) P1∈NP
and (b) for any problem P2in NP, there exists a polynomial-
time algorithm that converts (or “reduces”) any instance of P2
to an instance of P1such that the answer to the constructed
instance of P1provides the answer to the instance of P2.P1
is NP-hard if it satisfies (b), but not necessarily (a).
The above definition indicates that if one had a polynomial-
time algorithm for an NP-complete (or NP-hard) problem, then
one could solve every problem in NP in polynomial time.
Specifically, suppose we had a polynomial-time algorithm to
solve an NP-hard problem P1. Then, given any problem P2in
NP, one could first reduce any instance of P2to an instance of
P1in polynomial time (such that the answer to the constructed
instance of P1provides the answer to the given instance of
P2), and then use the polynomial-time algorithm for P1to
obtain the answer to P2.
The above discussion also reveals that to show that a given
problem P1is NP-hard, one simply needs to show that any
instance of some other NP-hard (or NP-complete) problem P2
can be reduced to an instance of P1in polynomial time (in
such a way that the answer to the constructed instance of P1
provides the answer to the given instance of P2). For then, an
algorithm for P1can be used to solve P2, and hence, to solve
all problems in NP (by NP-hardness of P2).
The following is a fundamental result in computational
complexity theory [39].
Lemma 2: If P 6=NP, there is no polynomial-time algorithm
for any NP-complete (or NP-hard) problem.
For NP-hard optimization problems, polynomial-time ap-
proximation algorithms are of particular interest. A constant-
factor approximation algorithm is defined as follows.
Definition 5: Aconstant-factor approximation algorithm for
an optimization problem is an algorithm that always returns a
solution within a constant (system-independent) factor of the
optimal solution.
We will discuss the notion of a constant-factor approxi-
mation algorithm in greater depth later in this section. As
described in the Introduction, the KFSS problem was shown
to be NP-hard in [18] for two classes of systems and sensor
costs. First, when the Amatrix is unstable, it was shown in
[2] that the problem of selecting a subset of sensors to make
the system detectable is NP-hard, which implies that KFSS
is NP-hard using Lemma 1as shown in [18]. Second, when
the Amatrix is stable (so that all sensor selections cause the
system to be detectable), [18] showed that when the sensor
selection costs can be arbitrary, the knapsack problem can be
encoded as a special case of the KFSS problem, thereby again
showing NP-hardness of the latter problem.
In this paper, we will show that the hardness of KFSS (resp.,
KFSA) does not solely come from selecting (resp., attacking)
sensors to make the system detectable (resp., undetectable)
or the sensor selection (resp., attack) costs. To do this, we
will show a stronger result that there is no polynomial-
time constant-factor approximation algorithm for KFSS (resp.,
KFSA) even when the corresponding system dynamics matrix
Ais stable (which guarantees the detectability of the system),
and all the sensors have the same selection (resp., attack) cost.
Specifically, we consider a known NP-complete problem, and
show how to reduce it to certain instances of KFSS (resp.,
KFSA) with stable Amatrices in polynomial time such that
hypothetical polynomial-time constant-factor approximation
algorithms for the latter problems can be used to solve the
known NP-complete problem. Since we know from Lemma
2that if P 6=NP, there does not exist a polynomial-time
algorithm for any NP-complete problem, we conclude that if P
6=NP, there is no polynomial-time constant-factor approxima-
tion algorithm for KFSS (resp., KFSA), which directly implies
that the KFSS (resp., KFSA) problem is NP-hard even under
the extra conditions described above. We emphasize that our
results do not imply that there is no polynomial-time constant-
factor approximation algorithm for specific instances of KFSS
(resp., KFSA). Rather, the result is that we cannot have such
an algorithm for all instances of KFSS (resp., KFSA).
B. Preliminary Results
The following results characterize properties of the KFSS
and KFSA instances that we will consider when proving the
inapproximability of the KFSS and KFSA problems. The
proofs are provided in Appendix A.
Lemma 3: Consider a discrete-time linear system defined in
(1) and (3). Suppose the system dynamics matrix is of the form
A=diag(λ1,...,λn)with 0≤ |λi|<1,∀i∈ {1,...,n}, the
system noise covariance matrix W∈Sn
+is diagonal, and the
sensor noise covariance matrix V∈Sq
+. Then, the following
hold for all sensor selections µ.
(a) For all i∈ {1,...,n},(Σ(µ))ii and (Σ∗(µ))ii satisfy
Wii ≤(Σ(µ))ii ≤Wii
1−λ2
i
,(7)
and
0≤(Σ∗(µ))ii ≤Wii
1−λ2
i
,(8)
respectively.
(b) If ∃i∈ {1,...,n}such that Wii 6= 0 and the ith column
of Cis zero, then (Σ(µ))ii = (Σ∗(µ))ii =Wii
1−λ2
i
.
(c) If V=0q×qand ∃i∈ {1, . . . , n}such that ei∈
rowspace(C(µ)), then (Σ(µ))ii =Wii and (Σ∗(µ))ii = 0.
Lemma 4: Consider a discrete-time linear system defined in
Eq. (1) and Eq. (3). Suppose the system dynamics matrix is of
the form A=diag(λ1,0,...,0) ∈Rn×n, where 0<|λ1|<1,
and the system noise covariance matrix is W=In.
(a) Suppose the measurement matrix is of the form C=
1γwith sensor noise variance V=σ2
v, where γ∈
R1×(n−1) and σv∈R≥0. Then, the MSEE of state 1, denoted
as Σ11, satisfies
Σ11 =1 + α2λ2
1−α2+p(α2−α2λ2
1−1)2+ 4α2
2,(9)
where α2=kγk2
2+σ2
v.
(b) Suppose the measurement matrix is of the form
C=1n−1ρIn−1with sensor noise covariance V=
0(n−1)×(n−1), where ρ∈R. Then, the MSEE of state 1,
denoted as Σ′
11, satisfies
Σ′
11 =λ2
1ρ2+n′−ρ2+p(ρ2−λ2
1ρ2−n′)2+ 4n′ρ2
2n′,
(10)
where n′=n−1.
Moreover, if we view Σ11 and Σ′
11 as functions of α2
and ρ2, denoted as Σ11(α2)and Σ′
11(ρ2), respectively, then
Σ11(α2)and Σ′
11(ρ2)are strictly increasing functions of
α2∈R≥0and ρ2∈R≥0, with limα→∞Σ11 (α2) = 1
1−λ2
1
and limρ→∞Σ′
11(ρ2) = 1
1−λ2
1
, respectively.
C. Inapproximability of the KFSS Problem
In this section, we characterize the achievable performance
of algorithms for the priori and posteriori KFSS problems.
For any given algorithm A(resp., A′) of the priori (resp.,
posteriori) KFSS problem, we define the following ratios:
rA(Σ) ,trace(ΣA)
trace(Σopt),(11)
and
rA′(Σ∗),trace(Σ∗
A′)
trace(Σ∗
opt),(12)
where Σopt (resp., Σ∗
opt) is the optimal solution to the priori
(resp., posteriori) KFSS problem and ΣA(resp., Σ∗
A′) is the
solution to the priori (resp., posteriori) KFSS problem given
by algorithm A(resp., A′).
In [18], the authors showed that there is an upper bound
for rA(Σ) (resp., rA′(Σ∗)) for any sensor selection algorithm
A(resp., A′), in terms of the system matrices. However, the
question of whether it is possible to find an algorithm A(resp.,
A′) that is guaranteed to provide an approximation ratio rA(Σ)
(resp., rA′(Σ∗)) that is independent of the system parameters
has remained open up to this point. In particular, it is desirable
to find polynomial-time constant-factor approximation algo-
rithms for the priori (resp., posteriori) KFSS problem, where
the ratio rA(Σ) (resp., rA′(Σ∗)) is upper-bounded by some
(system-independent) constant.1We provide a negative result
by showing that there are no polynomial-time algorithms that
can always yield a solution that is within any constant factor
of the optimal (unless P =NP), i.e., for all polynomial-time
algorithms A(resp., A′) and ∀K∈R≥1, there are instances of
the priori (resp., posteriori) KFSS problem where rA(Σ) > K
(resp., rA′(Σ∗)> K).
Remark 2: Note that the “constant” in “constant-factor
approximation algorithm” refers to the fact that the cost
of the solution provided by the algorithm is upper-bounded
by some (system-independent) constant times the cost of
the optimal solution. The algorithm can, however, use the
system parameters when finding the solution. For example, an
optimal algorithm for the KFSS problem will be a 1-factor
approximation, and would use the system matrices, sensor
costs, and budget to find the optimal solution. Similarly, a
polynomial-time K-factor approximation algorithm for KFSS
would use the system parameters to produce a solution whose
cost is guaranteed to be no more than Ktimes the cost of the
optimal solution. As indicated above, we will show that no
such algorithm exists for any constant K(unless P =NP).
To show the inapproximability of the priori KFSS problem,
we relate it to the EXACT COVER BY 3-SETS (X3C)
problem described below [39].
Definition 6: (X3C)Given a finite set D={d1,...,d3m}
and a collection C={c1,...,cτ}of 3-element subsets of D,
an exact cover for Dis a subcollection C′⊆ C such that every
element of Doccurs in exactly one member of C′.
We will use the following result [39].
1Polynomial-time constant-factor approximation algorithms have been
widely studied for NP-hard problems, e.g., [39].
Lemma 5: Given a finite set D={d1,...,d3m}and a
collection C={c1,...,cτ}of 3-element subsets of D, the
problem of determining whether Ccontains an exact cover for
Dis NP-complete.
As argued in Remark 1, in order to show that the priori
KFSS problem cannot be approximated within any constant
factor in polynomial time, it is sufficient for us to show that
certain special instances of this problem are inapproximable.
Specifically, consider any instance of the X3Cproblem. Using
the results in Lemma 3-4, we will first construct an instance
of the priori KFSS problem in polynomial time such that the
difference between the solution to KFSS when the answer to
X3Cis “yes” and the solution to KFSS when the answer to
X3Cis “no” is large enough. Thus, we can then apply any
hypothetical polynomial-time constant-factor approximation
algorithm for the priori KFSS problem to the constructed priori
KFSS instance and obtain the answer to the X3Cinstance.
Since we know from Lemma 5that the X3Cproblem is NP-
complete, we obtain from Lemma 2the following result; the
detailed proof is provided in Appendix B.
Theorem 1: If P 6=NP, then there is no polynomial-time
constant-factor approximation algorithm for the priori KFSS
problem.
The following result is a direct consequence of the above
arguments; the proof is also provided in Appendix B.
Corollary 1: If P 6=NP, then there is no polynomial-
time constant-factor approximation algorithm for the posteriori
KFSS problem.
D. Inapproximability of the KFSA Problem
In this section, we analyze the achievable performance
of algorithms for the priori and posteriori KFSA problems.
For any given algorithm A(resp., A′) for the priori (resp.,
posteriori) KFSA problem, we define the following ratios:
rA(˜
Σ) ,trace(˜
Σopt)
trace(˜
ΣA),(13)
and
rA′(˜
Σ∗),trace(˜
Σ∗
opt)
trace(˜
Σ∗
A′),(14)
where ˜
Σopt (resp., ˜
Σ∗
opt) is the optimal solution to the priori
(resp., posteriori) KFSA problem and ˜
ΣA(resp., ˜
Σ∗
A′) is the
solution to the priori (resp., posteriori) KFSA problem given
by algorithm A(resp., A′). It is worth noting that using the
arguments in [18], the same (system-dependent) upper bounds
for rA(˜
Σ) and rA′(˜
Σ∗)can be obtained as those for rA(Σ)
and rA′(Σ∗)in [18], respectively. Nevertheless, we show that
(if P6=NP) there is again no polynomial-time constant-factor
approximation algorithm for the priori (resp., posteriori) KFSA
problem, i.e., for all K∈R≥1and for all polynomial-time
algorithms A(resp., A′), there are instances of the priori
(resp., posteriori) KFSA problem where rA(˜
Σ) > K (resp.,
rA′(˜
Σ∗)> K). To establish this result, we relate the priori
KFSA problem to the X3Cproblem described in Definition 6
and Lemma 5. Similarly to the proof of Theorem 1, given any
instance of X3C, we will construct an instance of the priori
KFSA problem and show that any hypothetical polynomial-
time constant-factor approximation algorithm for the priori
KFSA problem can be used to solve the X3Cproblem. This
leads to the following result; the detailed proof is provided in
Appendix C.
Theorem 2: If P 6=NP, then there is no polynomial-time
constant-factor approximation algorithm for the priori KFSA
problem.
The arguments above also imply the following result whose
proof is provided in Appendix C.
Corollary 2: If P 6=NP, then there is no polynomial-
time constant-factor approximation algorithm for the posteriori
KFSA problem.
IV. FAIL UR E O F GRE EDY ALG ORI TH MS
Our results in Theorem 1 and Theorem 2 indicate that no
polynomial-time algorithm can be guaranteed to yield a solu-
tion that is within any constant factor of the optimal solution
to the priori (resp., posteriori) KFSS and KFSA problems. In
particular, these results apply to the greedy algorithms that are
often studied for sensor selection in the literature (e.g., [18],
[30]), where sensors are iteratively selected (resp., attacked) in
order to produce the greatest decrease (resp., increase) in the
error covariance at each iteration. In this section we will focus
on such greedy algorithms for the priori (resp., posteriori)
KFSS and KFSA problems, and show explicitly how these
greedy algorithms can fail to provide good solutions even for
small and fixed instances with only three states; this provides
additional insight into the factors that cause the KFSS and
KFSA problems to be challenging.
A. Failure of Greedy Algorithms for the KFSS Problem
It was shown via simulations in [18] that greedy algorithms
for KFSS work well in practice (e.g., for randomly generated
systems). In this section, we provide an explicit example
showing that greedy algorithms for the priori and posteriori
KFSS problems can perform arbitrarily poorly, even for small
systems (containing only three states). We consider the greedy
algorithm for the priori (resp., posteriori) KFSS problem
given in Algorithm 1, for instances where all sensors have
selection costs equal to 1, and the sensor selection budget
B∈ {1,...,q}(i.e., up to Bsensors can be chosen). For any
such instance of the priori (resp., posteriori) KFSS problem,
define rgre (Σ) = trace(Σg re )
trace(Σopt)(resp., rgre(Σ∗) = trace(Σ∗
gre )
trace(Σ∗
opt)),
where Σgre (resp., Σ∗
gre ) is the solution of Eq. (4) (resp., Eq.
(6)) corresponding to the sensors selected by Algorithm 1.
Algorithm 1 Greedy Algorithm for Problem 1
Input: An instance of priori (resp., posteriori) KFSS
Output: A set Sof selected sensors
1: k←1,S ← ∅
2: for k≤Bdo
3: j∈arg mini /∈S trace(Σ(S ∪ {i})) (resp., j∈
arg mini /∈S trace(Σ∗(S ∪ {i})))
4: S ← S ∪ {j},k←k+ 1
5: end for
Example 1: Consider an instance of the priori (resp., pos-
teriori) KFSS problem with matrices W=I3and V=03×3,
and A,Cdefined as
A=
λ10 0
0 0 0
0 0 0
, C =
1h h
1 0 h
0 1 1
,
where 0<|λ1|<1,λ1∈R, and h∈R>0. In addition, we
have the selection budget B= 2, the cost vector b= [1 1 1]T
and the set of candidate sensors Q={1,2,3}, where sensor
icorresponds to the ith row of matrix C, for i∈ {1,2,3}.
Based on the system defined in Example 1, we have the
following result whose proof is provided in Appendix D.
Theorem 3: For the instance of the priori (resp., posteriori)
KFSS problem defined in Example 1, the ratios rgre (Σ) =
trace(Σgre )
trace(Σopt)and rgre(Σ∗) = trace(Σ∗
gre )
trace(Σ∗
opt)satisfy
lim
h→∞ rgre (Σ) = 2
3+1
3(1 −λ2
1),(15)
and
lim
h→∞ rgre (Σ∗) = 1
1−λ2
1
,(16)
respectively.
Examining Eq. (15) (resp., Eq. (16)), we see that for the
given instance of the priori (resp., posteriori) KFSS problem,
we have rgre(Σ) → ∞ (resp., rgre (Σ∗)→ ∞) as h→ ∞
and λ1→1. Thus, rgre (Σ) (resp., rgr e(Σ∗)) can be made
arbitrarily large by choosing the parameters in the instance
appropriately. To explain the result in Theorem 3, we first note
that the only nonzero eigenvalue of the diagonal Adefined
in Example 1is λ1, and so we know from Lemma 3(a)
that state 2and state 3of the system defined in Example 1
each contributes at most 1to trace(Σ(µ)) (resp., trace(Σ∗(µ)))
for all µ. Hence, in order to minimize trace(Σ(µ)) (resp.,
trace(Σ∗(µ))), we need to minimize the MSEE of state 1.
Moreover, the measurements of state 2and state 3can be
viewed as measurement noise that corrupts the measurements
of state 1. It is then easy to observe from the form of matrix
Cdefined in Example 1that sensor 2is the single best sensor
among the three sensors since it provides measurements of
state 1with less noise than sensor 1(and sensor 3does not
measure state 1at all). Thus, the greedy algorithm for the priori
(resp., posteriori) KFSS problem defined in Algorithm 1se-
lects sensor 2in its first iteration. Nonetheless, we notice from
Cdefined in Example 1that the optimal set of two sensors that
minimizes trace(Σ(µ)) (resp., trace(Σ∗(µ))) contains sensor
1and sensor 3, which together give us exact measurements
(without measurement noise) on state 1(after some elementary
row operations). Since the greedy algorithm selects sensor
2in its first iteration, no matter which sensor it selects in
its second iteration, the two chosen sensors can only give a
noisy measurement of state 1(if we view the measurements
of state 2and state 3as measurement noise), and the variance
of the measurement noise can be made arbitrary large if we
take h→ ∞ in Cdefined in Example 1. Hence, the greedy
algorithm fails to perform well due to its myopic choice in
the first iteration.
It is also useful to note that the above behavior holds for any
algorithm that outputs a sensor selection that contains sensor
2for the above example.
B. Failure of Greedy Algorithms for the KFSA Problem
In [30], the authors showed that a simple greedy algo-
rithm can perform arbitrarily poorly for an instance of the
observation attack problem in Gaussian process regression.
Here, we consider a simple greedy algorithm for the priori
(resp., posteriori) KFSA problem given in Algorithm 2, for
instances where all sensors have an attack cost of 1, and the
sensor attack budget Ω∈ {1,...,q}(i.e., up to Ωsensors
can be attacked). For any such instance of the priori (resp.,
posteriori) KFSA problem, define rgr e(˜
Σ) = trace(˜
Σopt)
trace(˜
Σgre )(resp.,
rgre (˜
Σ∗) = trace(˜
Σ∗
opt)
trace(˜
Σ∗
gre )), where ˜
Σgre (resp., ˜
Σ∗
gre ) is the
solution to the priori (resp., posteriori) KFSA problem given
by Algorithm 2. We then show that Algorithm 2 can perform
arbitrarily poorly for a simple instance of the priori (resp.,
posteriori) KFSA problem described below.
Algorithm 2 Greedy Algorithm for Problem 2
Input: An instance of priori (resp., posteriori) KFSA
Output: A set Sof targeted sensors
1: k←1,S ← ∅
2: for k≤Ωdo
3: j∈arg maxi /∈S trace(Σ(Q \ (S ∪ {i}))) (resp., j∈
arg maxi /∈S trace(Σ∗(Q \ (S ∪ {i}))))
4: S ← S ∪ {j},k←k+ 1
5: end for
Example 2: Consider an instance of the priori (resp., poste-
riori) KFSA problem with matrices W=I3,V=04×4, and
A,Cdefined as
A=
λ10 0
0 0 0
0 0 0
, C =
1h h
1 0 h
0 1 0
0 0 1
,
where 0<|λ1|<1,λ1∈Rand h∈R>0. In addition, the
attack budget is Ω = 2, the cost vector is ω= [1 1 1 1]T, and
the set of sensors Q={1,2,3,4}has already been installed
on the system, where sensor icorresponds to the ith row of
matrix C, for i∈ {1,2,3,4}.
We then have the following result, whose proof is provided
in Appendix D.
Theorem 4: For the instance of the priori (resp., posteriori)
KFSA problem defined in Example 2, the ratios rgre (˜
Σ) =
trace(˜
Σopt)
trace(˜
Σgre )and rgre (˜
Σ∗) = trace(˜
Σ∗
opt)
trace(˜
Σ∗
gre )satisfy
lim
h→0rgre (˜
Σ) = 2
3+1
3(1 −λ2
1),(17)
and
lim
h→0rgre (˜
Σ∗) = 1
1−λ2
1
,(18)
respectively.
Inspecting Eq. (17) (resp., Eq. (18)), we observe that for the
given instance of the priori (resp., posteriori) KFSA problem,
we have rgre(˜
Σ) → ∞ (resp., rgre (˜
Σ∗)→ ∞) as h→0
and λ1→1. Thus, rgre (˜
Σ) (resp., rgre (˜
Σ∗)) can be made
arbitrarily large by choosing the parameters in the instance
appropriately. Here, we explain the results in Theorem 4as
follows. Using similar arguments to those before, we know
from the structure of matrix Adefined in Example 2that
in order to maximize trace(Σ(νc)) (resp., trace(Σ∗(νc))),
we need to maximize the MSEE of state 1, i.e., make the
measurements of state 1“worse”. Again, the measurements
of state 2and state 3can be viewed as measurement noise
that corrupts the measurements of state 1. No matter which
of sensor 1, sensor 2, or sensor 3is attacked, the resulting
measurement matrix C(νc)is full column rank, which yields
an exact measurement of state 1. We also observe that if
sensor 4is targeted, the surviving sensors can only provide
measurements of state 1that are corrupted by measurements
of states 2and state 3. Hence, the greedy algorithm for the
priori (resp., posteriori) KFSA problem defined in Algorithm
2targets sensor 4in its first iteration, since it is the single best
sensor to attack from the four sensors. Nevertheless, sensor 1
and sensor 2form the optimal set of sensors to be attacked
to maximize trace(Σ(νc)) (resp., trace(Σ∗(νc))), since the
surviving sensors provide no measurement of state 1. Since
the greedy algorithm targets sensor 4in its first iteration, no
matter which sensor it targets in the second step, the surviving
sensors can always provide some measurements of state 1with
noise (if we view the measurements of state 2and state 3as
measurement noise), and the variance of the noise will vanish
if we take h→0in matrix Cdefined in Example 2. Hence,
the myopic behavior of the greedy algorithm makes it perform
poorly.
Furthermore, it is useful to note that the above result holds
for any algorithm that outputs a sensor attack that does not
contain sensor 1or sensor 2for the above example.
Remark 3: Using similar arguments to those in the proof of
Theorem 3(resp., Theorem 4), we can also show that when we
set V=εI3(resp., V=εI4), where ε∈R>0, the results in
Eqs. (15)-(16) (resp., Eqs. (17)-(18)) hold if we let ε→0. This
phenomenon is also observed in [11], where the approximation
guarantees for the greedy algorithms provided in that paper get
worse as the sensor measurement noise tends to zero.
V. CO NC LUS ION S
In this paper, we studied sensor selection and attack prob-
lems for (steady state) Kalman filtering of linear dynamical
systems. We showed that these problems are NP-hard and have
no polynomial-time constant-factor approximation algorithms,
even under the assumption that the system is stable and each
sensor has identical cost. To illustrate this point, we provided
explicit examples showing how greedy algorithms can perform
arbitrarily poorly on these problems, even when the system
only has three states. Our results shed new insights into the
problem of sensor selection and attack for Kalman filtering
and show, in particular, that this problem is more difficult
than other variants of the sensor selection problem that have
submodular (or supermodular) cost functions. Future work on
extending the results to Kalman filtering over finite time hori-
zons, characterizing achievable (non-constant) approximation
ratios, identifying classes of systems that admit near-optimal
approximation algorithms, and investigating resilient sensor
selection problems under adversarial settings would be of
interest.
VI. ACKN OW LE D GM ENT S
The authors thank the anonymous reviewers for their in-
sightful comments that helped to improve the paper.
APP EN D IX A
Proof of Lemma 3:
Since Aand Ware diagonal, the system represents a set of
nscalar subsystems of the form
xi[k+ 1] = λixi[k] + wi[k],∀i∈ {1,...,n},
where xi[k]is the ith state of x[k]and wi[k]is a zero-mean
white Gaussian noise process with variance σ2
wi=Wii. As
Ais stable, the pair (A, C (µ)) is detectable and the pair
(A, W 1
2)is stabilizable for all sensor selections µ. Thus, the
limits lim
k→∞(Σk|k−1(µ))ii and lim
k→∞(Σk|k(µ))ii exist for all i
and for all µ(based on Lemma 1), and are denoted as (Σ(µ))ii
and (Σ∗(µ))ii, respectively.
Proof of (a): Since Aand Ware diagonal, we know from
Eq. (5) that
(Σ(µ))ii =λ2
i(Σ∗(µ))ii +Wii,
which implies (Σ(µ))ii ≥Wii,∀i∈ {1,...,n}. Moreover,
it is easy to see that (Σ(µ))ii ≤(Σ(0))ii ,∀i∈ {1,...,n}.
Since C(0) = 0, we obtain from Eq. (4) that
Σ(0) = AΣ(0)AT+W.
which implies that (Σ(0))ii =Wii
1−λ2
i
since Ais diagonal.
Hence, Wii ≤(Σ(µ))ii ≤Wii
1−λ2
i
,∀i∈ {1,...,n}. Similarly,
we also have (Σ∗(µ))ii ≤(Σ∗(0))ii, and we obtain from Eq.
(6) that
Σ∗(0) = AΣ∗(0)AT+W.
Thus, 0≤(Σ∗(µ))ii ≤Wii
1−λ2
i
,∀i∈ {1,...,n}.
Proof of (b): Assume without loss of generality that the first
column of C(µ)is zero, since we can simply renumber the
states to make this the case without affecting the trace of the
error covariance matrix. We then have C(µ)of the form
C(µ) = 0C1(µ).
Moreover, since Aand Ware diagonal, we obtain from Eq.
(4) that Σ(µ)is of the form
Σ(µ) = Σ1(µ)0
0Σ2(µ),
where Σ1(µ) = (Σ(µ))11 and satisfies
(Σ(µ))11 =λ2
i(Σ(µ))11 +W11,
which implies (Σ(µ))11 =W11
1−λ2
1
. Furthermore, we obtain from
Eq. (6) that Σ∗(µ)is of the form
Σ∗(µ) = Σ∗
1(µ)0
0Σ∗
2(µ),
where Σ∗
1(µ) = (Σ∗(µ))11 and satisfies
(Σ∗(µ))11 =λ2
1(Σ∗(µ))11 +W11,
which implies (Σ∗(µ))11 =W11
1−λ2
1
.
Proof of (c): We assume without loss of generality that
e1∈rowspace(C(µ)). If we further perform elementary row
operations on C(µ), which does not change the solution to Eq.
(4) (resp., Eq. (6)), we obtain a measurement matrix ˜
C(µ)of
the form
˜
C(µ) = 10
0˜
C1(µ)
with ˜
V(µ) = 0. Moreover, since Aand Ware diagonal, we
obtain from Eq. (4) that Σ(µ)is of the form
Σ(µ) = Σ1(µ)0
0Σ2(µ),
where Σ1(µ) = (Σ(µ))11 and satisfies
(Σ(µ))11 =λ2
1(Σ(µ))11 +W11 −λ2
1(Σ(µ))11,
which implies (Σ(µ))11 =W11. Furthermore, we obtain from
Eq. (6) that Σ∗(µ)is of the form
Σ∗(µ) = Σ∗
1(µ)0
0Σ∗
2(µ),
where Σ∗
1(µ) = (Σ∗(µ))11 and satisfies (Σ∗(µ))11 =
(Σ(µ))11 −(Σ(µ))11 = 0.
Proof of Lemma 4:
Proof of (a): We first note from Lemma 1that the limit
Σ(µ)exists for all µ(since Ais stable). Since A=
diag(λ1,0,...,0), we have xi[k+1] = wi[k],∀i∈ {2,...,n}
and ∀k∈Z≥0. Moreover, we have from Eq. (3) that
y[k] = [1 01×(n−1)]x[k]+v[k]+v′[k] = x1[k]+˜v[k],∀k∈Z≥0,
where v′[k] =
n−1
X
i=1
γixi+1[k]and ˜v[k],v[k] + v′[k]. Recall
that we have assumed with out loss of generality that ¯x0=0
and Π0=In. Moreover, noting that W=Inand that x[0] is
independent of w[k]and v[k]for all k∈Z≥0, where w[k]
and v[k]are uncorrelated zero-mean white Gaussian noise
processes (as assumed), we have that ˜v[k]is a zero-mean white
Gaussian noise process with E[(˜v[k])2] = kγk2
2+σ2
v. Thus,
to compute the MSEE of state 1of the Kalman filter, i.e.,
Σ11, we can consider a scalar discrete-time linear system with
A=λ1,C= 1,W= 1 and V=α2, and obtain from Eq.
(4) the scalar DARE
Σ11 =λ2
1(1 −Σ11
α2+ Σ11
)Σ11 + 1,(19)
where α2=kγk2
2+σ2
v. Solving for Σ11 in Eq. (19) and
omitting the negative solution lead to Eq. (9).
To show that Σ11 is strictly increasing in α2∈R≥0, we can
use the result of Lemma 6in [18]. For a discrete-time linear
system defined in Eq. (1) and Eq. (3), given A=λ1and
W= 1, suppose we have two sensors with the measurement
matrices as C1=C2= 1 and the variances of the (Gaussian)
measurement noise as V1=α2
1and V2=α2
2. Define R,
CTV−1Cto be the sensor information matrix corresponding
to a sensor with measurement matrix Cand measurement
noise covariance matrix V. The sensor information matrix of
these two sensors are denoted as R1and R2. We then have
R1=1
α2
1
and R2=1
α2
2
. If α2
1> α2
2, we know from Lemma 6
in [18] that Σ11(α2
1)<Σ11(α2
2). Hence, Σ11(α2)is a strictly
increasing function of α2∈R≥0. For α > 0, we can rewrite
Eq. (9) as
Σ11(α2) = 2
q(1 −λ2
1−1
α2)2+4
α2+ 1 −λ2
1−1
α2
.(20)
We then obtain from Eq. (20) that lim
α→∞ Σ11(α2) = 1
1−λ2
1
.
Proof of (b): Using similar arguments to those above, we
obtain from Eq. (3) that
y[k] = 1n−1x1[k] + v′[k],
where v′[k] = ρx2[k]···xn[k]Tis a zero-mean white Gaus-
sian noise process with E[v′[k](v′[k])T] = ρ2In−1. Hence, to
compute the MSEE of state 1of the Kalman filter, i.e., Σ′
11,
we can consider a system with A=λ1,C=1n−1,W= 1
and V=ρ2In−1. Solving Eq. (4) (using the matrix inversion
lemma [40]) yields Eq. (10). Similarly, we have Σ′
11 is strictly
increasing in ρ2∈R≥0and lim
ρ→∞ Σ′
11(ρ2) = 1
1−λ2
1
.
APP EN D IX B
We will use the following result in the proof of Theorem 1.
Lemma 6: Consider an instance of X3C: a finite set D
with |D|= 3m, and a collection C={c1,...,cτ}of τ3-
element subsets of D, where τ≥m. For each element ci∈ C,
define a column vector gi∈R3mto encode which elements
of Dare contained in ci, i.e., for i∈ {1,2,...,τ}and j∈
{1,2,...,3m},(gi)j= 1 if element jof set Dis in ci, and
(gi)j= 0 otherwise. Denote G,g1··· gτT. For any
l≤m(l∈Z) and L,{i1,...,il} ⊆ {1,...,τ}, define
GL,gi1··· gilTand denote rank(GL) = rL.2. If the
answer to the X3Cproblem is “no”, then for all Lwith |L| ≤
m, there exists an orthogonal matrix N∈R3m×3msuch that
1T
3m
GLN=γ β
0˜
GL,(21)
where ˜
GL∈Rl×ris of full column rank, γ∈R1×(3m−r)and
at least κ≥1(κ∈Z) elements of γare 1’s , and β∈R1×r.
Further elementary row operations on hγ β
0˜
GLitransform it into
the form hγ0
0˜
GLi.
Proof: Assume without loss of generality that there
are no identical subsets in C. Since rank(GL) = r, the
2We drop the subscript Lon rfor notational simplicity.
dimension of the nullspace of GLis 3m−r. We choose
an orthonormal basis of the nullspace of GLand let it form
the first 3m−rcolumns of N, denoted as N1. Then, we
choose an orthonormal basis of the columnspace of GT
Land
let it form the rest of the rcolumns of N, denoted as
N2. Clearly, N=N1N2∈R3m×3mis an orthogonal
matrix. Furthermore, since the answer to the X3Cproblem
is “no”, for any union of l≤m(l∈Z) subsets in C,
denoted as Cl, there exist κ≥1 (κ∈Z)elements in D
that are not covered by Cl, i.e., GLhas κzero columns.
Let these denote the j1th, ...,jκth columns of GL, where
{j1,...,jκ} ⊆ {1,...,3m}. Hence, we can always choose
ej1,...,ejκto be in the orthonormal basis of the nullspace of
GL, i.e., as columns of N1. Constructing Nin this way, we
have GLN1=0and GLN2=˜
GL, where ˜
GL∈Rl×ris of
full column rank since the columns of N2form an orthonormal
basis of the columnspace of GT
Land r≤l. Moreover, we have
1T
3mN1=γand 1T
3mN2=β, where at least κelements of
γare 1’s (since 1T
3meT
js= 1,∀s∈ {1,...,κ}). Combining
these results, we obtain Eq. (21). Since ˜
GLis of full column
rank, we can perform elementary row operations on hγ β
0˜
GLi
and obtain hγ0
0˜
GLi.
Proof of Theorem 1:
Assume that there exists such an approximation algorithm
A, i.e., ∃K∈R≥1such that rA(Σ) ≤Kfor all instances of
the priori KFSS problem, where rA(Σ) is defined in Eq. (11).
We will show that Acan be used to solve the X3Cproblem,
which will lead to a contradiction.
Given an arbitrary instance of the X3Cproblem described
in Definition 6and Lemma 5, for each element ci∈ C, we de-
fine gi∈R3mto encode which elements of Dare contained in
ci. Specifically, for i∈ {1,2,...,τ}and j∈ {1,2,...,3m},
(gi)j= 1 if dj∈Dis in ci, and (gi)j= 0 otherwise. Denote
G,g1··· gτT. Thus GTx=13mhas a solution
x∈ {0,1}τsuch that xhas mnonzero entries if and only
if the answer to the X3Cinstance is “yes” [41].
Given the above instance of X3C, we then construct an
instance of the priori KFSS problem as follows. Denote
Z=⌈K⌉(m+ 1)(σ2
v+ 3), where we set σv= 1. Define
the system dynamics matrix as A=diag(λ1,0,...,0) ∈
R(3m+1)×(3m+1), where λ1=Z−1/2
Z. Note that Z∈Z>1
and 0< λ1<1. The set Qis defined to contain τ+ 1 sensors
with collective measurement matrix
C=1ε1T
3m
0G,(22)
where Gis defined based on the given instance of X3Cas
above. The constant εis chosen as ε= 2Zl√Z−1m+ 1.
The system noise covariance matrix is set to be W=I3m+1 .
The measurement noise covariance matrix is set as V=
σ2
v10
01
ε2Iτ. The sensor selection cost vector is set as
b=1τ+1, and the sensor selection budget is set as B=m+1.
Note that the sensor selection vector for this instance is
denoted by µ∈ {0,1}τ+1. For the above construction, since
the only nonzero eigenvalue of Ais λ1, we know from Lemma
3(a) that P3m+1
i=2 (Σ(µ))ii =P3m+1
i=2 Wii = 3mfor all µ.
We claim that algorithm Awill return a sensor selection
vector µsuch that trace(Σ(µ)) ≤K(m+ 1)(σ2
v+ 3) if and
only if the answer to the X3Cproblem is “yes”.
We prove the above claim as follows. Suppose that the
answer to the instance of the X3Cproblem is “yes”. Then
GTx=13mhas a solution such that xhas mnonzero
entries. Denote the solution as x∗and denote supp(x∗) =
{i1,...,im}. Define ˜µto be the sensor selection vector
that indicates selecting the first and the (i1+ 1)th to the
(im+ 1)th sensors, i.e., sensors that correspond to rows
C1,Ci1+1,...,Cim+1 from (22). Since GTx∗=13m, we
have [1 −εx∗T]C=e1for Cdefined in Eq. (22).
Noting that supp(x∗) = {i1,...,im}, it then follows that
e1∈rowspace(C(˜µ)). We can then perform elementary row
operations on C( ˜µ)(which does not change the steady state
a priori error covariance matrix of the corresponding Kalman
filter) and obtain ΓC(˜µ),˜
C(˜µ) = 10
0∗with the corre-
sponding measurement noise covariance ΓV(µ)ΓT,˜
V(˜µ) =
σ2
v(m+ 1) ∗
∗ ∗, where Γ = 1−ε1T
m
0Im. Let ˜
Σdenote the
error covariance obtained from sensing matrix (˜
C(˜µ))1=e1
with measurement noise variance ˜σ2
v,σ2
v(m+ 1), which
corresponds to the first sensor in ˜
C(˜µ). We then know from
Lemma 4(a) that
˜
Σ11 =1 + ˜σ2
vλ2
1−˜σ2
v+p(˜σ2
v−˜σ2
vλ2
1−1)2+ 4˜σ2
v
2,
which further implies
˜
Σ11 ≤1 + p(˜σ2
v(1 −λ2
1))2−2˜σ2
v(1 −λ2
1) + 1 + 4˜σ2
v
2
≤1 + p(˜σ2
v(1 −λ2
1))2+ 1 + 4˜σ2
v
2
≤1 + p˜σ4
v+ 4˜σ2
v+ 4
2≤1 + ˜σ2
v+ 2
2.(23)
Using similar arguments to those above, we have that
P3m+1
i=2 ˜
Σii = 3m. We then obtain from (23) that
trace(˜
Σ) ≤˜σ2
v+ 3 + 3m= (m+ 1)(σ2
v+ 3).(24)
Since adding more sensors does not increase the MSEE of
the corresponding Kalman filter, we have from (24) that
trace(Σ(˜µ)) ≤(m+ 1)(σ2
v+ 3), which further implies
trace(Σ(µ∗)) ≤(m+ 1)(σ2
v+ 3), where µ∗is the optimal
sensor selection of the priori KFSS problem. Since Ahas
approximation ratio K, it returns a sensor selection µsuch
that trace(Σ(µ)) ≤K(m+ 1)(σ2
v+ 3).
Conversely, suppose that the answer to the X3Cinstance
is “no”. Then, for any union of l≤m(l∈Z≥0) subsets in C,
denoted as Cl, there exist κ≥1 (κ∈Z)elements in Dthat are
not covered by Cl, i.e., for any l≤mand L,{i1,...,il} ⊆
{1,...,τ},GL,gi1··· gilThas κ≥1zero columns. We
then show that trace(Σ(µ)) > K(m+1)(σ2
v+3) for all sensor
selections µ(that satisfy the budget constraint). We divide our
arguments into two cases.
First, for any sensor selection µ1that does not select the
first sensor, the first column of C(µ1)is zero (from the form
of Cdefined in Eq. (22)). We then know from Lemma 3(b)
that (Σ(µ1))11 =1
1−λ2
1
. Hence, by our choice of λ1, we have
(Σ(µ1))11 =Z2
Z−1/4> Z ≥K(m+ 1)(σ2
v+ 3)
⇒trace(Σ(µ1)) > K(m+ 1)(σ2
v+ 3),(25)
where (25) follows from P3m+1
i=2 (Σ(µ1))ii = 3m > 0for all
possible sensor selections.
Second, consider sensor selections µ2that select the
first sensor. To proceed, we first assume that the mea-
surement noise covariance is V=0(τ+1)×(τ+1). Denote
supp(µ2) = {1, i1,...,il}, where l≤mand define G(µ2) =
gi1−1··· gil−1T. We then have
C(µ2) = 1ε1T
3m
0G(µ2),
where G(µ2)has κ≥1zero columns. As argued in Lemma
6, there exists an orthogonal matrix E∈R(3m+1)×(3m+1) of
the form E= [ 1 0
0N]such that
˜
C(µ2),C(µ2)E=1εγ εβ
0 0 ˜
G(µ2).
In the above expression, ˜
G(µ2)∈Rl×ris of full column
rank, where r=rank(G(µ2)). Furthermore, γ∈R1×(3m−r)
and at least κof its elements are 1’s, and β∈R1×r. We
then perform a similarity transformation on the system with E,
which does not affect the trace of the steady state a priori error
covariance matrix of the corresponding Kalman filter,3and
does not change A,Wand V. We further perform additional
elementary row operations to transform ˜
C(µ2)into the matrix
˜
C′(µ2) = 1εγ 0
0 0 ˜
G(µ2).
Since Aand Ware both diagonal, and V=0, we can obtain
from Eq. (4) that the steady state a priori error covariance cor-
responding to the sensing matrix ˜
C′(µ2), denoted as ˜
Σ′(µ2),
is of the form
˜
Σ′(µ2) = ˜
Σ′
1(µ2)0
0˜
Σ′
2(µ2),
where ˜
Σ′
1(µ2)∈R(3m+1−r)×(3m+1−r)satisfies
˜
Σ′
1(µ2) = A1˜
Σ′
1(µ2)AT
1+W1−
A1˜
Σ′
1(µ2)˜
CT˜
C˜
Σ′
1(µ2)˜
CT−1˜
C˜
Σ′
1(µ2)AT
1,
where A1=diag(λ1,0,...,0) ∈R(3m+1−r)×(3m+1−r),˜
C=
[1 εγ]and W1=I3m+1−r. Denoting α2=ε2kγk2
2≥κε2≥
ε2, we then obtain from Lemma 4(a) that
(˜
Σ′(µ2))11 =1 + α2λ2
1−α2+p(α2−α2λ2
1−1)2+ 4α2
2
≥1 + ε2λ2
1−ε2+p(ε2−ε2λ2
1−1)2+ 4ε2
2.
(26)
3This can be easily verified using Eq. (4) as Eis an orthogonal matrix.
By our choices of λ1and ε, we have the following:
ε2>4Z2(Z−1) ⇒(1 −Z−1/4
Z)ε2> Z2−Z
⇒ε2> Z2+Zε2Z−1/4
Z2−Z
⇒ε2> Z2+Z(ε2(1 −λ2
1)−1)
⇒(ε2−ε2λ2
1−1)2+ 4ε2>
(ε2−ε2λ2
1−1)2+ 4Z2+ 4Z(ε2−ε2λ2
1−1)
⇒(ε2−ε2λ2
1−1)2+ 4ε2>(2Z+ε2−ε2λ2
1−1)2
⇒q(ε2−ε2λ2
1−1)2+ 4ε2>2Z+ε2−ε2λ2
1−1
⇒1 + ε2λ2
1−ε2+p(ε2−ε2λ2
1−1)2+ 4ε2
2> Z.
(27)
Since Z≥K(m+ 1)(σ2
v+ 3), (26) and (27) imply
(˜
Σ′(µ2))11 > K(m+ 1)(σ2
v+ 3), which further implies
trace(˜
Σ′(µ2)) > K(m+ 1)(σ2
v+ 3). Since trace(˜
Σ′(µ2)) =
trace(Σ(µ2)) as argued above, we obtain that trace(Σ(µ2)) >
K(m+ 1)(σ2
v+ 3). We then note the fact that the MSEE of
the Kalman filter with noiseless measurements is no greater
than that with any noisy measurements (for fixed A,Wand
C), when the system noise and the measurement noise are
uncorrelated. Therefore, for V=σ2
v10
01
ε2Iτ, we also have
that trace(Σ(µ2)) > K(m+ 1)(σ2
v+ 3) for all µ2.
It then follows from the above arguments that trace(Σ(µ)) >
K(m+ 1)(σ2
v+ 3) for all sensor selections µ, which implies
that algorithm Awould also return a sensor selection µsuch
that trace(Σ(µ)) > K(m+ 1)(σ2
v+ 3). This completes the
proof of the converse direction of the claim above.
Hence, it is clear that algorithm Acan be used to solve
the X3Cproblem by applying it to the above instance of the
priori KFSS problem. Since X3Cis NP-complete, there is
no polynomial-time algorithm for it if P 6=NP, and we get a
contradiction. This completes the proof of the theorem.
Proof of Corollary 1:
We have shown in Theorem 1that for any polynomial-time
algorithm Afor the priori KFSS problem and any K∈R≥1,
there exist instances of the priori KFSS problem such that
rA(Σ) > K (unless P =NP). Suppose that there exists a
polynomial-time constant-factor approximation algorithm A′
for the posteriori KFSS problem, i.e., ∃K′∈R≥1such
that rA′(Σ∗)≤K′for all instances of the posteriori KFSS
problem, where rA′(Σ∗)is defined in Eq. (12). We consider
an instance of the priori KFSS problem constructed in the
proof of Theorem 1. We then set the instance of the posteriori
KFSS problem to be the same as the constructed instance of
the priori KFSS problem. Since A=diag(λ1,0,...,0) ∈
R(3m+1)×(3m+1) and W=I3m+1, where 0< λ1<1, we
have from Eq. (5) that
(Σ(µ))11 =λ2
1(Σ∗(µ))11 + 1,∀µ. (28)
Since we know from Lemma 3(a) that (Σ(µ))ii = 1,∀i∈
{2,...,3m+ 1}and ∀µ, it then follows from Eq. (28) that
trace(Σ(µ)) = λ2
1(Σ∗(µ))11 + 3m+ 1,∀µ. (29)
We also know from Lemma 3(a) that 0≤(Σ∗(µ))ii ≤1,
∀i∈ {2,...,3m+ 1}and ∀µ, which implies that
trace(Σ∗(µ)) ≤(Σ∗(µ))11 + 3m, ∀µ. (30)
We then obtain from Eqs. (29)-(30) that
trace(Σ∗(µ)) ≤3mλ2
1+trace(Σ(µ)) −3m−1
λ2
1
≤trace(Σ(µ))
λ2
1
,∀µ, (31)
where the second inequality follows from the fact that 0<
λ1<1. Denote the optimal sensor selections of the priori
and the posteriori KFSS problems as µ∗
1and µ∗
2, respectively.
Denote the sensor selection returned by algorithm A′as µ′.
Note that Σopt = Σ(µ∗
1)and Σ∗
opt = Σ∗(µ∗
2)and Σ∗
A′=
Σ∗(µ′). We then have the following:
trace(Σ∗
A′)≤K′trace(Σ∗
opt)
⇒(Σ∗(µ′))11 +
3m+1
X
i=2
(Σ∗(µ′))ii ≤K′trace(Σ∗(µ∗
2))
⇒(Σ(µ′))11 −1
λ2
1≤K′trace(Σ∗(µ∗
2)) ≤K′trace(Σ∗(µ∗
1))
(32)
⇒(Σ(µ′))11 −1≤K′trace(Σ(µ∗
1)) (33)
⇒trace(Σ(µ′)) ≤K′trace(Σ(µ∗
1)) + 3m+ 1 (34)
⇒trace(Σ(µ′))
trace(Σ(µ∗
1)) ≤K′+3m+ 1
trace(Σ(µ∗
1)) ≤K′+ 1,(35)
where the first inequality in (32) follows from Eq. (28) and
(Σ∗(µ′))ii ≥0,∀i(from Lemma 3(a)), the second inequality
in (32) follows from the fact that µ∗
2is the optimal sensor
selection for the posteriori KFSS problem, (33) follows from
(31), (34) follows from the fact that P3m+1
i=2 (Σ(µ′))ii = 3m
(from Lemma 3(a)), and the second inequality in (35) uses the
fact that trace(Σ(µ∗
1)) ≥3m+1 (from Lemma 3(a)). Thus, we
have from (35) that rA′(Σ) ≤K′+ 1, which contradicts the
fact that the priori KFSS problem cannot have a polynomial-
time constant-factor approximation algorithm for instances of
the given form, and completes the proof of the corollary.
APP EN D IX C
Proof of Theorem 2:
Assume that there exists such a polynomial-time constant-
factor approximation algorithm A, i.e., ∃K∈R≥1such that
rA(˜
Σ) ≤Kfor all instances of the priori KFSA problem,
where rA(˜
Σ) is defined in Eq. (13). We will show that Acan
be used to solve the X3Cproblem, leading to a contradiction.
Consider any instance of the X3Cproblem to be a finite
set D={d1,···, d3m}and a collection C={c1,...,cτ}of
3-element subsets of D, where τ≥m. Recall in the proof of
Theorem 1that we use a column vector gi∈R3mto encode
which elements of Dare contained in ci, where (gi)j= 1 if
dj∈Dis in ci, and (gi)j= 0 otherwise, for i∈ {1,2,...,τ}
and j∈ {1,2,...,3m}. The matrix G∈Rτ×3mwas defined
in the proof of Theorem 1as G=g1··· gτT. In this
proof, we will make use of the matrix F,GT; note that each
column of Fcontains exactly three 1’s.
Given the above instance of the X3Cproblem, we then
construct an instance of the priori KFSA as follows. Denote
Z=⌈K⌉(τ+ 2)(δ2
v+ 1), where we set δv= 1. Define
the system dynamics matrix as A=diag(λ1,0,...,0) ∈
R(τ+1)×(τ+1), where λ1=Z−1/2
Z. Note that Z∈Z>1and
0< λ1<1. The set Qconsists of 3m+τsensors with
collective measurement matrix
C=13mρF
0Iτ,(36)
where Fis defined above and Iτis used to encode the
collection C, i.e., ejrepresents cj∈ C for all j∈ {1,2,...,τ}.
The constant ρis chosen as ρ= 2Zlpm(Z−1)m+ 1. The
system noise covariance matrix is set to be W=Iτ+1. The
measurement noise covariance is set as V=δ2
vI3m0
01
ρ2Iτ.
The sensor attack cost vector is set as ω=13m+τ, and the sen-
sor attack budget is set as Ω = m. Note that the sensor attack
vector is given by ν∈ {0,1}3m+τ. For the above construction,
since the only nonzero eigenvalue of Ais λ1, we know from
Lemma 3(a) that Pτ+1
i=2 (Σ(νc))ii =Pτ+1
i=2 Wii =τfor all ν.
We claim that algorithm Awill return a sensor attack vector
νsuch that trace(Σ(νc)) >(τ+ 2)(δ2
v+ 1) if and only if the
answer to the X3Cproblem is “yes”.
We prove the above claim as follows. Suppose that the
answer to the X3Cproblem is “yes”. Similarly to the proof
of Theorem 1, we first assume that V=0(3m+τ)×(3m+τ).
Denote an exact cover as C′={cj1,...,cjm}, where
{j1,...,jm} ⊆ {1,2,...,τ}. Define ˜νto be the sensor attack
such that supp(˜ν) = {3m+j1,...,3m+jm}. We then
renumber the states of the system from state 2to state τsuch
that for all i∈ {1,2,...,m}, the columns of the submatrix Iτ
of Cin Eq. (36) representing cjiin C′, i.e., the columns of Iτ
that correspond to supp(˜ν), come first. Note that renumbering
the states does not change the trace of the steady state a priori
error covariance of the corresponding Kalman filter. We then
have from Eq. (36) that
C(˜νc) = 13mρF1ρF2
0 0 Iτ−m,(37)
where F1∈R3m×mand F2∈R3m×(τ−m)satisfy F=
F1F2, and Iτ−mis the submatrix of Iτthat corresponds
to supp(˜νc)∩{3m+1,...,3m+τ}, i.e., the elements of Cthat
are not in C′.4Since the sensor attack ˜νtargets the rows of C
that correspond to the elements of the exact cover C′for D, we
have that F1, after some row permutations of C(˜νc), is given
by F1=eT
1eT
1eT
1··· eT
meT
meT
mT. We perform
additional elementary row operations and merge identical rows
(which does not change the steady state a priori error covari-
ance matrix of the corresponding Kalman filter) to transform
C(˜νc)into the matrix
˜
C(˜νc) = 1mρIm0
0 0 Iτ−m.(38)
4Note that if the submatrix of Iτcorresponding to supp(˜νc)∩ {3m+
1, . . . , 3m+τ}is not identity, we can always permute the rows of C(˜νc)
to make it identity.
Since Aand Ware both diagonal, and V=0, we can obtain
from Eq. (4) that the steady state a priori error covariance
corresponding to ˜
C(˜νc), denoted as ˜
Σ(˜νc), is of the form
˜
Σ(˜νc) = ˜
Σ1(˜νc)0
0˜
Σ2(˜νc),
where ˜
Σ1(˜νc)∈R(m+1)×(m+1) satisfies
˜
Σ1(˜νc) = A1˜
Σ1(˜νc)AT
1+W1−
A1˜
Σ1(˜νc)˜
CT˜
C˜
Σ1(˜νc)˜
CT−1˜
C˜
Σ1(˜νc)AT
1,
where A1=diag(λ1,0,...,0) ∈R(m+1)×(m+1),˜
C=
1mρImand W1=Im+1. We then know from Lemma
4(b) that (Σ(˜νc))11 = (˜
Σ(˜νc))11 satisfies
(Σ(˜νc))11 =λ2
1ρ2+m−ρ2+p(ρ2−λ2
1ρ2−m)2+ 4mρ2
2m.
(39)
By our choices of λ1and ρ, we have
ρ2>4Z2m(Z−1) ⇒(1 −Z−1/4
Z)ρ2> Z2m−Z m
⇒ρ2> mZ2+Zρ2Z−1/4
Z2−Zm
⇒4mρ2>4m2Z2+ 4mZ(ρ2(1 −λ2
1)−m)
⇒(ρ2−λ2
1ρ2−m)2+ 4mρ2
>4m2Z2+ 4mZ(ρ2−λ2
1ρ2−m) + (ρ2−λ2
1ρ2−m)2
⇒(ρ2−λ2
1ρ2−m)2+ 4mρ2>(2mZ +ρ2−λ2
1ρ2−m)2
⇒q(ρ2−λ2
1ρ2−m)2+ 4mρ2>2mZ +ρ2−λ2
1ρ2−m
⇒λ2
1ρ2+m−ρ2+p(ρ2−λ2
1ρ2−m)2+ 4mρ2
2m> Z.
(40)
Noting that Z≥K(τ+2)(δ2
v+1), we then know from (39) and
(40) that (Σ(˜νc))11 > K (τ+ 2)(δ2
v+1), which further implies
that trace(Σ(˜νc)) > K (τ+ 2)(δ2
v+ 1). Following the same
arguments as those in the proof of Theorem 1, we have that for
V=δ2
vI3m0
01
ρ2Iτ, trace(Σ(˜νc)) > K (τ+ 2)(δ2
v+ 1) also
holds, which implies trace(Σ(ν∗c)) > K(τ+2)(δ2
v+1), where
ν∗is the optimal sensor attack for the priori KFSA problem.
Since algorithm Ahas approximation ratio K, it would return
a sensor attack νsuch that trace(Σ(νc)) >(τ+ 2)(δ2
v+ 1).
Conversely, suppose the answer to the X3Cproblem is
“no”. For any union of l≤m(l∈Z≥0) subsets in C, denoted
as Cl, there exists at least one element in Dthat is not covered
by Cl. We then show that trace(Σ(νc)) ≤(τ+ 2)(δ2
v+ 1) for
all sensor attacks ν(that satisfy the budget constraint). We
split our discussion into three cases.
First, consider any sensor attack ν1that targets lsensors
merely from C1to C3min Eq. (36), i.e., |supp(ν1)|=land
supp(ν1)⊆ {1,...,3m}, where l≤m. We then obtain
C(νc
1) = 13m−lρF (νc
1)
0Iτ,
where F(νc
1)∈R(3m−l)×τis defined to be the submatrix of
Fthat corresponds to supp(νc
1)∩ {1, . . . , 3m}, i.e., the rows
of Fthat are left over by ν1. We perform elementary row
operations to transform C(νc
1)into
˜
C(νc
1),ΨC(νc
1) = 13m−l0
0Iτ(41)
with the corresponding measurement noise covariance
˜
V(νc
1),ΨV(νc
1)ΨT
=δ2
vI3m−l+F(νc
1)(F(νc
1))T−1
ρF(νc
1)
−1
ρ(F(νc
1))T1
ρ2Iτ,(42)
where Ψ = I3m−l−ρF (νc
1)
0Iτ. Since there are at most τ
nonzero elements (which are all 1’s) in the first row of F(νc
1),
it follows that (F(νc
1)(F(νc
1))T)11 (i.e., the element in the
first row and first column of the matrix F(νc
1)(F(νc
1))T) is at
most τ. We then have from Eq. (42) that (˜
V(νc
1))11, denoted
as ˜
δ2
v(νc
1), satisfies
˜
δ2
v(νc
1)≤(τ+ 1)δ2
v.(43)
Second, consider any sensor attack ν2that targets lsensors
merely from C3m+1 to C3m+τin Eq. (36), i.e., |supp(ν2)|=l
and supp(ν2)⊆ {3m+ 1,...,3m+τ}, where l≤m. Via
similar arguments to those for obtaining Eqs. (37), (41) and
(42), we can perform elementary row operations to transform
C(νc
2) = 13mρF ′
1ρF ′
2
0 0 Iτ−l
into
˜
C(νc
2) = 13mρF ′
10
0 0 Iτ−l
with the corresponding measurement noise covariance
˜
V(νc
2) = ˜
δ2
v(νc
2)∗
∗ ∗, where
˜
δ2
v(νc
2)≤(τ−l+ 1)δ2
v.(44)
Note that F′
1∈R3m×land F′
2∈R3m×(τ−l)satisfy F=
F′
1F′
2. Recall that for any union of l≤msubsets in C,
denoted as Cl, there exists at least one element in Dthat is not
covered by Cl. We can then assume without loss of generality
that one such element is d1, which implies that the first row
of F′
1is zero.
Third, consider any sensor attack ν3that targets sensors
from both C1to C3mand C3m+1 to C3m+τin Eq. (36).
Suppose that the attack ν3attacks l1sensors from C1to
C3mand l2sensors from C3m+1 to C3m+τ, i.e., supp(ν3) =
{j′
1,...,j′
l1,3m+j′′
1,...,3m+j′′
l2} ⊆ {1,2,...,3m+τ},
where l1, l2∈Z≥1,l1+l2=l≤m,{j′
1,...,j′
l1} ⊆
{1,...,3m}and {j′′
1,...,j′′
l2} ⊆ {1,...,τ}. By similar
arguments to those above, we can perform elementary row
operations to transform
C(νc
3) = 13m−l1ρF1(νc
3)ρF2(νc
3)
0 0 Iτ−l2
into
˜
C(νc
3) = 13m−l1ρF1(νc
3) 0
0 0 Iτ−l2,
where F1(νc
3)∈R(3m−l1)×l2and F2(νc
3)∈R(3m−l1)×(τ−l2)
satisfy F(νc
3) = F1(νc
3)F2(νc
3)with F(νc
3)defined in
the same way as F(νc
1). Moreover, the measurement noise
covariance corresponding to ˜
C(νc
3)is given by ˜
V(νc
3) =
˜
δ2
v(νc
3)∗
∗ ∗, where
˜
δ2
v(νc
3)≤(τ−l2+ 1)δ2
v.(45)
Since any l2subsets in Ccan cover at most 3l2elements in D,
there are at least 3m−3l2elements in Dthat are not covered
by the l2subsets in C. Also note that
3m−3l2−l1= 3m−2l2−l= 2(m−l2) + m−l > 0,
where the last inequality follows from the facts that l1+l2=
l≤mand l1, l2∈Z≥1. Hence, by attacking l1sensors from
C1to C3mand l2sensors from C3m+1 to C3m+τ, we have at
least 3m−3l2−l1>0row(s) of F1(νc
3)that are zero. Again,
we can assume without loss of generality that the first row of
F1(νc
3)is zero.
In summary, for any sensor attack ν, we let ˆ
Σ(νc
i)denote
the steady state a priori error covariance obtained from
measurement matrix (˜
C(νc
i))1=e1with measurement noise
variance ˜
δ2
v(νc
i)(which corresponds to the first sensor in
˜
C(νc
i)), ∀i∈ {1,2,3}, where ν1,ν2and ν3are given
as above. Following similar arguments to those for (23),
we have (ˆ
Σ(νc
i))11 ≤˜
δ2
v(νc
i) + 2,∀i∈ {1,2,3}. Since
Pτ+1
i=2 (ˆ
Σ(νc
i))ii =Pτ+1
i=2 Wii =τholds for all i∈ {1,2,3}
via similar arguments to those above, we obtain that
trace(ˆ
Σ(νc
i)) ≤˜
δ2
v(νc
i) + 2 + τ , ∀i∈ {1,2,3}.(46)
Again note that adding more sensors does not increase the
MSEE of the corresponding Kalman filter, and the above
operations performed on the sensing matrix Cdo not change
the trace of the steady state a priori error covariance of the cor-
responding Kalman filter as well. We then have from Eqs. (43)-
(46) that trace(Σ(νc)) ≤(τ+1)δ2
v+2+τ≤(τ+2)(δ2
v+1) for
all ν. It follows that algorithm Awould also return a sensor
attack νsuch that trace(Σ(νc)) ≤(τ+2)(δ2
v+1). This proves
the converse direction of the claim above.
Therefore, we know that Acan be used to solve the
X3Cproblem by applying it to the above instance of the
priori KFSA problem. Since X3Cis NP-complete, there is
no polynomial-time algorithm for it if P 6=NP, yielding a
contradiction. This completes the proof of the theorem.
Proof of Corollary 2:
Note that the Aand Wmatrices for the instance of KFSA
that we constructed in the proof of Theorem 2are the same
as those for the instance of KFSS that we constructed in the
proof of Theorem 1. We then follow the same arguments as
those in the proof of Corollary 1. Suppose that there exists a
polynomial-time constant-factor approximation algorithm A′
for the posteriori KFSA problem, i.e., ∃K′∈R≥1such
that rA′(˜
Σ∗)≤K′for all instances of the posteriori KFSA
problem, where rA′(˜
Σ∗)is defined in Eq. (14). We consider
an instance of the priori KFSA problem as constructed in the
proof of Theorem 2. We then set the instance of the posteriori
KFSA problem to be the same as the constructed instance of
the priori KFSA problem. Denote the optimal sensor attacks
of the priori and the posteriori KFSA problems as ν∗
1and ν∗
2,
respectively. Denote the sensor attack returned by algorithm A′
as ν′. Note that ˜
Σopt = Σ(ν∗c
1),˜
Σ∗
opt = Σ∗(ν∗c
2)and ˜
Σ∗
A′=
Σ∗(ν′c). Also note that trace(Σ∗(ν∗c
1)) ≤trace(Σ∗(ν∗c
2)),
since ν∗
2is the optimal sensor attack for the posteriori KFSA
problem. We then have the following:
trace(Σ∗(ν∗c
1)) ≤trace(Σ∗(ν∗c
2)) ≤K′trace(˜
Σ∗
A′)
⇒(Σ∗(ν∗c
1))11 +
3m+1
X
i=2
(Σ∗(ν∗c
1))ii ≤K′trace(Σ∗(ν′c))
⇒(Σ(ν∗c
1))11 −1
λ2
1≤K′trace(Σ∗(ν′c))
⇒(Σ(ν∗c
1))11 −1≤K′trace(Σ(ν′c))
⇒trace(Σ(ν∗c
1)) ≤K′trace(Σ(ν′c)) + 3m+ 1
⇒trace(Σ(ν∗c
1))
trace(Σ(ν′c)) ≤K′+3m+ 1
trace(Σ(ν′c)) ≤K′+ 1,
which implies rA′(˜
Σ) ≤K′+ 1, and yields a contradiction
with the fact that the priori KFSA problem cannot have a
polynomial-time constant-factor approximation algorithm for
the instances of the form given as above. This completes the
proof of the corollary.
APP EN D IX D
Proof of Theorem 3:
We first prove that Algorithm 1 for the priori KFSS problem
selects sensor 2and sensor 3in its first and second iterations,
respectively. Since the only nonzero eigenvalue of Ais λ1, we
know from Lemma 3(a) that (Σ(µ))22 = 1 and (Σ(µ))33 =
1,∀µ, which implies that (Σgre)22 = 1 and (Σgre )33 = 1.
Hence, we focus on determining (Σgre)11.
Denoting µ1= [1 0 0]Tand µ2= [0 1 0]T, we have
C(µ1) = [1 h h]and C(µ2) = [1 0 h]. Using the result
in Lemma 4(a), we obtain that σ1,(Σ(µ1))11 and σ2,
(Σ(µ2))11 satisfy
σ1=2
q(1 −λ2
1−1
2h2)2+2
h2+ 1 −λ2
1−1
2h2
,
and
σ2=2
q(1 −λ2
1−1
h2)2+4
h2+ 1 −λ2
1−1
h2
,
respectively. Similarly, denoting µ3= [0 0 1]T, we obtain
C(µ3) = [0 1 1]. Since the first column of C(µ3)is zero,
we know from Lemma 3(b) that σ3,(Σ(µ3))11 =1
1−λ2
1
. If
we view σ2as a function of h2, denoted as σ(h2), we have
σ1=σ(2h2). Since we know from Lemma 4(a) that σ(h2)is
a strictly increasing function of h2∈R>0and upper bounded
by 1
1−λ2
1
, we obtain σ2< σ1< σ3, which implies that the
greedy algorithm selects sensor 2in its first iteration.
Denote µ12 = [1 1 0]T. We have C(µ12) = 1h h
1 0 h,
on which we perform elementary row operations and obtain
˜
C(µ12) = 0h0
1 0 h. By direct computation from Eq. (4), we
obtain (Σ(µ12))11 =σ2. Moreover, we denote µ23 = [0 1 1]T
and obtain C(µ23) = [ 1 0 h
0 1 1 ]. By direct computation from Eq.
(4), we have (Σ(µ23))11, denoted as σ23 , to be
σ23 =2
q(1 −λ2
1−2
h2)2+8
h2+ 1 −λ2
1−2
h2
.
Similarly to the argument above, we have σ12 =σ(h2)and
σ23 =σ(h2
2), where σ(h2
2)< σ(h2), which implies that
the greedy algorithm selects sensor 3in its second iteration.
Hence, we have trace(Σgre ) = σ23 + 2.
Furthermore, it is easy to see that the optimal sensor selec-
tion (for the priori KFSS instance) is µ= [1 0 1]T, denoted
as µ13. Since if µ=µ13 , then e1∈rowspace(C(µ)) and thus
we know from Lemma 3(a) and (c) that trace(Σ(µ)) = 3 =
trace(W), which is also the minimum value of trace(Σ(µ))
among all possible sensor selections µ. Combining the results
above and taking the limit as h→ ∞ lead to Eq. (15).
We next prove that the greedy algorithm defined in Algo-
rithm 1 for the posteriori KFSS problem selects sensor 2and
sensor 3in its first and second iterations, respectively. Note
that it is easy to obtain from Eq. (5) that Σ(µ)is of the form
Σ(µ) = diag((Σ(µ))11,1,1),∀µ. Hence, we have from Eq.
(6) that trace(Σ∗(µ1)) = 2 + h2
σ1
2+h2(σ1−1), trace(Σ∗(µ2)) =
2 + h2
σ2+h2(σ2−1) and trace(Σ∗(µ3)) = 2 + 1
1−λ2
1−1, where
σ1=σ(2h2)and σ2=σ(h2)are defined above. Since σ(h2)
is a strictly increasing function of h2∈R>0with σ(h2)≥1
and upper bounded by 1
1−λ2
1
, and it is easy to obtain that
σ1
2< σ2, it then follows that Algorithm 1 for the posteriori
KFSS problem selects sensor 2in its first iteration.
Similarly, we have from Eq. (6) that trace(Σ∗(µ12)) = 1 +
h2
σ2+h2(σ2−1), trace(Σ∗(µ23)) = 1 + h2
2σ23+h2(σ23 −1) and
trace(Σ∗(µ13)) = 1, where σ23 =σ(h2
2)is defined above.
Since σ(h2)is strictly increasing in h2∈R>0with σ(h2)≥
1and upper bounded by 1
1−λ2
1
, and it is easy to check that
σ2<2σ23, it follows that the greedy algorithm selects sensor
3in its second iteration, and µ=µ13 is the optimal sensor
selection (for the posteriori KFSS instance). Combining the
results above and letting h→ ∞, we obtain Eq. (16).
Proof of Theorem 4 :
We first analyze Algorithm 2for the priori KFSA problem.
Since the only nonzero eigenvalue of Ais λ1, we know from
Lemma 3(a) that (Σ(νc))22 = 1 and (Σ(νc))33 = 1,∀ν, which
implies that (˜
Σgre )22 = 1 and (˜
Σgre )33 = 1. Hence, we only
need to determine (˜
Σgre )11.
First, denote ν1= [1 0 0 0]T,ν2= [0 1 0 0]Tand
ν3= [0 0 1 0]T. Then, it is easy to see that C(νc)is of
full column rank for all ν∈ {ν1, ν2, ν3}. This implies that
e1∈rowspace(C(νc)) for all ν∈ {ν1, ν2, ν3}. Thus, we know
from Lemma 3(c) that (Σ(νc))11 = 1,∀ν∈ {ν1, ν2, ν3}.
Moreover, denoting ν4= [0 0 0 1]T, we have C(νc
4)(after
some elementary row operations and merging identical rows)
is of the form C(νc
4) = [ 1 0 h
0 1 0 ]. Using the results from the
proof of Theorem 3, we obtain that σ′
4,(Σ(νc
4))11 satisfies
σ′
4=1 + h2λ2
1−h2+p(h2−h2λ2
1−1)2+ 4h2
2.
If we view σ′
4as a function of h2, denoted as σ′(h2), we
know from Lemma 4(a) that σ′(h2)is a strictly increasing
function of h2∈R≥0with σ′(0) = 1, which implies σ′
4>1.
Thus, Algorithm 2for priori KFSA targets sensor 4in its first
iteration.
Second, denote ν14 = [1 0 0 1]T,ν24 = [0 1 0 1]Tand
ν34 = [0 0 1 1]T. We obtain that C(νc)(after some elementary
row operations) is of the form C(νc) = [ 1 0 h
0 1 0 ], for all ν∈
{ν14, ν24 , ν34}. It follows that (Σ(νc))11 =σ′
4for all ν∈
{ν14, ν24 , ν34}, which implies that trace(˜
Σgre ) = σ′
4+ 2.
Furthermore, the optimal sensor attack (for the priori KFSA
instance) is ν=ν12, where ν12 = [1 1 0 0]T, since in this
case we know from Lemma 3(a) and (b) that Σ(νc)11 =1
1−λ2
1
,
which is also the maximum value of Σ(νc)11 that it can
achieve, i.e., ˜
Σopt =1
1−λ2
1
+ 2. Combining the results above
and taking the limit as h→0, we obtain Eq. (17).
We next analyze Algorithm 2 for the posteriori KFSA
problem. Since we know from previous arguments that C(νc)
is of full column rank for all ν∈ {ν1, ν2, ν3}, it follows
from Lemma 3(c) that trace(Σ∗(νc
1)) = trace(Σ∗(νc
2)) =
trace(Σ∗(νc
3)) = 0. Moreover, it is easy to obtain from Eq.
(5) that Σ(νc)is of the form Σ(νc) = diag((Σ(νc))11,1,1),
∀ν. We then have from Eq. (6) that trace(Σ∗(νc
4)) = 1 +
h2
σ′
4+h2(σ′
4−1), where σ′
4=σ′(h2)is defined above. Since
σ′(h2)is strictly increasing in h2∈R≥0with σ′(0) = 1, it
implies that Algorithm 2 for posteriori KFSA targets sensor
4in its first iteration. Similarly, we have from Eq. (6) that
trace(Σ∗(νc
14)) = trace(Σ∗(νc
24)) = trace(Σ∗(νc
34)) = 1 +
h2
σ′
4+h2(σ′
4−1), which implies trace(˜
Σ∗
gre ) = 1+ h2
σ′
4+h2(σ′
4−1).
Furthermore, denote ν23 = [0 1 1 0]Tand ν13 =
[1 0 1 0]T. It is easy to show, via similar arguments to
those above, that trace(Σ∗(νc)) = 1 + h2
σ′
4+h2(σ′
4−1)
for all ν∈ {ν34, ν24 , ν23 , ν14}, trace(Σ∗(νc
13)) = 1, and
trace(Σ∗(νc
12)) = 1 + 1
1−λ2
1−1. Since σ′
4=σ′(h2)is strictly
increasing in h2∈R≥0with σ′(0) = 1, and upper bounded
by 1
1−λ2
1
, it follows that the optimal sensor attack (for the
posteriori KFSA instance) is ν=ν12. Combining the results
above and taking the limit as h→0, we obtain Eq. (18).
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