The Edge Group Coloring Problem with Applications to Multicast Switching

Abstract

Washington University technical report - WUCSE-2015-02. Describes a generalization of the edge coloring problem in graphs, shows it to be NP-hard and evaluates several approximate algorithms.

Full text

The Edge Group Coloring Problem
with Applications to Multicast Switching
Jonathan Turner
WUCSE-2015-02
Abstract
This paper introduces a natural generalization of the classical edge
coloring problem in graphs that provides a useful abstraction for two
well-known problems in multicast switching. We show that the problem
is NP-hard and evaluate the performance of several approximation algo-
rithms, both analytically and experimentally. We find that for random
χ-colorable graphs, the number of colors used by the best algorithms
falls within a small constant factor of χ, where the constant factor is
mainly a function of the ratio of the number of outputs to inputs. When
this ratio is less than 10, the best algorithms produces solutions that
use fewer than 2χcolors. In addition, one of the algorithms studied
finds high quality approximate solutions for any graph with high prob-
ability, where the probability of a low quality solution is a function only
of the random choices made by the algorithm.
An instance of the edge group coloring problem is an undirected graph
G= (V, E ) and a partition of its edges into groups {g1, . . . , gk}, where the
edges forming each group all share a common endpoint. A coloring is a
function c, from the edges to the positive integers, which assigns different
values to pairs of edges that share a common endpoint and belong to different
edge groups. The objective of the problem is to find a coloring that uses the
smallest possible number of distinct colors. An example is shown in Figure 1;
here, the arcs joining selected edges define the groups (so for example, one
1

a
c e
b
d
2
2 1
1
2
1
3
Figure 1: Example of edge group coloring
group consists of the edges {b, e},{c, e}and {d, e}). The integers on the
edges form a valid assignment of colors.
In this paper, we focus on a restricted version of the problem in which
the graph is bipartite, with the vertices divided between inputs and outputs,
and all groups are “centered” at an input. An example of such a graph
appears in Figure 2. This version of edge group coloring can be applied to a
a
c
w
b
d
1
3
2
3
x
y
z
2
1
1
2
3
2
1
Figure 2: Restricted version of edge group coloring problem
version of the multicast packet scheduling problem in crossbar switches [5].
In this application, each edge-group represents a multicast packet, with each
group including an edge for each output that is to receive a copy. The colors
assigned to edges correspond to time-slots during which copies of multicast
packets are transferred to outputs. Copies of a multicast packets can be
2

transferred to multiple outputs at the same time, but there is no requirement
that all copies be transferred at the same time. This corresponds to the edge
coloring rule that allows edges in the same group to share a color.
The problem can also be applied to routing mulitcast connections in a
three stage Clos network [2], and indeed the multicast routing problem has
been studied extensively. This paper draws directly on two papers in the
multicast routing literature. The paper by Yang and Masson [6] includes a
multicast routing algorithm that can be adapted to the edge-coloring prob-
lem and provides the best worst-case approximation bound that is currently
known for edge-group coloring. Kirkpatrick, Klawe and Pippenger [4] formu-
late the problem of routing connections in a Clos network as a hypergraph
coloring problem and give bounds on the number of colors required. Our for-
mulation of the problem is equivalent to theirs, but is expressed in somewhat
simpler language.
We start by showing that the edge-group coloring probelm is NP-complete
and give a simple approximation method that can be refined into several spe-
cific algorithms. These are evaluated experimentally on random k-colorable
graphs. In section 4, we re-visit Yang and Masson’s algorithm in the context
of the edge-group coloring problem, provide a simplified analysis of its worst-
case performance and evaluate its performance experimentally. In section 5,
we discuss an algorithmic approach that is implicit in the proof of the main
theorem of [4]. We give an explicit statement of this approach and consider
two specific algorithms based on it. We show that one of these can be viewed
as a randomized approximation algorithm with a performance ratio that is
somewhat better than the one in [6].
1 Preliminaries
For the bipartite version of the problem studied here, we find it convenient
to represent edges as ordered pairs (u, v) where uis an input and vis an
output. For any vertex u, let δ(u) denote the number of edges incident to u
(the vertex degree) and let d(u) denote the number of distinct groups with
an edge incident to u(the group count). In general, d(u)≤δ(u) and for
the restricted graphs considered here, d(v) = δ(v) for outputs v. We also let
3

∆ = maxuδ(u) and D= maxud(u). In addition, we let ∆obe the largest
vertex degree among the outputs, and Dibe the maximum group count
among the inputs.
By Vizing’s theorem [1] for the ordinary edge coloring problem, we can
color any bipartite graph with ∆ colors. For edge group coloring we can
often do much better than this. Indeed, it’s tempting to think that we might
be able to color any simple bipartite graph with Dcolors. Unfortunately,
this is not true, as the example in Figure 3 demonstrates. In this figure, the
c b
y
z
d
e
f
a
x
Figure 3: Dcolors are not always enough
shaded vertices are the outputs. Note that D= 2, but this graph requires
three colors. We’ll show in the next section that Di∆ocolors are always
enough, and we note that in [4], the authors show that there are graphs that
require more than (Di−1)∆ocolors.
The edge group coloring problem can be shown to be NP-complete using
a reduction from the vertex coloring problem in graphs. An instance of
the vertex coloring problem is a graph G= (V, E) and an integer k. The
objective is to determine if the vertices of Gcan be colored with at most k
colors, with no two adjacent vertices having the same color.
Given an instance of the vertex coloring problem, we can create a cor-
responding instance of the edge group coloring problem that can be colored
with kcolors if and only if the vertices of the original graph can be colored
with kcolors. To construct the edge group coloring instance, we start with
the original graph and insert a new vertex in the middle of each of its edges.
These new vertices are outputs in the edge group coloring instance, while
the original vertices are inputs. For each input u, all the edges incident to
4

u(so far) form a single edge group. To complete the construction, we add
k−1 outputs for each of the inputs. Each of these outputs is connected to
its input by an edge. We refer to these edges as stubs. Each stub belongs to
a singleton group. An example of this reduction is shown in Figure 4.
c
b
a
e
d
c
b
a
e
d
ab
ac
ae
bc
cd
de
Figure 4: Reduction from graph coloring to edge group coloring
Note that in any valid edge group coloring, the stubs consume k−1
colors, leaving one color for the remaining group (corresponding to the edges
in the original graph). This means that there is a direct correspondence
between proper kcolorings of the original graph and edge-group colorings
of the constructed graph. This allows us to conclude that the edge group
coloring problem is NP-complete. Moreover, this is true even for fixed values
of kas small as 3.
2 A simple approximation algorithm
We start with a simple observation. Consider a graph in which each input
has a single group. Such a graph can be colored with ∆ocolors, since we
can independently assign colors to the edges incident to each output. This
leads to the following general method for coloring the edge groups. Repeat
the following phase until all edges have been colored.
5

Select a set of edge groups (from among those not previously selected),
with one group centered at each input (omitting inputs whose edge
groups have all been previously selected); let tbe the maximum number
of edges incident to any single output in the subgraph induced by
the edges in the selected groups; color all the selected edges using t
previously unused colors.
We call the groups selected in each phase a layer and the general method is
called the layering method. The thickness of a layer is the maximum number
of edges incident to any output in the subgraph defined by the layer. Since
the layer thickness cannot exceed ∆o, the number of colors used for a layer
is at most ∆o. Since the number of phases is Di, the number of colors used
is at most Di∆o. Moreover, since any valid coloring requires max{Di,∆o}
colors, this method produces solutions that use at most min{Di,∆o}times
as many colors as an optimal solution.
In the next section, we’ll consider ways to refine the basic layering method,
but in the remainder of this section we evaluate its performance experimen-
tally, using a simple random graph model. The model uses five parameters:
the number of inputs, ni; the number of outputs, no; an upper bound on
the group count at the inputs, Di; the vertex degree of the outputs, ∆o; and
an upper bound on the number of colors needed to color the edges, χ. The
generated graphs have input degree ∆i= ∆ono/ni(we require this quantity
to be an integer). The color bound χmust be at least max{Di,∆o}. The
generation process consists of three steps.
•Select a graph uniformly from among the bipartite graphs with the
specified number of inputs and outputs, and the specified input and
output degrees.
•At each output, randomly assign distinct colors in {1, . . . , χ}to its
incident edges.
•At each input, form groups from the edges that share the same color,
then for each input with more than Digroups, randomly merge groups
until the number of groups is Di.
6

The coloring used to generate the group graph is discarded once the genera-
tion process is complete. The minimum number of colors required for these
graphs falls in the range [max{Di,∆o}, χ]. An example of a group graph
generated using this method is shown below.
[a: (f i l) (g k) (e)]
[b: (i l) (h j) (g k)]
[c: (f h j) (e) (g h)]
[d: (f i) (e j) (k l)]
Each line in this representation shows the neighbors of an input, with the
parentheses identifying the groups. So for example, input ahas three groups,
the first consisting of the edges (a, f ), (a, i) and (a, l). This graph was
constructed using χ= 4 and a 4-coloring appears below.
1: a(f i.) b(h j) c(e) d(k l)
2: a(g k) b(i l) c(h j.) d(f.)
3: a(l.) b(g.) c(f.) d(e j)
4: a(e) b(k.) c(g h) d(i.)
Here, each line identifies the edges assigned a particular color, with paren-
theses again used to identify groups or partial groups (which are indicated
using a period). So for example, the first two edges in a’s first group are
assigned color 1, while the remaining edge is assigned color 3. The coloring
obtained by the basic layering method for this graph uses seven colors and
is shown below.
1: a(f i l) c(h j.)
2: b(i l) c(f.)
3: d(f i)
4: a(g k) b(h j) c(e)
5: d(e j)
6: a(e) b(g k) c(h.) d(l.)
7: c(g.) d(k.)
The first layer selected by the algorithm includes the first group at each input
and has a thickness of 3. Hence, the first three colors are used to color its
7

1
2
3
4
5
6
7
0 5 10 15 20
(colors used)/Δo
no/ni
ni=100, Δo=Di=50, χ=55
basic&layer!
Figure 5: Effect of input/output asymmetry on basic layering method
edges. The next layer is colored with the next two colors and the third layer
with the last two.
Our first experiment explores the effect of input/output asymmetry (no/ni)
on the performance of the basic layering method. For this experiment, we
use dense graphs, where the number of edges m=nino/2. The bound on
the number of groups per input is equal to the output degree (Di= ∆o) and
the color bound is slightly larger than the output degree (χ= 1.1∆o). The
chart in Figure 5 shows the ratio of the number of colors used to ∆o.
The curve shows the average of ten trials on different random group
graphs. The error bars show the minimum and maximum values from these
ten trials. We observe that the number of colors increases initially as the
graph becomes asymmetric, but then the rate of increase tapers off. For the
most asymmetric graphs, the number of colors used is just under six times
the lower bound, ∆o. (The upper bound on the number of colors is 50∆oin
this case.)
Our next experiment examines how the performance varies as a function
of the ratio of Dito ∆o, which we refer to as the skew. The results are
8

2.00
3.00
4.00
5.00
6.00
0.25 0.5 1 2 4
(colors used)/max(Di,Δo)
skew (Di /Δo)
basic layers
ni=500, no=1000, Δo=50 , χ=1.1max(Di,Δo)
Figure 6: Effect of skew on basic layering method
shown in Figure 6. Observe that the performance ratio is highest (worst)
when Di= ∆o. In general, we can expect better performance (relative to
the lower bound) when Diand ∆odiffer substantially.
In our next experiment, we examine how the performance varies with
the density of the graph. In Figure 7 we hold the number of vertices fixed,
while increasing the degree of the inputs and outputs. Observe that there is
a modest increase in the performance ratio as ∆oincreases from 5 to 40, but
after that, it remains roughly constant. Since the number of phases of the
algorithm increases directly with the graph density, the performance ratio is
just a reflection of the layer thickness, which is only weakly dependent on
density.
Our final experiment for the basic layer method examines the impact of
the color bound χ. One might expect that increasing χwould also increase
the number of colors required, and the number used by the algorithm. Fig-
ure 8 shows that, there is no clear relationship between χand the number
of colors used by the algorithm. On reflection, this is not surprising. Recall
that the number of iterations is Diand the number of colors used is Ditimes
9

4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
0 25 50 75 100 125 150
(colors used)/Δo
Δo
ni=200, no=2000, Di=Δo, χ=1.1Δo
basic&layers!
Figure 7: Effect of graph density on basic layering method
the average layer thickness. Neither Di, nor the layer thickness is directly
affected by the choice of χ, so neither is the performance of the algorithm.
(Note that the error bars in this chart are exaggerated by the limited scale
of the y-axis.)
The overall conclusion is that the performance of the basic layering
method is primarily a function of the layer thickness, which is most strongly
influenced by input/output asymmetry, although even in that case the de-
pendence tapers off as asymmetry grows beyond 5. We also observe that
group graphs with Di= ∆orequire the largest number of colors.
3 Refining the layering method
In this section we introduce several refinements to the basic layering method
and observe how they affect its performance. Our first refinement is called
the thin layers method, since it seeks to select the groups that form each layer
with the objective of minimizing the overall layer thickness. More precisely,
10

4.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
5.8
6.0
1 1.2 1.4 1.6
(colors used)/Δo
χ/Δo
ni=100, no=500, Di=Δo=50
basic&layers!
no/ni=5
no/ni=2
Figure 8: Effect of color bound on basic layering method
the thin layers method forms each layer by repeating the following step at
each input u.
Select the group at uthat yields the smallest thickness value when
added to the layer formed by the groups that have been selected so far.
Once the groups forming a layer have been selected, they are colored using
tnew colors, where tis the thickness of the new layer. When applied to the
example graph from the last section, the thin layers method produces the
coloring using six colors shown below.
1: a(l f i) b(j h) c(e)
2: d(i f)
3: a(k g) b(l i) c(j f h)
4: d(j e)
5: a(e) b(k g) c(h.) d(l.)
6: c(g.) d(k.)
11

Here, each of the three layers has a thickness of 2.
Before describing our next algorithm, we need a definition. Given a
partial coloring of a graph, a color cand an uncolored edge e= (u, v), we
say that cis viable for eat uif cis not being used by any other edge at u
that is in a different group than e;cis viable for eat vif it is not being used
by any other edge at v. We say that cis viable for eif it is viable at both
endpoints.
Our next method extends the thin layers method by relaxing the require-
ment that each layer use a distinct set of colors. The min color method colors
each selected group using colors allocated for previous layers whenever pos-
sible. In particular, when coloring a group, each of its edges eis colored
according to the first of the three cases listed below, that is applicable.
1. If there is a color cthat is viable for eand has already been used by
some edge in e’s group, color eusing the smallest such color (that is,
the color with the smallest index).
2. If there is a color cthat is viable for eand has already been used by
any edge, color eusing the smallest such color.
3. If there is no previously used color that is viable for e, allocate the next
unused color and use it to color e.
Colors are allocated in increasing order of their positive integer index. When
applied to the example graph from the last section, the min color method
produces the coloring using five colors shown below.
1: a(f i l) b(h j) c(e) d(k.)
2: a(g k) b(l.) c(h j.) d(f i)
3: b(i.) c(f.) d(e j)
4: a(e) b(g k) c(h.) d(l.)
5: c(g.)
Here, the first and second layers each require two new colors, but the third
layer is colored using just one additional color.
Our last refinment to the basic layers method is inspired by the classical
proof of Vizing’s theorem for the standard edge coloring problem. The proof
12

describes an algorithm that uses augmenting paths to find a coloring. That
method cannot be used directly for the edge group coloring problem, but we
can adapt it to accommodate edge groups.
Let e= (u, v) be an edge to be colored and let iand jbe two edge colors
where iis viable for eat u,jis viable at v, but neither is viable at both
endpoints. An augmenting path is a path pthat starts at v, ends at a vertex
w, has edges that alternate in color between iand j(starting with i) and
that satisfies the following conditions.
•The two path edges incident to any intermediate vertex xbelong to
different groups.
•No non-path edge incident to an intermediate vertex xhas color ior j.
•If colors iand jare both in use at w, the edges with those colors are
all in the same group.
Observe that if pis an augmenting path, we can reverse the colors of the
edges on pand still have a valid coloring. Performing this color reversal
makes color iviable for e. Figure 9 shows an example of a graph with two
augmenting paths, a 1-2 augmenting path from vto band a 3-4-augmenting
path from vto e.
u
a
v
e
b
c d
1
2
3
4 3
3
4 6
5
1
6
3
6
5
5
5
3
2
2
6
5
1
2
4
Figure 9: Example of augmenting paths
13

Note that when attempting to construct an augmenting path, we may
arrive at a vertex that does not satisfy either the conditions for an interme-
diate vertex x, or the conditions for the terminal vertex w. In this case, there
is no ij-augmenting path for eand the attempted path construction fails.
The recolor method is a modification of the min color method. In par-
ticular, when coloring edge e= (u, v), it replaces the third case in the min
color method with the following.
If there is no previously used color that is viable for e, color eusing the
first applicable sub-case listed below (this may involve testing multiple
color pairs).
•If color iis used by some other edge in e’s edge group at u, color
jis a previously used color that is unused at vand there is an
ij-augmenting path from v, reverse the colors of the edges on the
path and and color eusing i.
•If color iis a previously used color that is unused at u, color j
is a previously used color that is unused at vand there is an ij-
augmenting path from v, reverse the colors of the edges on the
path and color eusing i.
•If neither of the previous sub-cases apply for any pair of previously
used colors, allocate the next unused color and use it to color e.
Observe that in the example in Figure 9, the recolor algorithm will choose
the 1-2 augmenting path in preference to the 3-4 path.
Next, we examine the performance of our refinements to the basic layering
method. Figure 10 shows how the performance varies with asymmetry (we
have omitted the error bars for clarity). Observe that for the most asymmet-
ric graphs, the thin layers method brings the performance ratio down from
about 5.8 to about 3.3. The min color and recolor methods bring it down to
about 2.2. The recolor method provides only a very small improvement over
the min color method (a dashed line style is used for recolor to make the
small difference more apparent). Finally note that for asymmetries smaller
than 10, the min color and recoloring methods produce solutions that are
within a factor of two of optimal.
14

1
2
3
4
5
6
7
0 5 10 15 20
(colors used)/δo
no/ni
ni=100,%χ=55,%δi=no/2,%δο=di=ni/2%
basic(layers%
thin(layers%
min(color%
recolor%
Figure 10: Effect on input/output asymmetry on the layering algorithms
Figure 11 shows how the performance varies with skew. In all cases, we
see a peak when Di= ∆o, although the height of the peaks is smaller for
the more complex variations. Figure 12 shows how the performance ratio
is affected by the graph density. Here, we observe that for the algorithms
introduced in this section, the average number of colors used per layer de-
creases (by small amounts) as the density increases. There appear to be two
factors accounting for this. First, as density increases, we have more groups
per input, which provides a wider range of available choices when forming
the layers. This leads to thinner layers, particularly in the early phases.
The second factor that plays a role arises only in the min color and recolor
methods. Because dense graphs require more colors, they offer a wider range
of color choices that the algorithms can exploit when coloring the individual
edges.
15

0
1
2
3
4
5
6
0.25 0.5 1 2 4
(colors used)/max(Di,Δo)
skew (Di /Δo)
basic layers
ni=100, no=1000, Δo=50 , χ=1.1max(Di,Δo)
recolor
min color
thin layers
Figure 11: Effect of skew on the layering algorithms
4 Re-visiting an algorithm by Yang and Masson
In [6], Yang and Masson describe an algorithm for routing a single multi-
cast connection in a three stage Clos network. They show that a suitably
configured Clos network is wide-sense nonblocking for new multicast con-
nections, if their algorithm is used. Here, we adapt their algorithm for the
edge-group coloring problem. The algorithm has a parameter kthat limits
the number of colors used for any single edge group (we’ll discuss the choice
of klater). We say that a color is eligible for selection at some stage, if its
index is ≤max{Di,∆o}or it has already been used at least once (possibly
in an earlier step). The algorithm attempts to color each edge group with at
most kcolors, by selecting colors using a greedy strategy.
While some edge in the group remains uncolored, select an eligible color
cthat is viable for the largest number of uncolored edges remaining in
the group; use cto color all edges in the group for which it is viable.
16

1.00
2.00
3.00
4.00
5.00
6.00
0 25 50 75 100 125 150
(colors used)/Δo
Δo
ni=200, no=2000, Di=Δo, χ=1.1Δo
basic&layers!
recolor!
min&color!
thin&layers!
Figure 12: Effect of graph density on the layering algorithms
If this procedure fails to color all edges in a group using ≤kcolors, we
allocate a new color and use it to color all edges of the group, instead.
We refer to this as the few colors method. When applied to the graph
[a: (f i l) (g k) (e)]
[b: (i l) (h j) (g k)]
[c: (f h j) (e) (g h)]
[d: (f i) (e j) (k l)]
the few colors method produces the 5-coloring shown below, when k= 2,
assuming that the edge groups are colored in the order in which they appear
in the graph.
1: a(f i l) b(h j) c(e) d(k.)
2: a(g k) b(i l) c(f h j)
3: a(e) b(g k) d(f i)
4: c(g h) d(e j)
5: d(l.)
17

The number of colors used by the few colors method is bounded in the
following theorem, which is an adaptation of a theorem in [6].
Theorem 1 Let G= (V, E )be a bipartite graph in which the edges incident
to each input are divided into edge groups to produce an instance of the edge
group coloring problem. The few colors algorithm with parameter kcolors
the edges of Gusing at most
(Di−1)k+ (∆o−1)n1/k
o+ 1
colors.
We give a simplified proof of the theorem, based on a lemma that bounds the
number of colors used by a greedy algorithm for the set covering problem,
when applied to instances that satisfy a special condition. An instance of the
set covering problem consists of a base set Aand a collection of subsets Si
of A. The objective is to find a smallest possible collection of subsets whose
union includes all elements of A. The greedy algorithm for the set covering
problem repeatedly selects a subset that covers the largest possible number
of previously uncovered elements. It terminates as soon as all elements are
included in at least one of the selected subsets.
Lemma 1 Let A={a1, . . . , at}and S={S1, . . . , Sp}be an instance of the
set covering problem in which every aiappears in at least p−qsubsets for
some integer q. If integer ksatisfies p > qt1/k then the greedy algorithm finds
a covering of Ausing at most ksubsets.
proof. Let hbe the number of subsets in the greedy solution and assume
the subsets are numbered so that for 1 ≤i≤h,Siis the subset chosen in
step i. Define Ui=S1∪. . . ∪Siand let Di=Si−Ui−1,si=|Si|,ui=|Ui|,
di=|Di|. Then,
ps1≥
p
X
i=1
si≥(p−q)t
So, u1=s1≥(1 −q/p)t= (1 −xo)t, where xj= (q−j)/(p−j). Next, note
that
(p−1)d2≥
p
X
i=2
|Si−U1| ≥ (p−q)(t−u1)
18

So, d2≥(1 −x1)(t−u1) and
u2=u1+d2≥(1 −x1)t+x1u1≥(1 −x1)t+x1(1 −x0)t= (1 −x1x0)t
Extending this reasoning using induction, we find that for i≤h,ui≥(1 −
xi−1· · · x1x0)t. In particular
uk≥(1 −xk−1· · · x1x0)t≥(1 −xk
0)t > (1 −1/t)t=t−1
So, Ukhas more than t−1 elements and since |A|=t,|Uk|=t.
Now, let’s proceed to the proof of the theorem. We view each step of the
few colors method as a set covering problem, in which the base set is the set
of edges in the current edge group. There is a subset for each color that is
not already being used by an edge incident to the group’s input u, and for
each such color c, its subset consists of those edges (u, v) for which cis not
being used by any edge incident to v. Let tbe the number of edges in the
group and let pbe the number of subsets.
Now, let Cbe the number of colors that have been used so far. We claim
that if C > (Di−1)k+(∆o−1)n1/k
o, the group can be colored with ≤kcolors,
without any further increase in C. So, assume C > (Di−1)k+ (∆o−1)n1/k
o
at the start of the step. Let ube the group’s input and note that the number
of colors already in use at uis ≤(Di−1)k. Hence, the number of colors that
are available to color the group is ≥C−(Di−1)k > (∆o−1)n1/k
o. That is
p > (∆o−1)n1/k
o.
Next, consider any edge (u, v) in the group. At most ∆o−1 colors
are already being used at v. Consequently, (u, v) is a member of at least
p−(∆o−1) subsets. Letting q= (∆o−1), we have p > qn1/k
oand since
t≤no,p > qt1/k . The lemma then implies that the group can be colored
using at most kcolors. 
We may choose the value of kthat minimizes the bound in the theorem.
When Di= ∆o, the best choice of kis the smallest integer for which
n1/k
o−n1/k+1
o≤1
The required value is approximately 2 ln no/ln ln no, as shown by Yang and
Masson. This also leads to a worst-case performance ratio of approximately
19

(2 ln no/ln ln no) + (ln no)1/2. For all but the sparsest graphs, this is better
than the trivial ratio for the layering method.
Note that the performance ratio is independent of the order in which the
edge groups are selected. We find that coloring the groups in decreasing order
of their size gives better performance than other simple ordering strategies.
This approach allows the largest groups to be colored with fewer colors,
leaving more colors available for groups that come later. We can also refine
the method in another way. Instead of using a fixed parameter k, we can
allow the number of colors per group to vary with the number of groups at
an input. Specifically, we allow a group centered at an input uto use up to
d(the number of eligible colors at u)/d(u)e
colors. Recall that the number of eligible colors is initially max{Di,∆o}and
it increases by one whenever a previously unused color is used. We evaluate
this version of the few colors method experimentally at the end of the next
section.
5 Using matchings to color group graphs
In [4], the authors give several results that can be interpreted as bounds on
the number of colors needed to color a group graph. Their main result is
that the number of colors required for any graph is at most
(Di−1)blog2(2no)c+ 2∆o
For most parameter choices, this is not quite as strong as Yang and Masson’s
result, but the method used to prove it leads to an interesting and distinctly
different algorithmic method. In this section, we describe this general method
and evaluate two specific algorithms based on it. The method divides the
coloring process into two parts. First, at each input, we divide the available
colors among the groups at that input. The colors assigned to a given group
are referred to as its menu. Next, we color the edges incident to each output
using colors selected from the menus of the groups that the edges belong to.
We can illustrate this procedure for the example graph
20

[a: (f i l) (g k) (e)]
[b: (i l) (h j) (g k)]
[c: (f h j) (e) (g h)]
[d: (f i) (e j) (k l)]
Suppose we are attempting to color this using four colors. In the first step,
we create menus for each group.
[a: (f i l)[1,2] (g k)[3] (e)[4]]
[b: (i l)[1] (h j)[2] (g k)[3,4]]
[c: (f h j)[1,2] (e)[3] (g h)[4]]
[d: (f i)[4] (e j)[2,3] (k l)[1]]
The menus are shown within the square brackets, so for example the menu
of the group a(f i l) contains colors 1 and 2. In the next step, we attempt
to color the edges at each output using the colors in the menus associated
with its edge groups. For example, output ehas three edges (a, e), (c, e) and
(d, e) whose groups have menus [4], [3] and [2,3]; so, they can be colored
using colors 4, 3 and 2. Similarly, the edges incident to output fcan be
colored using colors 1, 2 and 4. When we get to output g, we find that the
groups for its three edges have menus [3], [3,4] and [4]. Since these menus
give us only two colors to choose from, we cannot color all three edges, given
these menus. This raises the question of how we can best select the menus in
the first place. We’ll explore two different ways to answer this question and
the algorithms based on those answers. First however, let’s consider how to
color the edges once the menus have been chosen. This involves finding a
matching in a graph.
For each output v, define M(v) to be the menu graph of v.M(v) has
an input for each group with an edge incident to vand an output for every
color. It includes edges joining each input to the outputs corresponding to
colors in the group’s menu. So in the earlier example, M(j) has the menu
graph in Figure 13. A matching that includes an edge incident to every input
of M(v) defines a valid coloring of the edges incident to v. We call such a
matching complete. So given an assignment of colors to menus, we can color
all the edges of the graph if we can find complete matchings for all the menu
21

b(hj)
d(ej)
1
c(fhj) 2
3
Figure 13: Menu graph for output j
graphs. If any menu graph does not have a complete matching, then there
is no valid coloring using the given menus.
We start by considering random menus. Here, we assign kcolors to each
group, for some integer k. We can show that for large enough values of k, this
approach succeeds with high probability. By Hall’s Theorem [1], a random
menu graph with kedges incident to each input has a complete matching if
and only if, for every non-empty set Xof the inputs, the set of neighbors
of Xis at least as large as X. For large enough k, the probability that a
random menu graph contains a set of inputs Xfor which the set of neighbors
is smaller than Xis small. The required analysis is a variation on well-known
results for expander graphs [3].
Theorem 2 Let Gbe a group graph and let C=kmax(Di,∆o), where
k≥2 (ln 2∆on0/ln ln 2∆on0)1/2. A set of random menus using Ccolors and
kcolors per group defines a set of menu graphs that all contain a complete
matching with probability ≥1/2.
proof. Consider the menu graph M(v) for some vertex v. Let Xbe a
subset of the inputs of M(v) and Ybe a subset of its outputs. Let NX,Y be
the event that all of X’s neighbors are in Y. Then
Pr{NX,Y }= (|Y|/C)k|X|
22

The probability that there exists a pair of such subsets X,Ywhere |Y|<|X|
is
≤
δ(v)
X
i=k+1 δ(v)
i C
i−1i−1
Cik
≤
∆o
X
i=k+1 ∆o
iC
i i
Cik
≤
∆o
X
i=k+1 e∆o
i
eC
i
ik
Cki
≤
∆o
X
i=k+1 e2∆oik−2
Ck−1i
≤∆o e2∆o
Ck−1!k+1
≤∆oe2/kk−1k+1
The right side of this last expression is ≤1/2noif ksatisfies the condition
in the theorem. Since the number of menu graphs is no, the probability that
the random menu method fails to find full matchings for all menu graphs is
≤1/2. 
The random menu method selects random menus repeatedly until it finds
a set for which the menu graphs all have complete matchings. If ksatisfies
the bound in the theorem, we can expect to succeed after a small number of
attempts. In our experimental implementation of the random menu method,
we allow inputs with < Digroups to have more colors per group. In par-
ticular, at each input, we assign colors to groups in a round-robin fashion,
starting with the largest group (consequently, larger groups may get one
more color than the smallest groups). For the most asymmetric graphs used
in our experiments, the bound in the theorem requires k≥5. We find that
for random graphs, 2 or 3 colors per group is generally sufficient.
We can also construct menus in a more systematic way. For each output,
we maintain a menu graph and a maximum matching on that graph. We
define the deficit of a group g, to be the number of menu graphs in which
there is a vertex for gwhich is not currently matched. We define the gain
23

of a color cnot in g’s menu, to be the number of menu graphs in which the
vertices for gand care both unmatched. The gain is a lower bound on the
reduction in g’s deficit that will result if cis added to its menu.
The greedy menu method repeatedly performs the following step, until
all groups have a zero deficit.
Select a group gwith a positive deficit. Let ube the input on which gis
centered and let kg=d(number of eligible colors)/d(u)e. While ghas a
positive deficit, its menu has fewer than kgcolors, and some previously
used color has positive gain for g, select one such color and add it to
g’s menu; update all menu graphs containing a vertex for gand update
their maximum matchings. If gstill has a positive deficit, allocate a
previously unused color, remove all colors currently in g’s menu, add
the new color and update all menu graphs containing a vertex for g
and their maximum matchings.
We select groups in decreasing order of their size and we select colors that
yield the largest gain for the group.
Observe that this method is operates similarly to the few colors method
discussed in the previous section. The key difference is that while colors
are assigned to a group’s menu, they are not rigidly assigned to the group’s
edges, allowing a little more flexibility in the choice of the final edge colors.
Figure 14 shows how the two menu-based methods compare to the recolor
and few colors methods. We observe that the greedy menu method has the
best performance overall, although the few colors method is very close for
smaller asymmetries. For random menus, the performance ratio increases
very slowly for asymmetries of two or more,. Indeed, for larger asymmetries
than are shown on the chart, it out-performs both the recolor and few colors
methods. Specifically, when the asymmetry is 100, the performance ratio
for the recolor method is 2.82, for the few colors method it is 2.72, for the
random menu method it is 2.44 and for the greedy menu method it is 2.1.
Figure 15 shows how the performance ratio varies with skew. For the
most part, these results are consisent with what we have seen before. How-
ever, the behavior of the random menu method does seem a little unusual.
Its superior performance at small values of skew may reflect the fact that
24

1
1.25
1.5
1.75
2
2.25
2.5
0 5 10 15 20
(colors used)/Δo
no/ni
ni=100, Di=Δo=50, χ=55
recolor
few colors
greedy menu
random menu
Figure 14: Effect of input/output asymmetry on the menu methods
the random menu method assigns roughly the same number of colors to all
groups, providing lots of flexiblity when coloring edges, especially when Diis
small. In contrast, the greedy menu method assigns just one or two colors to
some groups, in order to leave more available for groups that require greater
flexibility. This is advantageous when the number of colors per group is very
limited, but may not help when Diis small and there is no need to limit
any group to just one or two colors. For larger values of skew, the random
menu method underperforms the others by a large margin. Apparently, with
random menus, it’s difficult to do much better two colors per group, while
the other methods are able to use just one color for most groups, since they
choose the colors in a more informed way.
We conclude this section by examining how the performance of the few
colors and menu methods varies with graph density (Figure 16). The rel-
ative ordering of the methods remains consistent with what we have seen
before. As with the layering methods, we observe that the performance ratio
improves as the density increases. Since dense graphs have larger values of
∆o, the algorithms have more colors to choose from and this added flexibility
25

0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
0.25 0.5 1 2 4
(colors used)/max(Di,Δo)
skew (Di /Δo)
ni=100, no=1000, Δo=50 , χ=1.1max(Di,Δo)
recolor
greedy menu
random menu
few colors
Figure 15: Effect of skew on the menu methods
allows them to get a bit closer to the lower bound.
6 Closing Remarks
The edge group coloring problem is a natural generalization of the classi-
cal edge coloring problem in graphs. While all the algorithms studied here
perform well on random graphs, only the few colors method and random
menu method have good analytical bounds on their performance. It seems
likely that similar bounds might be possible for the thin layers and min color
methods, but currently our only bound is the trivial one of Di∆o. It remains
open whether there exists an approximation algorithm with a constant per-
formance ratio. The case of non-bipartite graphs appears to be entirely
unexplored.
The menu-based methods can be applied directly to the problem of rout-
ing multicast connections in a three stage Clos network. They could also be
applied to the problem of routing connections in an online manner. They
26

1.00
1.25
1.50
1.75
2.00
2.25
2.50
0 25 50 75 100 125 150
(colors used)/Δo
Δo
ni=200, no=2000, Di=Δo, χ=1.1Δo
random'menu!
recolor!
greedy'menu!
few'colors!
Figure 16: Effect of density on the menu methods
do require the ability to rearrange existing connections, when used in this
application, but the impact of the rearrangement on existing connections
would be fairly limited.
Our results apply to the version of the multicast packet scheduling prob-
lem in which the objective is to transfer a set of packets from inputs to
outputs in a minimum amount of time. It does not apply directly to the
more practically interesting problem of work-conserving scheduling. How-
ever, it might be possible to adapt the menu methods to this problem. The
main challenge here is that multicast packets may involve copies going to
both lightly-loaded outputs and heavily-loaded outputs. These outputs im-
pose conflicting requirements on the scheduler, making it difficult to achieve
strict work-conservation. However, the flexibility inherent in the menu meth-
ods may allow for some approximate form of work-conservation.
27

References
[1] Bondy, J. A. and U. S. R. Murty. Graph Theory and Applications,
Elsevier North Holland, 1976.
[2] Clos, C. “A study of nonblocking switching networks,” Bell System
Technical Journal, 1953.
[3] Hoory, Shlomo, Nathan Linial and Avi Wigderson. “Expander graphs
and their applications,” Bulletin of the American Mathematical Society,
10/2006.
[4] Kirkbatrick, D. G., M. Klawe and N. Pippenger. “Some graph coloring
theorems with applications to generalized connection networks,” SIAM
J. Alg. Disc. Math, 10/1985.
[5] Prabhakar, B., N. McKeown and R. Ahuja. “Multicast scheduling for
input-queued switches,” IEEE Journal on Selected Aeas in Commuica-
tions, 6/1997.
[6] Yang, Yuanyan and Gerald Masson. “Nonblocking broadcast switching
networks,” IEEE Transactions on Computers, 9/91.
28