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Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree
Abstract
Various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O(logn) (minimum front size and treewidth) and O(log^2 n) (pathwidth and minimum elimination tree height) times the optimal values. In addition, we show that unless P = NP there are no absolute approximation algorithms for any of the parameters.
... Our first lemma bounds the treedepth of a graph G in terms of the width of one of its tree decomposition .T; W / and the treedepth of T . This lemma may be considered folklore; it is implicit in proofs of the inequality td.G/ Ä .tw.G/ C 1/ log jV .G/j [7,27]. We could not find a proof in the literature, so we include one for completeness. ...
... Results of Bodlaender et al. [6,7] give a polynomial-time algorithm which, given a graph G, outputs a tree decomposition of G of width O.tw.G/ 2 /. (This is actually a combination of two polynomial-time approximation algorithms for treewidth: an O.log n/-approximation for arbitrary n-vertex graphs and 5-approximation when tw.G/ Ä log n [7].) Combining this algorithm with Corollary 7.2, we get To obtain the algorithmic version of Theorem 1.2, we combine Corollary 7.2 with the randomized polynomial-time algorithm of Chekuri and Chuzhoy [10] which, given a graph G, outputs a minor embedding of the k k grid where k D tw.G/ .1/ . ...
... Results of Bodlaender et al. [6,7] give a polynomial-time algorithm which, given a graph G, outputs a tree decomposition of G of width O.tw.G/ 2 /. (This is actually a combination of two polynomial-time approximation algorithms for treewidth: an O.log n/-approximation for arbitrary n-vertex graphs and 5-approximation when tw.G/ Ä log n [7].) Combining this algorithm with Corollary 7.2, we get To obtain the algorithmic version of Theorem 1.2, we combine Corollary 7.2 with the randomized polynomial-time algorithm of Chekuri and Chuzhoy [10] which, given a graph G, outputs a minor embedding of the k k grid where k D tw.G/ .1/ . ...
... We thus move to co-tripartite graphs G with tripartition (A, B, C) (into cliques) where A encodes the clauses, and B ∪ C encodes the variables, with B associated to their positive form, and C, their negation. More precisely, a blow-up (where vertices are replaced by clique modules) of a semi-induced matching 6 between B and C constitutes our variable gadgets. For A, we add 7 = 2 3 − 1 vertices for each clause, one for each partial satisfying assignment of the clause, each adjacent to the three modules corresponding to its literals; see Figure 1. ...
... The ETH asserts that there is a λ > 0 such that no algorithm solves n-variable 3-SAT in O(λ n ) time[13].5 We will not need the definition of cutwidth.6 Here, an induced matching if not for the cliques B and C. ...
We present a simple, self-contained, linear reduction from 3-SAT to Treewidth. Specifically, it shows that 1.00005-approximating Treewidth is NP-hard, and solving Treewidth exactly requires $2^{\Omega(n)}$ time, unless the Exponential-Time Hypothesis fails. We further derive, under the latter assumption, that there is some constant $\delta > 1$ such that $\delta$-approximating Treewidth requires time $2^{n^{1-o(1)}}$.
... If G has treewidth τ , then any subgraph of G has a 1/2-balanced separator of size τ + 1; conversely, if G is τ -separable, then G has treewidth at most τ log n (c.f. [BGHK95]). By leveraging properties of separable graphs, the sequence of three papers [DLY21, DGG + 22, DGL + 23] have iteratively refined the robust IPM framework and associated data structures for solving structured linear programs. ...
... It's well known that given a O(log n) approximation algorithm for finding a balanced vertex separator, one can construct a tree decomposition of width O(tw(G) log n). Specifically, the algorithm of [BGHK95] find a tree decomposition by recursively using a balanced vertex separator algorithm. Now, it suffices to show we can find a log(n) pseudo-approximation balanced vertex separator in O(m · tw(G) 3 ) expected time. ...
We present a $\widetilde{O}(m\sqrt{\tau}+n\tau)$ time algorithm for finding a minimum-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $\tau$ and polynomially bounded integer costs and capacities. This improves upon the current best algorithms for general linear programs bounded by treewidth which run in $\widetilde{O}(m \tau^{(\omega+1)/2})$ time by [Dong-Lee-Ye,21] and [Gu-Song,22], where $\omega \approx 2.37$ is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods. As a corollary, for any graph $G$ with $n$ vertices, $m$ edges, and treewidth $\tau$, we obtain a $\widetilde{O}(\tau^3 \cdot m)$ time algorithm to compute a tree decomposition of $G$ with width $O(\tau \cdot \log n)$.
... As branchwidth and treewidth are linearly equivalent, every approximation algorithm for the one is also an approximation algorithm for the other. As a first step, Bodlaender, Gilbert, Hafsteinsson, and Kloks proved in [6] that both parameters admit a polynomial time ...
... Lemma 2.10 permits us to prove that one may not expect, in general, any polynomial time additive approximation for branchwidth. The following proof of Theorem 1.2 uses a reduction employed by [6] for deriving the analogous result for treewidth. ...
The \textsl{branchwidth} of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph classes, further than planar graphs as follows: Let $H_{0}$ be a graph embeddedable in the projective plane and $H_{1}$ be a graph embeddedable in the torus. We prove that every $\{H_{0},H_{1}\}$-minor free graph $G$ contains a subgraph $G'$ where the difference between the branchwidth of $G$ and the branchwidth of $G'$ is bounded by some constant, depending only on $H_{0}$ and $H_{1}$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving an EPTAS for branchwidth: For $\{H_{0},H_{1}\}$-minor free graphs, there is a function $f\colon\mathbb{N}\to\mathbb{N}$ and a $(1+\epsilon)$-approximation algorithm for branchwidth, running in time $\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n),$ for every $\epsilon>0$.
... This current work introduces the unlabeled dependency graph. We observe that, in the labeled setting, through the dependency graph, the running buffer problems naturally connect to graph layout problems [53,54,55,56,57,58], where an optimal linear ordering of graph vertices is sought. Graph layout problems find a vast number of important applications including VLSI design, scheduling [59], and so on. ...
... We show that this is indeed the case, by examining the interesting relationship between MRB and the vertex separation problem (VSP), which is equivalent to path width, gate matrix layout, and search number problems as described in Theorem 3.1 in [53], resulting from a series of studies [67,68,69]. Unless P = N P , there cannot be an absolute approximation algorithm for any of these problems [58]. First, we describe the vertex separation problem. ...
For rearranging objects on tabletops with overhand grasps, temporarily relocating objects to some buffer space may be necessary. This raises the natural question of how many simultaneous storage spaces, or "running buffers", are required so that certain classes of tabletop rearrangement problems are feasible. In this work, we examine the problem for both labeled and unlabeled settings. On the structural side, we observe that finding the minimum number of running buffers (MRB) can be carried out on a dependency graph abstracted from a problem instance, and show that computing MRB is NP-hard. We then prove that under both labeled and unlabeled settings, even for uniform cylindrical objects, the number of required running buffers may grow unbounded as the number of objects to be rearranged increases. We further show that the bound for the unlabeled case is tight. On the algorithmic side, we develop effective exact algorithms for finding MRB for both labeled and unlabeled tabletop rearrangement problems, scalable to over a hundred objects under very high object density. More importantly, our algorithms also compute a sequence witnessing the computed MRB that can be used for solving object rearrangement tasks. Employing these algorithms, empirical evaluations reveal that random labeled and unlabeled instances, which more closely mimics real-world setups, generally have fairly small MRBs. Using real robot experiments, we demonstrate that the running buffer abstraction leads to state-of-the-art solutions for in-place rearrangement of many objects in tight, bounded workspace.
... Thus, a p-centered coloring can be said to be a "low treedepth coloring". Since the treedepth of a graph is always at least one more than its treewidth [1] (in fact, one more than its pathwidth) and the treewidth of any graph is equal to the maximum of the treewidth of its connected components, low treedepth colorings are a generalization of low treewidth colorings. The following generalization of the result of DeVos et al. was shown in [5]: for every proper minor closed class of graphs, there exists a function f : N → N such that for every integer p ≥ 1, every graph in the class has a p-centered coloring using at most f (p) colors. ...
... We now show another coloring of the grid G which is again not p-centered, but this time has no "large violators"; in particular, it has the property that any violator in G spans at most 2(p + 1) rows and at most 1 For a more precise calculation of |µ(V (G))|, note that we can have ρ(x, y) = 0 only if l(x, y) = lg(4p) and ...
A $p$-centered coloring of a graph $G$, where $p$ is a positive integer, is a coloring of the vertices of $G$ in such a way that every connected subgraph of $G$ either contains a vertex with a unique color or contains more than $p$ different colors. We give an explicit construction of a $p$-centered coloring using $O(p)$ colors for the planar grid.
... Further tree-decomposition algorithms can be found in [2,8,9,22,34]. The basic strategy of repeated separator searches is the foundation of all of these treewidth approximation algorithms, as mentioned by Bodlaender et al. [11]. ...
... 4, where we also show a space-efficient computa- Table 1 The space requirements in bits to run the different parts of the algorithm to compute a tree decomposition (t.d.) with applications for an n-vertex graph with sufficiently small treewidth k NP-hard problems above Ω(kn log n) Θ ( n) (Sect. 8) tion of a balanced vertex separator using O(n) bits. In Sect. 5 we present an iterator that outputs the bags of a tree decomposition using O(kn) bits. ...
For n -vertex graphs with treewidth $$k = O(n^{1/2-\epsilon })$$ k = O ( n 1 / 2 - ϵ ) and an arbitrary $$\epsilon >0$$ ϵ > 0 , we present a word-RAM algorithm to compute vertex separators using only O ( n ) bits of working memory. As an application of our algorithm, we give an O (1)-approximation algorithm for tree decomposition. Our algorithm computes a tree decomposition in $$c^k n (\log \log n) \log ^* n$$ c k n ( log log n ) log ∗ n time using O ( n ) bits for some constant $$c > 0$$ c > 0 . Together with the result of Banerjee et al. (Proceedings of 21st international conference on computing and combinatorics (COCOON 2015). LNCS, vol 9198, Springer, pp 349–360, 2015. https://doi.org/10.1007/978-3-319-21398-9_28 ) we are able to compute a solution for all monadic-second-order problems (MSO) with $$O(n + \tau (k) \cdot p (\log _{p} n) \log n)$$ O ( n + τ ( k ) · p ( log p n ) log n ) bits in $$O(\tau (k) \cdot n^{2 + (2/\log p)})$$ O ( τ ( k ) · n 2 + ( 2 / log p ) ) time where k is the treewidth of the given graph, p is some arbitrary parameter with $$2 \le p \le n$$ 2 ≤ p ≤ n and $$\tau $$ τ is some function depending on the MSO formula. We finally use the tree decomposition obtained by our algorithm to solve Vertex Cover , Independent Set , Dominating Set , MaxCut and q - Coloring by using polynomial time and O ( n ) bits as long as the treewidth of the graph is smaller than $$c' \log n$$ c ′ log n for some problem dependent constant $$0< c' < 1$$ 0 < c ′ < 1 .
... In graph-theoretic terminology, we would say thatḠ is a chordal completion of G, and, to minimize the time and space complexity of the hierarchy above, we need that the size of the maximal clique ofḠ be as small as possible. Finding the chordal completion of an arbitrary graph with minimum maximum clique size is known to be an NP-complete problem [21]. However, for fixed k ∈ N, deciding whether a graph admits a chordal completion of maximum clique size k or less is a problem that can be solved in linear time on |A| [22]. ...
... such as local compatibility, or relations (21), (25), then so will the ensemble of measures {p I } I . Finally, due to eq. (39), ...
We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find that we can achieve an arbitrarily good approximation to the set of nearest-neighbour marginals of separable states with a time (space) complexity polynomial (linear) on the system size. Our results even extend to infinite systems, such as translation-invariant systems in 1D, as well as higher spatial dimensions with extra symmetries.
... First consider the treewidth of the cartesian and strong products. The following upper bound is well-known (see [6] for an implicit proof). ...
... The proof of Lemma 19 in [6] shows the following, more general result which we use in Section 4.3. For all graphs G 1 , G 2 and H, if G 1 has an H-decomposition with width at most k, then G 1 ⊠G 2 has an H-decomposition with width at most (k + 1)v(G 2 ) −1. ...
Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this result, this paper systematically studies various structural properties of cartesian, direct and strong products. In particular, we characterise when these graph products contain a given complete multipartite subgraph, determine tight bounds for their degeneracy, establish new lower bounds for the treewidth of cartesian and strong products, and characterise when they have bounded treewidth and when they have bounded pathwidth.
... Bodlaender et al. [BGHK95] have shown that if G 2 is a clique, the upper bounds for tw(G 1 [G 2 ]) and pw(G 1 [G 2 ]) as stated above are tight. ...
Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems allow polynomial-time solutions, when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width and path-width under various unary and binary graph transformations. Doing so, for considered transformations we provide upper and lower bounds for the tree-width and path-width of the resulting graph in terms of the tree-width and path-width of the initial graphs or argue why such bounds are impossible to specify. Among the studied, unary transformations are vertex addition, vertex deletion, edge addition, edge deletion, subgraphs, vertex identification, edge contraction, edge subdivision, minors, powers of graphs, line graphs, edge complements, local complements, Seidel switching, and Seidel complementation. Among the studied, binary transformations we consider the disjoint union, join, union, substitution, graph product, 1-sum, and corona of two graphs.
... Treedepth is closely related to other sparsity metrics like treewidth [9]. In particular, graphs with bounded treewidth have at most logarithmic treedepth [2,9]. ...
Sometimes local search algorithms cannot efficiently find even local peaks. To understand why, I look at the structure of ascents in fitness landscapes from valued constraint satisfaction problems (VCSPs). Given a VCSP with a constraint graph of treedepth $d$, I prove that from any initial assignment there always exists an ascent of length $2^{d + 1} \cdot n$ to a local peak. This means that short ascents always exist in fitness landscapes from constraint graphs of logarithmic treedepth, and thus also for all VCSPs of bounded treewidth. But this does not mean that local search algorithms will always find and follow such short ascents in sparse VCSPs. I show that with loglog treedepth, superpolynomial ascents exist; and for polylog treedepth, there are initial assignments from which all ascents are superpolynomial. Together, these results suggest that the study of sparse VCSPs can help us better understand the barriers to efficient local search.
... The latter objective suggests to proceed according to an ordering with the smallest vertex separation number 4 . However, the vertex separation number of a graph is equal to its pathwidth, and thus it is NP-hard to determine exactly, or approximate up to a constant additive term [4]; and while pathwidth is fixed-parameter tractable [5], the corresponding algorithm is not useful in practice. Moreover, following this ordering may conflict with the desire to keep the processed part of the graph as dense (and thus hard to orient) as possible. ...
As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi proved that if certain coefficients of the graph polynomial are non-zero, then the graph is choosable, i.e., colorable from any assignment of lists of prescribed size. We show that in case all relevant coefficients are zero, then further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. We also describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges.
... All four parameters tw, pw, cw, tcw have been extensively studied both from the combinatorial and the algorithmic point of view [3][4][5]18,30,52]. For all of them, the corresponding decision problem is NP-complete [1,8,13,29,52]. Moreover, all of them enjoy nice closeness properties under known partial ordering relations on graphs: treewidth and pathwidth are minor-closed, while cutwidth and tree-cut width are immersion-closed. ...
New parameter for graph algorithms.This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of edge-treewidth. We first observe that edge-treewidth does not enjoy any closeness properties under the known partial ordering relations on graphs. We introduce a variant of the topological minor relation, namely, the weak topological minor relation and we prove that edge-treewidth is monotone with respect to weak topological minors. Based on this new relation we are able to provide universal obstructions for edge-treewidth. The proofs are based on the fact that edge-treewidth of a graph is parametrically equivalent with the maximum over the treewidth and the maximum degree of the blocks of the graph.
... For a distance-5 independent set I in a graph G, let I(G) be the 2-shallow minor of G obtained by contracting each of the balls of radius 2 that are centred at vertices in I with corresponding model (X v : v ∈ V (I(G)). Observe that if G has maximum degree ∆, then |X v | ∆ 2 + 1 for all v ∈ V (I(G)) and so pw(I(G)) (pw(G) + 1)/(∆ 2 + 1) − 1 (see Theorem 21 in [12] for an implicit proof). We use the following lemma implicitly proved by Korhonen [21]. ...
A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Similarly, we show that every graph excluding a fixed minor and with sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Finally, we present a characterisation for when a hereditary class defined by a finite set of forbidden induced subgraphs has bounded pathwidth.
... Bauer et al. [BCRW16] showed that in Customizable Contraction Hierarchies, the maximum search space size corresponds to the height of the associated elimination tree. The minimum elimination tree height of a graph G equals the treedepth td of G, and moreover it holds that b − 1 td [BGHK95]. This means that the balanced separator number b yields a lower bound on the maximum search space size S max . ...
... The latter objective suggests to proceed according to an ordering with the smallest vertex separation number 4 . However, the vertex separation number of a graph is equal to its pathwidth, and thus it is NP-hard to determine exactly, or approximate up to a constant additive term [4]; and while pathwidth is fixed-parameter tractable [5], the corresponding algorithm is not useful in practice. Moreover, following this ordering may conflict with the desire to keep the processed part of the graph as dense (and thus hard to orient) as possible. ...
As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi gave a way to prove that a graph is choosable (colorable from any lists of prescribed size). We describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges. We also show that in case that Alon-Tarsi method fails to show that the graph is choosable, further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. The implementation can be found at https://gitlab.mff.cuni.cz/dvorz9am/alon-tarsi-method.
... Proposition 1 (Bodlaender et al. 1995;Courcelle and Olariu 2000;Gajarský et al. 2013;Ganian 2015;Lampis 2012) For any graph G, the following inequalities hold: ...
Densestk-Subgraph is the problem to find a vertex subset S of size k such that the number of edges in the subgraph induced by S is maximized. In this paper, we show that Densestk-Subgraph is fixed parameter tractable when parameterized by neighborhood diversity, block deletion number, distance-hereditary deletion number, and cograph deletion number, respectively. Furthermore, we give a 2-approximation 2tc(G)/2nO(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{{{\texttt{tc}}(G)}/2}n^{O(1)}$$\end{document}-time algorithm where tc(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\texttt{tc}}(G)}$$\end{document} is the twin cover number of an input graph G.
... Next, we define the tree decomposition and treewidth (see [BGHK95,Dav06,LMS13,DLY21]), Definition 1.3. A tree decomposition is a mapping of graphs into trees. ...
Computing John Ellipsoid is a fundamental problem in machine learning and convex optimization, where the goal is to compute the ellipsoid with maximal volume that lies in a given convex centrally symmetric polytope defined by a matrix $A \in \mathbb{R}^{n \times d}$. In this work, we show two faster algorithms for approximating the John Ellipsoid. $\bullet$ For sparse matrix $A$, we can achieve nearly input sparsity time $\mathrm{nnz}(A) + d^{\omega}$, where $\omega$ is exponent of matrix multiplication. Currently, $\omega \approx 2.373$. $\bullet$ For the matrix $A$ which has small treewidth $\tau$, we can achieve $n \tau^2$ time. Therefore, we significantly improves the state-of-the-art results on approximating the John Ellipsoid for centrally symmetric polytope [Cohen, Cousins, Lee, and Yang COLT 2019] which takes $nd^2$ time.
... There is a rich history of approximation algorithms for treewidth. In terms of polynomial time approximation algorithms, the best known approximation algorithm [FHL08] by Feige, Hajiaghayi and Lee has approximation factor O( √ log k), improving upon a O(log n)-approximation algorithm [BGHK95] and a O(log k)-approximation algorithm [Ami01]. On the other hand, Wu, Austrin, Pitassi and Liu [WAPL14] showed that assuming the Small Set Expansion Conjecture (and P = NP), there is no constant factor approximation algorithm for treewidth. ...
We give an algorithm that takes as input an $n$-vertex graph $G$ and an integer $k$, runs in time $2^{O(k^2)} n^{O(1)}$, and outputs a tree decomposition of $G$ of width at most $k$, if such a decomposition exists. This resolves the long-standing open problem of whether there is a $2^{o(k^3)} n^{O(1)}$ time algorithm for treewidth. In particular, our algorithm is the first improvement on the dependency on $k$ in algorithms for treewidth since the $2^{O(k^3)} n^{O(1)}$ time algorithm given by Bodlaender and Kloks [ICALP 1991] and Lagergren and Arnborg [ICALP 1991]. We also give an algorithm that given an $n$-vertex graph $G$, an integer $k$, and a rational $\varepsilon \in (0,1)$, in time $k^{O(k/\varepsilon)} n^{O(1)}$ either outputs a tree decomposition of $G$ of width at most $(1+\varepsilon)k$ or determines that the treewidth of $G$ is larger than $k$. Prior to our work, no approximation algorithms for treewidth with approximation ratio less than $2$, other than the exact algorithms, were known. Both of our algorithms work in polynomial space.
... Proof. This basically follows from [BGHK95]. We present the construction here for completeness. ...
Semidefinite programming is a fundamental tool in optimization and theoretical computer science. It has been extensively used as a black-box for solving many problems, such as embedding, complexity, learning, and discrepancy. One natural setting of semidefinite programming is the small treewidth setting. The best previous SDP solver under small treewidth setting is due to Zhang-Lavaei '18, which takes $n^{1.5} \tau^{6.5}$ time. In this work, we show how to solve a semidefinite programming with $n \times n$ variables, $m$ constraints and $\tau$ treewidth in $n \tau^{2\omega+0.5}$ time, where $\omega < 2.373$ denotes the exponent of matrix multiplication. We give the first SDP solver that runs in time in linear in number of variables under this setting. In addition, we improve the running time that solves a linear programming with tau treewidth from $n \tau^2$ (Dong-Lee-Ye '21) to $n \tau^{(\omega+1)/2}$.
... inequality [2,11] (1.1) tw(G) ≤ td(G) ≤ (1 + log 2 n) tw(G). ...
... STTs and, more generally, search trees on graphs arise in several different contexts and have been studied under different names: tubings [CD06], vertex rankings [DKKM94, BDJ + 98, ES14], ordered colorings [KMS95], elimination trees [Liu90,PSL90,AH94,BGHK95]. STTs have been crucial in many algorithmic applications, e.g., in pattern matching and counting [Fer13, KPR + 14, GHLW15], cache-oblivious data structures [BFCK06,FV16], tree clustering [FJ83], geometric visibility [GHL + 87], planar point location [GT98], distance oracles [CGMW21]. ...
Search trees on trees (STTs) generalize the fundamental binary search tree (BST) data structure: in STTs the underlying search space is an arbitrary tree, whereas in BSTs it is a path. An optimal BST of size $n$ can be computed for a given distribution of queries in $O(n^2)$ time [Knuth 1971] and centroid BSTs provide a nearly-optimal alternative, computable in $O(n)$ time [Mehlhorn 1977]. By contrast, optimal STTs are not known to be computable in polynomial time, and the fastest constant-approximation algorithm runs in $O(n^3)$ time [Berendsohn, Kozma 2022]. Centroid trees can be defined for STTs analogously to BSTs, and they have been used in a wide range of algorithmic applications. In the unweighted case (i.e., for a uniform distribution of queries), a centroid tree can be computed in $O(n)$ time [Brodal et al. 2001; Della Giustina et al. 2019]. These algorithms, however, do not readily extend to the weighted case. Moreover, no approximation guarantees were previously known for centroid trees in either the unweighted or weighted cases. In this paper we revisit centroid trees in a general, weighted setting, and we settle both the algorithmic complexity of constructing them, and the quality of their approximation. For constructing a weighted centroid tree, we give an output-sensitive $O(n\log h)\subseteq O(n\log n)$ time algorithm, where $h$ is the height of the resulting centroid tree. If the weights are of polynomial complexity, the running time is $O(n\log\log n)$. We show these bounds to be optimal, in a general decision tree model of computation. For approximation, we prove that the cost of a centroid tree is at most twice the optimum, and this guarantee is best possible, both in the weighted and unweighted cases. We also give tight, fine-grained bounds on the approximation-ratio for bounded-degree trees and on the approximation-ratio of more general $\alpha$-centroid trees.
... Proposition 3 (Bodlaender et al. [5]). It holds tw(G) ≤ pw(G) ≤ td(G) − 1 for any graph G. ...
Deciding feasibility of large systems of linear equations and inequalities is one of the most fundamental algorithmic tasks. However, due to data inaccuracies or modeling errors, in practical applications one often faces linear systems that are infeasible. Extensive theoretical and practical methods have been proposed for post-infeasibility analysis of linear systems. This generally amounts to detecting a feasibility blocker of small size $k$, which is a set of equations and inequalities whose removal or perturbation from the large system of size $m$ yields a feasible system. This motivates a parameterized approach towards post-infeasibility analysis, where we aim to find a feasibility blocker of size at most $k$ in fixed-parameter time $f(k) \cdot m^{O(1)}$. We establish parameterized intractability ($W[1]$- and $NP$-hardness) results already in very restricted settings for different choices of the parameters maximum size of a deletion set, number of positive/negative right-hand sides, treewidth, pathwidth and treedepth. Additionally, we rule out a polynomial compression for MinFB parameterized by the size of a deletion set and the number of negative right-hand sides. Furthermore, we develop fixed-parameter algorithms parameterized by various combinations of these parameters when every row of the system corresponds to a difference constraint. Our algorithms capture the case of Directed Feedback Arc Set, a fundamental parameterized problem whose fixed-parameter tractability was shown by Chen et al. (STOC 2008).
... We write tw(G), pw(G) and td(G) for the treewidth, path-width and tree-depth of a graph G, respectively; see [47] for more information, in particular on tree-depth. It is known that tw(G) pw(G) td(G) fr(G) ds c (G)(fixed c) vc(G) |V (G)|, where the second relationship is proven in [8] and the others follow immediately from their definitions (see also Section 2). It is readily seen that tw(G) fv(G) ds 2 (G) and that fv(G) is incomparable with the parameters pw(G), td(G), fr(G) and ds c (G) for every fixed c ≥ 3 (consider e.g. a tree of large path-width and the disjoint union of many triangles). ...
... We write tw(G), pw(G) and td(G) for the treewidth, path-width and tree-depth of a graph G, respectively; see [59] for more information, in particular on tree-depth. It is known that tw(G) pw(G) td(G) fr(G) ds c (G)(fixed c) vc(G) |V (G)|, where the second relationship is proven in [8] and the others follow immediately from their definitions (see also Section 2.2). It is readily seen that tw(G) fv(G) ds 2 (G) and that fv(G) is incomparable with the parameters pw(G), td(G), fr(G) and ds c (G) for every fixed c ≥ 3 (consider e.g. a tree of arbitrarily large path-width and the disjoint union of arbitrarily many triangles). ...
A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.
... [NdM12, § 6] for a comprehensive treatment). In other contexts, search trees on graphs have been studied as tubings [CD06], vertex rankings [DKKM94, BDJ + 98, ES14], ordered colorings [KMS95], or elimination trees [Liu90,PSL90,AH94,BGHK95] with applications in matrix factorization, see e.g. [DER17,§ 12]. ...
... All four parameters tw, pw, cw, tcw have been extensively studied both from the combinatorial and the algorithmic point of view [3][4][5]15,27,45]. For all of them, the corresponding decision problem is NP-complete [1,8,10,26,45]. Moreover, all of them enjoy nice closeness properties under known partial ordering relations on graphs: treewidth and pathwidth are minor-closed, while cutwidth and tree-cut are immersion-closed. ...
We introduce the graph theoretical parameter of edge treewidth. This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of edge-treewidth. We first observe that edge-treewidth does not enjoy any closeness properties under the known partial ordering relations on graphs. We introduce a variant of the topological minor relation, namely, the weak topological minor relation and we prove that edge-treewidth is closed under weak topological minors. Based on this new relation we are able to provide universal obstructions for edge-treewidth. The proofs are based on the fact that edge-treewidth of a graph is parametetrically equivalent with the maximum over the treewidth and the maximum degree of the blocks of the graph. We also prove that deciding whether the edge-treewidth of a graph is at most k is an NP-complete problem.
... The strong product G 1 ⊠G 2 consists of edges of the form (a, v)(b, u) where either ab ∈ E(G 1 ) and v = u or uv ∈ E(G 2 ), or a = b and uv ∈ E(G 2 ). We frequently make use of the well-known fact that tw(G ⊠ K n ) (tw(G) + 1)n − 1 for every graph G and integer n 1 (see [9] for an implicit proof). For a graph product • ∈ { , •, ⊠}, graph class G and graph H, let G •H denote the class of graphs isomorphic to graphs in {G • H : G ∈ G}. ...
The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work is that many beyond planar graphs can be described as a shallow minor of the strong product of a planar graph with a small complete graph. In particular, we show that power of planar graphs, $k$-planar, $(k,p)$-cluster planar, $k$-semi-fan-planar graphs and $k$-fan-bundle planar graphs can be described in this manner. Using a combination of old and new results, we deduce that these classes have bounded queue-number, bounded nonrepetitive chromatic number, polynomial $p$-centred chromatic numbers, linear strong colouring numbers, and cubic weak colouring numbers. In addition, we show that $k$-gap planar graphs have super-linear local treewidth and, as a consequence, cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path.
... Theorem A.2. [Theorem 2 of [Gru12] and Theorem 12 of [BGHK95]]s(G) − 1 ≤ tw(G). ...
Consider property testing on bounded degree graphs and let $\varepsilon>0$ denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on $\varepsilon$. Recent advances in testing minor-freeness have proven that all additive and monotone properties of planar graphs can be tested in $poly(\varepsilon^{-1})$ queries. Some properties falling outside this class, such as Hamiltonicity, also have a similar complexity for planar graphs. Motivated by these results, we ask: can all properties of planar graphs can be tested in $poly(\varepsilon^{-1})$ queries? Is there a uniform query complexity upper bound for all planar properties, and what is the "hardest" such property to test? We discover a surprisingly clean and optimal answer. Any property of bounded degree planar graphs can be tested in $\exp(O(\varepsilon^{-2}))$ queries. Moreover, there is a matching lower bound, up to constant factors in the exponent. The natural property of testing isomorphism to a fixed graph needs $\exp(\Omega(\varepsilon^{-2}))$ queries, thereby showing that (up to polynomial dependencies) isomorphism to an explicit fixed graph is the hardest property of planar graphs. The upper bound is a straightforward adapation of the Newman-Sohler analysis that tracks dependencies on $\varepsilon$ carefully. The main technical contribution is the lower bound construction, which is achieved by a special family of planar graphs that are all mutually far from each other. We can also apply our techniques to get analogous results for bounded treewidth graphs. We prove that all properties of bounded treewidth graphs can be tested in $\exp(O(\varepsilon^{-1}\log \varepsilon^{-1}))$ queries. Moreover, testing isomorphism to a fixed forest requires $\exp(\Omega(\varepsilon^{-1}))$ queries.
In a previous work, the authors showed that the maximum number of infinitely long synchronous directed walks that never meet is equal to the dimension of the no‐meet matroid, namely the largest order of a collection of vertex‐disjoint cycles. Given , we want to compute the meeting time of walks: the first time step such that, given any set of walks, at least two of them must meet no later than . We precisely prove that the meeting time is at most , where is the number of vertices. A connection is established with a cops and robber game on directed graphs with helicopter cops and an invisible slow robber. The meeting time of walks equals the capture time in this game, when at most capture attempts are allowed. While this capture time can be computed in polynomial time, we show that it is NP‐hard to compute the minimum number of cops needed to catch the robber. More insights are also given on the number and its relation to pathwidth and other graph parameters. Finally we analyze these game measures on digraph tensor products.
The investigation of width parameters in both graph theory and algebraic contexts has attracted considerable attention. Among these parameters, tree-cut-decomposition has emerged as a crucial metric. The "Edge-Tangle" concept is deeply tied to the width parameter known as "tree-cut width" in graphs. In this paper, we establish a new definition called Edge-Ultrafilters on graphs and demonstrate their equivalence to Edge-Tangles.
Dujmovć, Joret, Micek, Morin, Ueckerdt, and Wood established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this result, this paper systematically studies various structural properties of cartesian, direct and strong products. In particular, we characterise when these graph products contain a given complete multipartite subgraph, determine tight bounds for their degeneracy, establish new lower bounds for the treewidth of cartesian and strong products, and characterise when they have bounded treewidth and when they have bounded pathwidth.
For rearranging objects on tabletops with overhand grasps, temporarily relocating objects to some buffer space may be necessary. This raises the natural question of how many simultaneous storage spaces, or “running buffers,” are required so that certain classes of tabletop rearrangement problems are feasible. In this work, we examine the problem for both labeled and unlabeled settings. On the structural side, we observe that finding the minimum number of running buffers (MRB) can be carried out on a dependency graph abstracted from a problem instance and show that computing MRB is NP-hard. We then prove that under both labeled and unlabeled settings, even for uniform cylindrical objects, the number of required running buffers may grow unbounded as the number of objects to be rearranged increases. We further show that the bound for the unlabeled case is tight. On the algorithmic side, we develop effective exact algorithms for finding MRB for both labeled and unlabeled tabletop rearrangement problems, scalable to over a hundred objects under very high object density. More importantly, our algorithms also compute a sequence witnessing the computed MRB that can be used for solving object rearrangement tasks. Employing these algorithms, empirical evaluations reveal that random labeled and unlabeled instances, which more closely mimic real-world setups generally have fairly small MRBs. Using real robot experiments, we demonstrate that the running buffer abstraction leads to state-of-the-art solutions for the in-place rearrangement of many objects in a tight, bounded workspace.
The circumference of a graph G is the length of a longest cycle in G, or \(+\infty \) if G has no cycle. Birmelé (J Graph Theory 43(1):24–25, 2003) showed that the treewidth of a graph G is at most its circumference minus 1. We strengthen this result for 2-connected graphs as follows: If G is 2-connected, then its treedepth is at most its circumference. The bound is best possible and improves on an earlier quadratic upper bound due to Marshall and Wood (J Graph Theory 79(3):222–232, 2015).
We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online O (log log n )-competitive search tree data structure in this model, where n is the number of vertices. This matches the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.
Computing optimal transport (OT) distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. In this paper, we study the problem of approximating the general OT distance between two discrete distributions of size $n$. Given the cost matrix $C=AA^\top$ where $A \in \mathbb{R}^{n \times d}$, we proposed a faster Sinkhorn's Algorithm to approximate the OT distance when matrix $A$ has treewidth $\tau$. To approximate the OT distance, our algorithm improves the state-of-the-art results [Dvurechensky, Gasnikov, and Kroshnin ICML 2018] from $\widetilde{O}(\epsilon^{-2} n^2)$ time to $\widetilde{O}(\epsilon^{-2} n \tau)$ time.
A homomorphism \(\phi \) from a guest graph G to a host graph H is locally bijective, injective or surjective if for every \(u\in V(G)\), the restriction of \(\phi \) to the neighbourhood of u is bijective, injective or surjective, respectively. The corresponding decision problems, LBHom, LIHom and LSHom, are well studied both on general graphs and on special graph classes. We prove a number of new \(\textsf{FPT}\), \(\textsf{W}\)[1]-hard and para-\(\textsf{NP}\)-complete results by considering a hierarchy of parameters of the guest graph G. For our \(\textsf{FPT}\) results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove \(\textsf{FPT}\) results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.Keywords(locally constrained) graph homomorphismparameterized complexityfracture number
\textsc{Densest $k$-Subgraph} is the problem to find a vertex subset $S$ of size $k$ such that the number of edges in the subgraph induced by $S$ is maximized. In this paper, we show that \textsc{Densest $k$-Subgraph} is fixed parameter tractable when parameterized by neighborhood diversity, block deletion number, distance-hereditary deletion number, and cograph deletion number, respectively. Furthermore, we give a $2$-approximation $2^{\tc(G)/2}n^{O(1)}$-time algorithm where $\tc(G)$ is the twin cover number of an input graph $G$.
Given an undirected graph, a balanced node separator is a set of nodes whose removal splits the graph into connected components of limited size. Balanced node separators are used for graph partitioning, for the construction of graph data structures, and for measuring network reliability. It is NP-hard to decide whether a graph has a balanced node separator of size at most [Formula: see text]. Therefore, practical algorithms typically try to find small separators in a heuristic fashion. In this paper, we present a branching algorithm that for a given value [Formula: see text] either outputs a balanced node separator of size at most [Formula: see text] or certifies that [Formula: see text] is a valid lower bound. Using this algorithm iteratively for growing values of [Formula: see text] allows us to find a minimum balanced node separator. To make this algorithm scalable to real-world (road) networks of considerable size, we first describe pruning rules to reduce the graph size without affecting the minimum balanced separator size. In addition, we prove several structural properties of minimum balanced node separators which are then used to reduce the branching factor and search depth of our algorithm. We experimentally demonstrate the applicability of our algorithm to graphs with thousands of nodes and edges. We showcase the usefulness of having minimum balanced separators for judging the quality of existing heuristics, for improving preprocessing-based route planning techniques on road networks, and for lower bounding important graph parameters. Finally, we discuss how our ideas and methods can also be leveraged to accelerate the heuristic computation of balanced node separators.
We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as $P^{NP}$ and $P^{QMA}$, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes $P^{NP[\log]}$ and $P^{QMA[\log]}$, defined identically to $P^{NP}$ and $P^{QMA}$, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a $P^{NP}$ machine have a "query graph" which is a tree, then this computation can be simulated in $P^{NP[\log]}$. In this work, we first show that for any verification class $C\in\{NP,MA,QCMA,QMA,QMA(2),NEXP,QMA_{\exp}\}$, any $P^C$ machine with a query graph of "separator number" $s$ can be simulated using deterministic time $\exp(s\log n)$ and $s\log n$ queries to a $C$-oracle. When $s\in O(1)$ (which includes the case of $O(1)$-treewidth, and thus also of trees), this gives an upper bound of $P^{C[\log]}$, and when $s\in O(\log^k(n))$, this yields bound $QP^{C[\log^{k+1}]}$ (QP meaning quasi-polynomial time). We next show how to combine Gottlob's "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding $P^C$ computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial $p$ specified via an arithmetic circuit, if one can "weakly compress" $p$ so that its optimal value requires $m$ bits to represent, then $P^{NP}$ can be decided with only $m$ queries to an NP-oracle.
We show that the following problem is NP-complete. Given a graph, find the minimum number of edges (fill-in) whose addition makes the graph chordal. This problem arises in the solution of sparse symmetric positive definite systems of linear equations by Gaussian elimination.
Graph Problems Related to Gate Matrix Layout and PLA Folding. This paper gives a survey on graph problems occuring in linear VLSI layout architectures such as gate matrix layout, folding of programmable logic arrays, and Weinberger arrays. These include a variety of mostly independently investigated graph problems such as augmentation of a given graph to an interval graph with small clique size, node search of graphs, matching problems with side constraints, and other. We discuss implications of graph theoretic results for the VLSI layout problems and survey new research directions. New results presented include NP-hardness of gate matrix layout on chordal graphs, efficient algorithms for trees, cographs, and certain chordal graphs, Lagrangean relaxation and approximation algorithms based on on-line interval graph augmentation.
It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the pathwidth of a cograph and build a corresponding path-decomposition is given.
We describe an algorithm, which for fixed k ≥ 0 has running time O(|V(G)|3), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.
This paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems. The method is formulated in terms of frontal matrices, update matrices, and an assembly tree. Formal definitions of these notions are given based on the sparse matrix structure. Various advances to the basic method are surveyed. They include the role of matrix reorderings, the use of supernodes, and other implementation techniques. The use of the method in different computational environments is also described.
Let M be a mesh consisting of $n^2 $ squares called elements, formed by subdividing the unit square $(0,1) \times (0,1)$ into $n^2 $ small squares of side ${1 / h}$, and having a node at each of the $(n + 1)^2 $ grid points. With M we associate the $N \times N$ symmetric positive definite system $Ax = b$, where $N = (n + 1)^2 $, each $x_i $ is associated with a node of M, and $A_{ij} \ne 0$ if and only if $x_i $ and $x_j $ are associated with nodes of the same element. If we solve the equations via the standard symmetric factorization of A, then $O(n^4 )$ arithmetic operations are required if the usual row by row (banded) numbering scheme is used, and the storage required is $O(n^3 )$. In this paper we present an unusual numbering of the mesh (unknowns) and show that if we avoid operating on zeros, the $LDL^T $ factorization of A can be computed using the same standard algorithm in $O(n^3 )$ arithmetic operations. Furthermore, the storage required is only $O(n^2 \log _2 n)$. Finally, we prove that all ord...
J. A. George has discovered a method, called nested dissection, for solving a system of linear equations defined on an n = k $\times$ k square grid in O(n log n) space and O($n{3/2}$) time. We generalize this method without degrading the time and space bounds so that it applies to any system of equations defined on a planar or almost-planar graph. Such systems arise in the solution of two-dimensional finite element problems. Our method uses the fact that planar graphs have good separators. More generally, we show that sparse Gaussian elimination is efficient for any class of graphs which have good separators, and conversely that graphs without good separators (including almost all sparse graphs) are not amenable to sparse Gaussian elimination.
An efficient parallel algorithm for the tree-decomposition problem for fixed width w is presented. The algorithm runs in time O(log/sup 3/ n) and uses O(n) processors on a concurrent-read, concurrent-write parallel random access machine (CRCW PRAM). This result can be used to construct efficient parallel algorithms for three important classes of problems: MS (monadic second-order) properties, linear EMS (extended monadic second-order) extremum problems, and enumeration problems for MS properties, for graphs of tree width at most w. The sequential time complexity of the tree-composition problem for fixed w is improved, and some implications for this improvement are stated.
The frontal method for solving linear systems of equations is solved by permitting more than one front to occur at the same time. This enables a code to be developed for general symmetric systems. The organization and implemetation of a multifrontal code which uses the minimum-degree ordering is discussed and it is indicated how one can solve indefinite systems in a stable manner. The performance of the code is illustrated both on the IBM 3033 and on the CRAY-1.
In this paper, the role of elimination trees in the direct solution of large sparse linear systems is examined. The notion of elimination trees is described and its relation to sparse Cholesky factorization is discussed. The use of elimination trees in the various phases of direct factorization are surveyed: in reordering, sparse storage schemes, symbolic factorization, numeric factorization, and different computing environments.
We address the problem of determining whether a graph is a partial k- tree. We determine the complexity status of two problems related to finding the smallest number k such that a given graph is a partial k- tree. First, finding the minimum such value in NP-hard (the corresponding decision problem is NP-complete). Second, for a fixed (predetermined) value of k, a dynamic programming approach gives algorithms with polynomially bounded (but exponential in k) worst case time complexity. Previously, this problem had only been solved for k=1,2,3.
In this survey I will give an overview of recent developments in methods for solving combinatorially difficult, i.e., NP-hard, problems defined on simple, labelled or unlabelled, graphs or hypergraphs. Special instances of these methods are repeatedly being invented in applications. They can be characterized as table-based reduction methods, because they work by successively eliminating the vertices of the problem graph, while building tables with the information about the eliminated part of the graph required to solve the problem at hand. Bertele and Brioschi [9] give a fUll account of the state-of-the-art in 1972 of these methods applied to non-serial optimization problems. A preliminary taste of the family of algorithms I consider is given by the most well-known of all graph algorithms: the series-parallel reduction method for computing the resistance between two terminals of a network of resistors. It was invented by Ohm (1787-1854) and is contained in most high-school physics curricula. However, since resistance computation is not combinatorially difficult (the resistance is easily obtained, via Kirchhol~s laws, from the solution tO a linear system of equations), I will illustrate the method on a different problem, that of computing the probability that a connection exists between two terminals of a series-parallel network of communication links that are unreliable and fail independently of each other, with given probabilities. This problem is NP-hard for arbitrary graphs, see Garey and Johnson [17, problem ND20], basically because all states of the set of communication links must be considered. On a series-parallel network, however, a simple reduction method will work. The method successively eliminates non-terminal nodes which are adjacent to at most two links, by a local transformation of the network. The elimination operation can be defined as in Figure 1.1., where the quantities Pi labelling edges are the probabilities that the corresponding links work correctly. If a link of the
This paper investigates the computational complexity of binary sequences as measured by the rapidity of their generation by multitape Turing machines. A "translational" method which escapes some of the limitations of earlier approaches leads to a refinement of the established hierarchy. The previous complexity classes are shown to possess certain translational properties. An related hierarchy of complexity classes of monotonic functions is examined
A triangulated graph is a graph in which for every cycle of length ℓ > 3, there is an edge joining two nonconsecutive vertices. In this paper we study triangulated graphs and show that they play an important role in the elimination process. The results have application in the study of the numerical solution of sparse positive definite systems of linear equations by Gaussian elimination.
Using a variation of the interpretability concept we show that all graph properties definable in monadic second-order logic (MS properties) with quantification over vertex and edge sets can be decided in linear time for classes of graphs of fixed bounded treewidth given a tree-decomposition. This gives an alternative proof of a recent result by Courcelle. We allow graphs with directed and/or undirected edges, labeled on edges and/or vertices with labels taken from a finite set. We extend MS properties to extended monadic second-order (EMS) problems involving counting or summing evaluations over sets definable in monadic second-order logic. Our technique allows us also to solve some EMS problems in linear time or in polynomial or pseudopolynomial time for classes of graphs of fixed bounded treewidth. Moreover, it is shown that each EMS problem is in NC for graphs of bounded treewidth. Most problems for which linear time algorithms for graphs of bounded treewidth were previously known to exist, and many others, are EMS problems.
The minimum fill problem is the problem to re-order the rows and columns of a given sparse symmetric matrix so that its triangular factor is as sparse as possible. Equivalently, the problem is to find the smallest set of edges whose addition makes a given undirected graph chordal. The problem is known to be NP-complete and no polynomial-time approximation algorithms are known that provide any nontrivial guarantee for arbitrary graphs (matrices), although some heuristics well in practice.Nested dissection is one such heuristic. In this article we prove that every graph with a fixed bound on the vertex degree has a nested dissection order that achieves fill within a factor of O(log n) ofminimum. This does not lead to a polynomial-time approximation algorithm, however, because the proof does not give an efficient method for finding the separators required by nested dissection.
In this paper we show that graph isomorphism and chromatic index are solvable in polynomial time when restricted to the class of graphs with treewidth ≤ k (k a constant) (or equivalently, the class of partial k-trees).
We introduce an invariant of graphs called the tree-width, and use it to obtain a polynomially bounded algorithm to test if a graph has a subgraph contractible to H, where H is any fixed planar graph. We also nonconstructively prove the existence of a polynomial algorithm to test if a graph has tree-width ≤ w, for fixed w. Neither of these is a practical algorithm, as the exponents of the polynomials are large. Both algorithms are derived from a polynomial algorithm for the DISJOINT CONNECTING PATHS problem (with the number of paths fixed), for graphs of bounded tree-width.
We show that for any fixed k, there is a linear-time algorithm which given a graph G either: (i) finds a cutset X of G with |X| ≤ k such that no component of G–X contains more than 3/4|G–X| vertices, or (ii) determines that for any set X of vertices of G with |X| ≤ k, there is a component of G–X which contains more than 2/3|G–X| vertices.This approximate separator algorithm can be used to develop and O(n log n algorithm for determining if G has a tree decomposition of width at most k (for fixed k) and finding such a tree decomposition if it exists.
We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations. We give a new linear-time algorithm to calculate the fill-in produced by any elimination ordering, and we give two new related algorithms for finding orderings with special properties. One algorithm, based on breadth-first search, finds a perfect elimination ordering, if any exists, in $O(n + e)$ time, if the problem graph has n vertices and e edges. An extension of this algorithm finds a minimal (but not necessarily minimum) ordering in $O(ne)$ time. We conjecture that the problem of finding a minimum ordering is NP-complete
An lmplementat]on of sparse LDL T and LU factorlzatlon and back substitution, based on a new scheme for storing sparse matrmes, is presented. The new method appears to be as efficmnt m terms of work and storage as existing schemes It is more amenable to efficmnt implementation on fast plpehned scmntlfic computers
Roughly, a graph has small “tree-width” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: 1.(i) a minimax formula relating tree-width with the largest such obstructions2.(ii) an association between such obstructions and large grid minors of the graph3.(iii) a “tree-decomposition” of the graph into pieces corresponding with the obstructions.
These results will be of use in later papers.
The first approximate max-flow-min-cut theorem for general
multicommodity flow is proved. It is used to obtain approximation
algorithms for minimum deletion of clauses of a 2-CNF≡formula, via
minimization problems, and other problems. Also presented are
approximation algorithms for chordalization of a graph and for register
sufficiency that are based on undirected and directed node separators
A multicommodity flow problem is considered where for each pair of
vertices ( u , v ) it is required to send f
half-units of commodity ( u , v ) from u to
v and f half-units of commodity ( v ,
u ) from v to u without violating capacity
constraints. The main result is an algorithm for performing the task
provided that the capacity of each cut exceeds the demand across the cut
by a Θ(log n ) factor. The condition on cuts is required
in the worst case, and is trivially within a Θ(log n )
factor of optimal for any flow problem. The result can be used to
construct the first polylog-times optimal approximation algorithms for a
wide variety of problems, including minimum quotient separators, 1/3-2/3
separators, bifurcators, crossing number, and VLSI layout area. It can
also be used to route packets efficiently in arbitrary distributed
networks
We consider the gate matrix layout problem for VLSI circuits, which is known to be NP-complete. We present an efficient algorithm for determining whether two tracks suffice. For the general problem of minimizing the number of tracks (and, hence, the area) needed, we design an attractive dynamic programming formulation to guarantee optimality. We also investigate the performance of fast heuristic algorithms published in the literature and demonstrate that there exist families of problem instances for which the ratio of the number of tracks required by these heuristics to the optimal value is unbounded. Moreover, we show that this result holds for any on-line layout algorithm. We additionally prove that, unless P = NP, no polynomial-time layout algorithm can ensure that the number of tracks it requires never exceeds k plus the optimum, for any constant k.
Call routing, rat catching, and planar branch width
- P D Seymour
- R Thomas
P. D. Seymour and R. Thomas. Call routing, rat catching, and planar
branch width. In DIMACS Workshop{ Planar Graphs: Structures and
Algorithms. Center for Discrete Mathematics & Theorical Computer
Science, Nov 1991.