Fixed-Parameter Tractability of Treewidth and Pathwidth

Bodlaender, Hans L.

doi:10.1007/978-3-642-30891-8_12

Cited by 16 publications

(11 citation statements)

References 141 publications

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“…Besides this, many efficient algorithms exist to approximate the treewidth of a graph to some constant factor. A detailed survey of these results is available elsewhere [ 7 , 8 ]. Open-source packages such as HTD can compute treewidth for graphs with thousands of nodes [ 1 ].…”

Section: Preliminariesmentioning

confidence: 99%

Reachability Analysis Using Message Passing over Tree Decompositions

Sankaranarayanan

2020

Computer Aided Verification

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In this paper, we study efficient approaches to reachability analysis for discrete-time nonlinear dynamical systems when the dependencies among the variables of the system have low treewidth. Reachability analysis over nonlinear dynamical systems asks if a given set of target states can be reached, starting from an initial set of states. This is solved by computing conservative over approximations of the reachable set using abstract domains to represent these approximations. However, most approaches must tradeoff the level of conservatism against the cost of performing analysis, especially when the number of system variables increases. This makes reachability analysis challenging for nonlinear systems with a large number of state variables. Our approach works by constructing a dependency graph among the variables of the system. The tree decomposition of this graph builds a tree wherein each node of the tree is labeled with subsets of the state variables of the system. Furthermore, the tree decomposition satisfies important structural properties. Using the tree decomposition, our approach abstracts a set of states of the high dimensional system into a tree of sets of lower dimensional projections of this state. We derive various properties of this abstract domain, including conditions under which the original high dimensional set can be fully recovered from its low dimensional projections. Next, we use ideas from message passing developed originally for belief propagation over Bayesian networks to perform reachability analysis over the full state space in an efficient manner. We illustrate our approach on some interesting nonlinear systems with low treewidth to demonstrate the advantages of our approach.

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Section: Preliminariesmentioning

confidence: 99%

Reachability Analysis Using Message Passing over Tree Decompositions

Sankaranarayanan

2020

Computer Aided Verification

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“…We claim again that α m (P ) < L α m (P). As in Case 1, it suffices to show that (7) holds for all i = 1, . .…”

Section: Clean Path Decompositions Are Well-arrangedmentioning

confidence: 99%

“…In this section we turn the generic exponential-time algorithm from Section 3.1 into a polynomialtime algorithm that finds a minimum-length path decomposition of width at most k of G for given integer k ≤ 3 and connected graph G. Recall that it can be checked in linear-time whether pw(G) ≤ k, see [4,7]. We formulate the algorithm in this section and prove its correctness in the next section.…”

Section: An Algorithm For Connected Graphsmentioning

confidence: 99%

The complexity of minimum-length path decompositions

Dereniowski

Kubiak

Zwólš

2015

Journal of Computer and System Sciences

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We consider a bi-criteria generalization of the pathwidth problem, where, for given integers k, l and a graph G, we ask whether there exists a path decomposition P of G such that the width of P is at most k and the number of bags in P, i.e., the length of P, is at most l.We provide a complete complexity classification of the problem in terms of k and l for general graphs. Contrary to the original pathwidth problem, which is fixed-parameter tractable with respect to k, we prove that the generalized problem is NP-complete for any fixed k ≥ 4, and is also NP-complete for any fixed l ≥ 2. On the other hand, we give a polynomial-time algorithm that, for any (possibly disconnected) graph G and integers k ≤ 3 and l > 0, constructs a path decomposition of width at most k and length at most l, if any exists.As a by-product, we obtain an almost complete classification of the problem in terms of k and l for connected graphs. Namely, the problem is NP-complete for any fixed k ≥ 5 and it is polynomial-time for any k ≤ 3. This leaves open the case k = 4 for connected graphs.In the optimization problem MLPD (Minimum Length Path Decomposition) the goal is to compute, for a given simple graph G and an integer k, a minimum length path decomposition P of G such that width(P) ≤ k. In the corresponding decision problem LCPD (Length-Constrained Path Decomposition), a simple graph G and integers k, l are given, and we ask whether there exists a path decomposition P of G such that width(P) ≤ k and len(P) ≤ l.Finally, in the optimization problem MLPD k the goal is to compute, for a given simple graph G, a minimum length path decomposition P of G such that width(P) ≤ k. The corresponding decision problem LCPD k the input consists of a simple graph G and an integer l and the question is whether there exists a path decomposition P of G such that width(P) ≤ k and len(P) ≤ l.Note that the difference between MLPD and MLPD k (and, similarly, LCPD and LCPD k ) is that in the former k is a part of the input while in the latter the value of k is fixed.2 MLPD k is NP-hard for k ≥ 4In this section we prove that the problem of finding a minimum length path decomposition of width k ≥ 4 is NP-hard. To that end it suffices to show that the decision problem LCPD 4 is NP-complete, since this implies that LCPD k is NP-complete for all k ≥ 4 which follows from the following lemma.Lemma 2.1. Let k ≥ 1. If LCPD k−1 is NP-complete for general graphs, then LCPD k is NPcomplete for connected graphs.

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“…This is why we feel that a restriction to a certain type of algorithm is not necessarily inferior to a complexity-based approach. Indeed, most algorithms leveraging treewidth are dynamic programming algorithms or can be equivalently expressed as such [19][20][21][22][23][24]. Even before dynamic programming on tree-decompositions became an important subject in algorithm design, similar concepts were already used implicitly [25,26].…”

Section: Introductionmentioning

confidence: 99%

“…Our tool of choice will be a family of boundaried graphs that are distinct under Myhill-Nerode equivalence. The perspective of viewing graph decompositions as an "algebraic" expression of boundaried graphs that allow such equivalences is well-established [21,23].…”

Section: Introductionmentioning

confidence: 99%

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Chen¹,

Reidl

Rossmanith

et al. 2018

Algorithms

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Treedepth is a well-established width measure which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded treeor pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show with a novel approach that the space consumption of any (single-pass) dynamic programming algorithm on treedepth decompositions of depth d cannot be bounded by (2 −) d • log O(1) n for VERTEX COVER, (3 −) d • log O(1) n for 3-COLORING and (3 −) d • log O(1) n for DOMINATING SET for any > 0. This formalizes the common intuition that dynamic programming algorithms on graph decompositions necessarily consume a lot of space and complements known results of the time-complexity of problems restricted to low-treewidth classes. We then show that treedepth lends itself to the design of branching algorithms. Specifically, we design two novel algorithms for DOMINATING SET on graphs of treedepth d: A pure branching algorithm that runs in time d O(d 2) • n and uses space O(d 3 log d + d log n) and a hybrid of branching and dynamic programming that achieves a running time of O(3 d log d • n) while using O(2 d d log d + d log n) space.

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Fixed-Parameter Tractability of Treewidth and Pathwidth

Cited by 16 publications

References 141 publications

Reachability Analysis Using Message Passing over Tree Decompositions

Reachability Analysis Using Message Passing over Tree Decompositions

The complexity of minimum-length path decompositions

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

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