Linear time algorithms for NP-hard problems restricted to partial k-trees

Arnborg, Stefan; Proskurowski, Andrzej

doi:10.1016/0166-218x(89)90031-0

Cited by 480 publications

(372 citation statements)

References 13 publications

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“…See [HKNO09] for a general reduction. An important fact here is that for bounded-treewidth graphs, there are linear-time algorithms for computing the exact solutions to these problems [AP89]. This result adds to a long line of research on this kind of approximation algorithm [PR07, MR09, NO08, YYI09, HKNO09, Ele10, NS11].…”

Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 94%

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

Edelman

Hassidim

Nguyen

et al. 2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constant-time algorithms. For any ε > 0, a partitioning oracle provides query access to a fixed partition of the input bounded-degree minor-free graph, in which every component has size poly(1/ε), and the number of edges removed is at most εn, where n is the number of vertices in the graph.However, the oracle of Hassidim et al. makes an exponential number of queries to the input graph to answer every query about the partition. In this paper, we construct an efficient partitioning oracle for graphs with constant treewidth. The oracle makes only O(poly(1/ε)) queries to the input graph to answer each query about the partition.Examples of bounded-treewidth graph classes include k-outerplanar graphs for fixed k, seriesparallel graphs, cactus graphs, and pseudoforests. Our oracle yields poly(1/ε)-time property testing algorithms for membership in these classes of graphs. Another application of the oracle is a poly(1/ε)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive εn in graphs with bounded treewidth. Finally, the oracle can be used to test in poly(1/ε) time whether the input bounded-treewidth graph is k-colorable or perfect.

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Section: The Partition Described By F Is a Function Of G And Random Bmentioning

confidence: 94%

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

Edelman

Hassidim

Nguyen

et al. 2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…Adapting the treewidth techniques (see Arnborg and Proskurowski [6], and Bodlaender [11]) on branch decompositions, a typical approach computing an optimal solution to a problem is as follows:…”

Section: Further Optimizationsmentioning

confidence: 99%

Subexponential Parameterized Algorithms

Fomin¹

2015

Encyclopedia of Algorithms

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We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch-(or tree-) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of sub-exponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2 O( √ k) · n O(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter.

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“…Proof. This follows from the polynomial time algorithm to compute the reliability of a graph with treewidth k for some fixed k [2].…”

Section: Propositionmentioning

confidence: 99%

Power Indices in Spanning Connectivity Games

Aziz

Lachish

Paterson

et al. 2009

Lecture Notes in Computer Science

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Abstract. The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values is #P-complete and computing ShapleyShubik indices or values is NP-hard for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. This answers (positively) an open question of whether computing Shapley-Shubik indices for a simple game represented by the set of minimal winning coalitions is NP-hard.

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Linear time algorithms for NP-hard problems restricted to partial k-trees

Cited by 480 publications

References 13 publications

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

Subexponential Parameterized Algorithms

Power Indices in Spanning Connectivity Games

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