Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time

Cygan, Marek; Nederlof, Jesper; Pilipczuk, Marcin; Pilipczuk, Michał; Rooij, Joham M.M. van; Wojtaszczyk, Jakub Onufry

doi:10.1109/focs.2011.23

Cited by 216 publications

(304 citation statements)

References 62 publications

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“…There is a quite extended bibliography on how to do fast dynamic programming on graphs of bounded treewidth; as a sample of this, we just mention [5,6,9,11,13,16,22,35,35,36,37,37,38,114,120,120]. t: Bounds are much better for the function t. For most natural graph parameters, it holds that t(k) = O(k) while for some of them, including tw and pw, it holds that t(k) = Θ(k).…”

Section: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 99%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique.

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Section: Theorem 5 ( [105]) There Exists a Recursive Functionmentioning

confidence: 99%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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“…For getting this approach to work for connectivity problems we require an upper bound on the rank of the matrix M defined as follows: Fix a ground set U and let p and q be partitions of U (p and q represent connectivity induced by partial solutions), define M[p, q] to be 1 if and only if the meet of p and q is the trivial partition, that is, if the union of the partial solutions induce a connected solution. Although this matrix has dimensions of order 2 θ(|U| log |U|) , we exploit a simple factorization in GF(2) of matrices with inner dimension 2 |U| using an idea of [8].…”

Section: Main Ideas Of the Approachmentioning

confidence: 99%

Deterministic Single Exponential Time Algorithms for Connectivity Problems Parameterized by Treewidth

Bodlaender

Cygan

Kratsch

et al. 2013

Automata, Languages, and Programming

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Abstract.It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2 O(tw) n O(1) time for graphs with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time it was unknown how to do this faster than tw O(tw) n O(1) until recently, when Cygan et al. (FOCS 2011) introduced the Cut&Count technique that gives randomized algorithms for a wide range of connectivity problems running in time c tw n O(1) for a small constant c.In this paper, we show that we can improve upon the Cut&Count technique in multiple ways, with two new techniques. The first technique (rank-based approach) gives deterministic algorithms with O(c tw n) running time for connectivity problems (like Hamiltonian Cycle and Stei-ner Tree) and for weighted variants of these; the second technique (determinant approach) gives deterministic algorithms running in time c tw n O(1) for counting versions, e.g., counting the number of Hamiltonian cycles for graphs of treewidth tw.The rank-based approach introduces a new technique to speed up dynamic programming algorithms which is likely to have more applications. The determinant-based approach uses the Matrix Tree Theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.

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“…The algorithm is based on the fact that planar graphs have the diameter-treewidth property [15], the fact that Steiner Tree can be solved in polynomial time on graphs of bounded treewidth [9] along with a simple preprocessing step. In this step, one contracts adjacent terminals to single vertices and removes all vertices at distance at least k + 1 from any terminal.…”

Section: Dominating Setmentioning

confidence: 99%

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

Jones

Lokshtanov

Ramanujan

et al. 2013

Lecture Notes in Computer Science

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Abstract. We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]-hard on general graphs, and hence unlikely to be fixed parameter tractable (FPT). The undirected Steiner Tree problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs. In this article we precisely chart the tractability border for Directed Steiner Tree (DST) on sparse graphs parameterized by the number of non-terminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W[2]-hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy. We further show that our algorithm achieves the best possible running time dependence on the solution size and degeneracy of the input graph, under standard complexity theoretic assumptions. Using the ideas developed for DST, we also obtain improved algorithms for Dominating Set on sparse undirected graphs. These algorithms are asymptotically optimal.

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Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time

Cited by 216 publications

References 62 publications

Graph Minors and Parameterized Algorithm Design

Graph Minors and Parameterized Algorithm Design

Deterministic Single Exponential Time Algorithms for Connectivity Problems Parameterized by Treewidth

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

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