An FPT 2-Approximation for Tree-Cut Decomposition

Kim, Eun-Jung; Oum, Sang‐il; Paul, Christophe; Sau, Ignasi; Thilikos, Dimitrios M.

doi:10.1007/s00453-016-0245-5

“…Proof. The proof is closely based on the ideas in [23,Section 3]. Namely, in the case that C does not contain two edges sharing the same endpoints, the proof follows immediately from [23,Lemma 3 and 4].…”

Section: Treecut Widthmentioning

confidence: 93%

“…The proof is closely based on the ideas in [23,Section 3]. Namely, in the case that C does not contain two edges sharing the same endpoints, the proof follows immediately from [23,Lemma 3 and 4]. Moreover, if C contains two edges sharing the same endpoints, say a ∈ A and b ∈ B, it follows from [23,Lemma 3]…”

Section: Treecut Widthmentioning

confidence: 93%

See 1 more Smart Citation

SAT-Encodings for Treecut Width and Treedepth

Ganian¹,

Lodha²,

Ordyniak³

et al. 2019

2019 Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments (ALENEX)

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The decomposition of graphs is a prominent algorithmic task with numerous applications in computer science. A graph decomposition method is typically associated with a width parameter (such as treewidth) that indicates how well the given graph can be decomposed. Many hard (even #P-hard) algorithmic problems can be solved efficiently if a decomposition of small width is provided; the runtime, however, typically depends exponentially on the decomposition width. Finding an optimal decomposition is itself an NP-hard task. In this paper we propose, implement, and test the first practical decomposition algorithms for the width parameters treecut width and treedepth. These two parameters have recently gained a lot of attention in the theoretical research community as they offer the algorithmic advantage over treewidth by supporting so-called fixed-parameter algorithms for certain problems that are not fixed-parameter tractable with respect to treewidth. However, the existing research has mostly been theoretical. A main obstacle for any practical or experimental use of these two width parameters is the lack of any practical or implemented algorithm for actually computing the associated decompositions. We address this obstacle by providing the first practical decomposition algorithms.Our approach for computing treecut width and treedepth decompositions is based on efficient encodings of these decomposition methods to the propositional satisfiability problem (SAT). Once an encoding is generated, any satisfiability solver can be used to find the decomposition. This allows us to leverage the surprising power of todays state-of-the art SAT solvers. The success of SAT-based decomposition methods crucially depends on the used characterisation of the decomposition method, as not every characterisation is suitable for that task. For instance, the successful leading SAT encoding for treewidth is based on a characterisation of treewidth in terms of elimination orderings. For treecut width and treedepth, however, we propose new characterisations that are based on sequences of partitions of the vertex set, a method that was pioneered for clique-width. We implemented and systematically tested our encodings on various benchmark instances, including famous named graphs and random graphs of various density. It turned out that for the

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“…Theorem 5 (Kim et al [16]). Given a graph G on n vertices, a tree-cut decomposition of G of width at most 2tcw(G) can be computed in time 2…”

Section: Fpt Algorithm For Star Tree-cutwidthmentioning

confidence: 99%

Parameterized Complexity of the MINCCA Problem on Graphs of Bounded Decomposability

Gözüpek

¹

,

Özkan

²

,

Paul

³

et al. 2016

Graph-Theoretic Concepts in Computer Science

Self Cite

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Abstract. In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence (MinCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. [14] that the MinCCA problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for MinCCA:• the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of [14]; • it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; • it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in [14]; • it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.

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“…1, tree-cutwidth provides an intermediate measurement which allows either to push the boundary of fixed parameter tractability or strengthen W[1]-hardness result (cf. [11,16,20]). Furthermore, Fig.…”

Section: Preliminariesmentioning

confidence: 99%

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

Gözüpek

¹

,

Özkan

²

,

Paul

³

et al. 2017

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence (MinCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. [14] that the MinCCA problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for MinCCA:• the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of [14]; • it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; • it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in [14]; • it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.

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An FPT 2-Approximation for Tree-Cut Decomposition

Cited by 20 publications

References 29 publications

SAT-Encodings for Treecut Width and Treedepth

SAT-Encodings for Treecut Width and Treedepth

Parameterized Complexity of the MINCCA Problem on Graphs of Bounded Decomposability

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

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