A logical approach to multicut problems

Gottlob, Georg; Lee, Stephanie Tien

doi:10.1016/j.ipl.2007.03.005

Cited by 27 publications

(21 citation statements)

References 12 publications

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“…Since we are dealing with undirected graphs G = (V, E), we consider terminal pairs h ∈ H as sets h ⊆ V of cardinality two. In [13], the treewidth w * of a structure representing both the graph G and the set H of pairs of terminals was introduced as a parameter of the multicut problems. Alternatively, w * can also be defined as w * = tw (G ′ ) with G ′ = (V, E ∪ H).…”

Section: Preliminariesmentioning

confidence: 99%

“…a parameter k, if a problem is solvable in time f (k) · n O (1) , where n denotes the size of the input instance. The function f is usually exponential but only depends on k. In case of multicut problems, various such parameters have been studied like solution size [7,8], cardinality |H| plus solution size [9,10], |H| plus the treewidth of the graph G [11,12], or the treewidth of the structure representing both G and H [13]. The result in [13] was proved by showing that the solutions to any of the above multicut problems can be described by a monadic second-order (MSO) formula over the structure representing G and H. The FPT follows by an extension of Courcelle's Theorem [14] proved in [15]: optimization problems expressible by an MSO formula over the input structures are fixed-parameter tractable w.r.t.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of this result in [15] is constructive -yielding algorithms whose running time is non-elementary in terms of the number of quantifier alternations of the MSO formula. For the MSO formula in [13], we would thus end up with time and space requirements which are at least double exponential in the treewidth. The goal of our paper is to construct more efficient algorithms in case of bounded treewidth of the structure representing G and H. Our main results are as follows (Due to space constraints, we omit proofs in this paper; full proofs are given in [16]):…”

Section: Introductionmentioning

confidence: 99%

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Multicut Algorithms via Tree Decompositions

Pichler

Rümmele

Woltran

2010

Lecture Notes in Computer Science

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Abstract. Various forms of multicut problems are of great importance in the area of network design. In general, these problems are intractable. However, several parameters have been identified which lead to fixed-parameter tractability (FPT). Recently, Gottlob and Lee have proposed the treewidth of the structure representing the graph and the set of pairs of terminal vertices as one such parameter. In this work, we show how this theoretical FPT result can be turned into efficient algorithms for optimization, counting, and enumeration problems in this area.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multicut Algorithms via Tree Decompositions

Pichler

Rümmele

Woltran

2010

Lecture Notes in Computer Science

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“…Many NP-hard problems in different areas such as AI [42], Database Systems [7,81], Game theory [45,31,20], and Network Design [34], are known to be efficiently solvable when restricted to instances whose underlying structures can be modeled via acyclic graphs or acyclic hypergraphs. For such restricted classes of instances, solutions can usually be computed via dynamic programming.…”

Section: Introductionmentioning

confidence: 99%

Treewidth and Hypertree Width

Gottlob¹,

Greco²,

Scarcello³

2014

Tractability

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The chapter covers methods for identifying islands of tractability for NP-hard combinatorial problems by exploiting suitable properties of their graphical structure. Acyclic structures are considered, as well as nearly acyclic ones identified by means of so-called structural decomposition methods. In particular, the chapter focuses on the tree decomposition method, which is the most powerful decomposition method for graphs, and on the hypertree decomposition method, which is its natural counterpart for hypergraphs. These problem-decomposition methods give rise to corresponding notions of width of an instance, namely, treewidth and hypertree width. It turns out that many NP-hard problems can be solved efficiently over classes of instances of bounded treewidth or hypertree width: deciding whether a solution exists, computing a solution, and even computing an optimal solution (if some cost function over solutions is specified) are all polynomial-time tasks. Example applications include problems from artificial intelligence, databases, game theory, and combinatorial auctions.

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“…The LCA algorithm computes for each vertex i a [3]. Other papers [4] studied the vertex multicut problem, but their focus was on making a distinction between classes of problems which are solvable in polynomial time, and not on developing efficient polynomial time algorithms, like we did in this paper.…”

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confidence: 99%

Reliability analysis of tree networks applied to balanced content replication

Andreica

Țăpuș

2008

2008 IEEE International Conference on Automation, Quality and Testing, Robotics

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Abstract-The fault tolerance of the communication topology of a distributed system is a very important feature which needs to be analyzed carefully. In this paper we propose a new reliability me-tric for tree topologies, based on the unrestricted vertex multicut problem on trees, for which we present the first optimal linear time algorithm. We present evaluation results of the reliability metric on tree networks used for balanced content replication. For this problem, we also developed a new O(n1k) algorithm solving the k-equitable coloring problem on trees, based on a hierarchy of color relabeling permutations.

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A logical approach to multicut problems

Cited by 27 publications

References 12 publications

Multicut Algorithms via Tree Decompositions

Multicut Algorithms via Tree Decompositions

Treewidth and Hypertree Width

Reliability analysis of tree networks applied to balanced content replication

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