Fixed-parameter tractability of multicut parameterized by the size of the cutset

Marx, Dániel; Razgon, Igor

doi:10.1145/1993636.1993699

Cited by 73 publications

(102 citation statements)

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“…There is a simple k-approximation in polynomial time obtained by solving each terminal pair as a separate instance of min s t cut and then taking the union of all the k cuts. Chekuri and Madan [8] and later Lee [31] showed that this is tight: assuming the Unique Games Conjecture of Khot [29], it is not possible to approximate DIRECTED MULTICUT better than factor k in polynomial time, for any fixed k. On the FPT side, Marx and Razgon [35] showed that DIRECTED MULTICUT is W[1]-hard paramterized by p. For the case of bounded k, Chitnis et al [13] showed that DIRECTED MULTICUT is FPT parameterized by p when k = 2, but Pilipczuk and Wahlstrom [38] showed that the problem remains W[1]-hard parameterized by p when k = 4. The status of DIRECTED MULTICUT parameterized by p when k = 3 is an outstanding open question.…”

Section: Previous Work and Our Resultsmentioning

confidence: 99%

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

Chitnis

Cygan

Hajiaghayi

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

138

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We introduce a new technique for designing fixed-parameter algorithms for cut problems, called randomized contractions. We apply our framework to obtain the first FPT algorithm for the UNIQUE LABEL COVER problem and new FPT algorithms with exponential speed up for the STEINER CUT and NODE MULTIWAY CUT-UNCUT problems. More precisely, we show the following:• We prove that the parameterized version of the UNIQUE LABEL COVER problem, which is the base of the UNIQUE GAMES CONJECTURE, can be solved in 2 O(k 2 log |Σ|) n 4 log n deterministic time (even in the stronger, vertex-deletion variant) where k is the number of unsatisfied edges and |Σ| is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of UNIQUE GAMES where the number of edges allowed not to be satisfied is upper bounded by O( √ log n) to optimality, which improves over the trivial O(1) upper bound.• We prove that the STEINER CUT problem can be solved in 2 O(k 2 log k) n 4 log n deterministic time andrandomized time where k is the size of the cutset. This result improves the double exponential running time of the recent work of Kawarabayashi and Thorup (FOCS'11).• We show how to combine considering 'cut' and 'uncut' constraints at the same time. More precisely, we define a robust problem NODE MULTIWAY CUT-UNCUT that can serve as an abstraction of introducing uncut constraints, and show that it admits an algorithm running in 2 O(k 2 log k) n 4 log n deterministic time where k is the size of the cutset. To the best of our knowledge, the only known way of tackling uncut constraints was via the approach of Marx, O'Sullivan and Razgon (STACS'10, ACM Trans. Alg. 2013), which yields algorithms with double exponential running time.An interesting aspect of our algorithms is that they can handle positive real weights.

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Section: Previous Work and Our Resultsmentioning

confidence: 99%

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

Chitnis

Cygan

Hajiaghayi

et al. 2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

138

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“…• Bounded fragmentation [20]: a graph G = (V , E) has bounded fragmentation if after removing any S ⊆ V of size at most s the number of components in the obtained graph is bounded by a function of s. • Multiway cut [7, 11-13, 17, 29, 33]: given a graph G, a set of k terminals and an integer s, find a set of at most s edges (or vertices) whose removal separates the terminals from each other. • Multicut [6,9,10,16,19,30]: given a graph G, a set of k terminal-pairs and an integer s, find a set of at most s edges (or vertices) whose deletion separates the vertices in each terminal-pair from each other.…”

Section: Related Problemsmentioning

confidence: 99%

Complexity and approximability of the k‐way vertex cut

2013

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In this article, we consider k -way vertex cut: the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of components. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs. Also, we show that k -way vertex cut is polynomially solvable on graphs of bounded treewidth and fixed-parameter tractable on planar graphs with the size of the separator as the parameter.

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“…With this technique, several cut related problems were shown to be FPT. Some of them, e.g., DIRECTED FEEDBACK VERTEX SET [6] and MULTICUT [19,2], had been open for quite a long time. Also related is the concept extreme set, which is very well-known in the study of cuts enumerations [20].…”

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confidence: 99%

An O *(1.84 k ) Parameterized Algorithm for the Multiterminal Cut Problem

Cao

Chen

Fan

2013

Lecture Notes in Computer Science

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We study the multiterminal cut problem, which, given an n-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in a distinct part, and the total weight of crossing edges is at most k. Our weapons shall be two classical results known for decades: maximum volume minimum (s,t)-cuts by [Ford and Fulkerson, Flows in Networks, 1962] and isolating cuts by [Dahlhaus et al., SIAM J. Comp. 23(4):864-894, 1994]. We sharpen these old weapons with the help of submodular functions, and apply them to this problem, which enable us to design a more elaborated branching scheme on deciding whether a non-terminal vertex is with a terminal or not. This bounded search tree algorithm can be shown to run in 1.84 k · n O(1) time, thereby breaking the 2 k · n O(1) barrier. As a by-product, it gives a 1.36 k · n O(1) time algorithm for 3-terminal cut. The preprocessing applied on non-terminal vertices might be of use for study of this problem from other aspects.

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Fixed-parameter tractability of multicut parameterized by the size of the cutset

Cited by 73 publications

References 54 publications

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

Complexity and approximability of the k‐way vertex cut

An O *(1.84 k ) Parameterized Algorithm for the Multiterminal Cut Problem

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