On multiway cut parameterized above lower bounds

Cygan, Marek; Pilipczuk, Marcin; Pilipczuk, Michał; Wojtaszczyk, Jakub Onufry

doi:10.1145/2462896.2462899

Cited by 94 publications

(116 citation statements)

References 20 publications

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“…A more desirable and stronger result would be a constant-factor approximation of the difference d between the optimal solution cost and a lower bound or a fixed-parameter tractability result with respect to the parameter d [6,11,17,24]. However, we show that such algorithms (presumably) do not exist.…”

Section: Parameterized Hardness and Inapproximabilitymentioning

confidence: 85%

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Bentert

Bevern

Nichterlein

et al. 2017

Algorithms for Sensor Systems

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We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless sensor communication network. Given an edge-weighted n-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. We provide an algorithm that works in polynomial time if one can find a set of obligatory edges that yield a spanning subgraph with O(log n) connected components. We also provide a linear-time algorithm that reduces any input graph that consists of a tree together with g additional edges to an equivalent graph with O(g) vertices. Based on this, we obtain a polynomial-time algorithm for g ∈ O(log n). On the negative side, we show that o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard and that there are presumably no exact algorithms running in 2 o(n) time or in f (d) · n O(1) time for any computable function f .

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Section: Parameterized Hardness and Inapproximabilitymentioning

confidence: 85%

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Bentert

Bevern

Nichterlein

et al. 2017

Algorithms for Sensor Systems

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“…Raman et al [29] gave a parameterized algorithm of running time O * (9 k ), using important separator and iterative compression technique. Based on LP-relaxation, Cygan et al [30] presented an O * (4 k ) time parameterized algorithm. By analyzing the LP value in LPrelaxation, Narayanaswamy et al [31] gave a parameterized algorithm of running time O * (2.6181 k ).…”

Section: Resultsmentioning

confidence: 99%

On the parameterized vertex cover problem for graphs with perfect matching

Wang

et al. 2014

Sci. China Inf. Sci.

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“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php fiable. Lokshtanov et al [32] (improving earlier work [42,14,41]) gave an O * (2.3146 k ) time algorithm for the problem of deciding if a 2CNF formula can be made satisfiable by the deletion of at most k clauses; they call this problem Almost 2SAT. We need a variant of the result here, where instead of deleting at most k clauses, we are allowed to delete at most k variables.…”

Section: Finding a Shadowless Solution By Reduction Tomentioning

confidence: 99%

Fixed-Parameter Tractability of Multicut Parameterized by the Size of the Cutset

Marx¹,

Razgon²

2014

SIAM J. Comput.

140

198

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Given an undirected graph G, a collection {(s 1 , t 1 ), . . . , (s k , t k )} of pairs of vertices, and an integer p, the Edge Multicut problem asks if there is a set S of at most p edges such that the removal of S disconnects every s i from the corresponding t i . Vertex Multicut is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2 O(p 3 ) · n O(1) , i.e., fixed-parameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · n O(1) exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset. Introduction. From the classical results of Ford and Fulkerson on minimum s − t cuts [20] to the more recent O(√ log n)-approximation algorithms for sparsest cut problems [44,1,18], the study of cut and separation problems has a deep and rich theory. One well-studied problem in this area is the Edge Multicut problem: given a graph G and pairs of vertices (s 1 , t 1 ), . . . , (s k , t k ), remove a minimum set of edges such that every s i is disconnected from its corresponding t i for every 1 ≤ i ≤ k. For k = 1, Edge Multicut is the classical s − t cut problem and can be solved in polynomial time. For k = 2, Edge Multicut remains polynomial-time solvable [46], but it becomes NP-hard for every fixed k ≥ 3 [15]. Edge Multicut can be approximated within a factor of O(log k) in polynomial time [22] (even in the weighted case, where the goal is to minimize the total weight of the removed edges). However, under the unique games conjecture of Khot [29], no constant factor approximation is possible [7]. One can analogously define the Vertex Multicut problem, where the task is to remove a minimum set of vertices. An easy reduction shows that the vertex version is more general than the edge version.Using brute force, one can decide in time n O(p) if a solution of size at most p exists. Our main result is a more efficient exact algorithm for small values of p (the O * notation hides factors that are polynomial in the input size).

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On multiway cut parameterized above lower bounds

Cited by 94 publications

References 20 publications

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks

On the parameterized vertex cover problem for graphs with perfect matching

Fixed-Parameter Tractability of Multicut Parameterized by the Size of the Cutset

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