New algorithms for the minimum coloring cut problem

Bordini, Augusto; Protti, Fábio; Silva, Thiago Gouveia da; Filho, Gilberto Farias de Sousa

doi:10.1111/itor.12494

Cited by 10 publications

(17 citation statements)

References 10 publications

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“…On the positive side, by considering all possibilities for choosing the k colors that shall be removed, COLORED (s, t)-CUT can be solved in n O(k) time. This brute-force algorithm can most likely not be improved to an FPT-algorithm, that is, to an algorithm with running time f (k) • n O (1) since the above-mentioned reduction from HITTING SET also implies that COLORED (s, t)-CUT is W [2]-hard when parameterized by k [10]. The brute-force algorithm also implies further running time bounds for COLORED (s, t)-CUT: First, the problem has an n O( ) -time algorithm, where is the maximum degree of G, since instances with ≤ k are trivial yesinstances.…”

Section: Colored (S T)-cutmentioning

confidence: 99%

Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs

et al. 2022

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In the NP-hard Colored (s,t)-Cut problem, the input is a graph G = (V,E) together with an edge-coloring ℓ : E → C, two vertices s and t, and a number k. The question is whether there is a set $S\subseteq C$ S ⊆ C of at most k colors such that deleting every edge with a color from S destroys all paths between s and t in G. We continue the study of the parameterized complexity of Colored (s,t)-Cut. First, we consider parameters related to the structure of G. For example, we study parameterization by the number ξi of edge deletions that are needed to transform G into a graph with maximum degree i. We show that Colored (s,t)-Cut is W[2]-hard when parameterized by ξ3, but fixed-parameter tractable when parameterized by ξ2. Second, we consider parameters related to the coloring ℓ. We show fixed-parameter tractability for three parameters that are potentially smaller than the total number of colors |C| and provide a linear-size problem kernel for a parameter related to the number of edges with rare edge colors.

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Section: Colored (S T)-cutmentioning

confidence: 99%

Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs

et al. 2022

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“…Silva et al [22] designed exact algorithms for Global Label Cut us-ing the branch-and-cut and branch-and-bound approaches based on integer programming formulations for the problem. Bordini et al [3] designed exact algorithms for Global Label Cut using the variable neighborhood search technique. Both of the authors [22,3] evaluated their algorithms on many concrete instances of the problem.…”

Section: More Related Workmentioning

confidence: 99%

“…Bordini et al [3] designed exact algorithms for Global Label Cut using the variable neighborhood search technique. Both of the authors [22,3] evaluated their algorithms on many concrete instances of the problem.…”

Section: More Related Workmentioning

confidence: 99%

Minimum Label s-t Cut has large integrality gaps

Zhang

Tang

2020

Information and Computation

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Given a graph G = (V, E) with a label set L = { 1 , 2 , . . . , q }, in which each edge has a label from L, a source s ∈ V , and a sink t ∈ V , the Min Label s-t Cut problem asks to pick a set L ⊆ L of labels with minimized cardinality, such that the removal of all edges with labels in L from G disconnects s and t. This problem comes from many applications in real world, for example, information security and computer networks. In this paper, we study two linear programs for Min Label s-t Cut, proving that both of them have large integrality gaps, namely, Ω(m) and Ω(m 1/3− ) for the respective linear programs, where m is the number of edges in the graph and > 0 is any arbitrarily small constant. As Min Label s-t Cut is NP-hard and the linear programming technique is a main approach to design approximation algorithms, our results give negative answer to the hope that designs better approximation algorithms for Min Label s-t Cut that purely rely on linear programming.

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“…An important progress was made by Ghaffari et al [12], who gave a quasi-polynomial time Monte-Carlo algorithm for unweighted Global Label Cut, and a PTAS with high probability for weighted Global Label Cut. Bordini et al [3] gave some heuristics for Global Label Cut. At the current time, the most intriguing open problem is whether Global Label Cut is in P or NP-hard.…”

Section: More Related Workmentioning

confidence: 99%

Approximating the Weighted Minimum Label $s$-$t$ Cut Problem

Zhang

2020

Preprint

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In the weighted (minimum) Label s-t Cut problem, we are given a (directed or undirected) graph G = (V, E), a label set L = {ℓ 1 , ℓ 2 , . . . , ℓ q } with positive label weights {w ℓ }, a source s ∈ V and a sink t ∈ V . Each edge edge e of G has a label ℓ(e) from L. Different edges may have the same label. The problem asks to find a minimum weight label subset L ′ such that the removal of all edges with labels in L ′ disconnects s and t.The unweighted Label s-t Cut problem (i.e., every label has a unit weight) can be approximated within O(n 2/3 ), where n is the number of vertices of graph G. However, it is unknown for a long time how to approximate the weighted Label s-t Cut problem within o(n). In this paper, we provide an approximation algorithm for the weighted Label s-t Cut problem with ratio O(n 2/3 ). The key point of the algorithm is a mechanism to interpret label weight on an edge as both its length and capacity.

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New algorithms for the minimum coloring cut problem

Cited by 10 publications

References 10 publications

Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs

Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs

Minimum Label s-t Cut has large integrality gaps

Approximating the Weighted Minimum Label $s$-$t$ Cut Problem

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