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

Given an undirected graph $G$, a collection $\{(s_1,t_1), \dots, (s_{k},t_{k})\}$ 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 Mult...

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Bibliographic Details
Published in:SIAM journal on computing Vol. 43; no. 2; pp. 355 - 388
Main Authors: Marx, Dániel, Razgon, Igor
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
Subjects:
Algorithms
Approximation
Collection
Computation
Computer science
Disengaging
Editing
Graphs
Integers
Mathematical models
Running
ISSN:0097-5397, 1095-7111
Online Access:Get full text
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Summary:Given an undirected graph $G$, a collection $\{(s_1,t_1), \dots, (s_{k},t_{k})\}$ 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 $2^{O({{p}}^3)}\cdot n^{O(1)}$, 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 $f({{p}})\cdot n^{O(1)}$ exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset. [PUBLICATION ABSTRACT]
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ISSN:0097-5397
1095-7111
DOI:10.1137/110855247