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arXiv · 1311.2563

Minimum Bisection is fixed parameter tractable

Marek Cygan·Daniel Lokshtanov·Marcin Pilipczuk·Michał Pilipczuk·Saket Saurabh
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Abstract

In the classic Minimum Bisection problem we are given as input a graph $G$ and an integer $k$. The task is to determine whether there is a partition of $V(G)$ into two parts $A$ and $B$ such that $||A|-|B|| \leq 1$ and there are at most $k$ edges with one endpoint in $A$ and the other in $B$. In this paper we give an algorithm for Minimum Bisection with running time $O(2^{O(k^{3})}n^3 \log^3 n)$. This is the first fixed parameter tractable algorithm for Minimum Bisection. At the core of our algorithm lies a new decomposition theorem that states that every graph $G$ can be decomposed by small separators into parts where each part is "highly connected" in the following sense: any cut of bounded size can separate only a limited number of vertices from each part of the decomposition. Our techniques generalize to the weighted setting, where we seek for a bisection of minimum weight among solutions that contain at most $k$ edges.

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Marek Cygan, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Saket Saurabh. 2014-03-18. Minimum Bisection is fixed parameter tractable. https://arxiv.org/abs/1311.2563

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