Global and fixed-terminal cuts in digraphs

Kristóf Bérczi; Karthekeyan Chandrasekaran; Tamás Király; Euiwoong Lee; Chao Xu

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arXiv · 1612.00156

Global and fixed-terminal cuts in digraphs

Kristóf Bérczi·Karthekeyan Chandrasekaran·Tamás Király·Euiwoong Lee·Chao Xu

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Abstract

The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut. 1. The fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show a tight approximability factor of $2$ for the fixed-terminal node-weighted double cut. We show that the global node-weighted double cut cannot be approximated to a factor smaller than $3/2$ under the Unique Games Conjecture (UGC). 2. The fixed-terminal edge-weighted bicut is known to have a tight approximability factor of $2$. We show that the global edge-weighted bicut is approximable to a factor strictly better than $2$, and that the global node-weighted bicut cannot be approximated to a factor smaller than $3/2$ under UGC. 3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of $4/3$ for the node-weighted $3$-cut problem. Second, we show that for constant $k$, there exists an efficient algorithm to solve the minimum $\{s,t\}$-separating $k$-cut problem. Our techniques for the algorithms are combinatorial, based on LPs and based on enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.

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Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Euiwoong Lee, Chao Xu. 2016-12-01. Global and fixed-terminal cuts in digraphs. https://arxiv.org/abs/1612.00156

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