Global and fixed-terminal cuts in digraphs

TL;DR

This paper studies the complexity and approximation algorithms for multicut-like problems in directed graphs, focusing on fixed versus global terminal scenarios, and presents new bounds and algorithms for double cut and bicut problems.

Contribution

It provides tight approximation bounds for fixed-terminal and global versions of double cut and bicut problems, and introduces new algorithms and hardness results, including for undirected graphs.

Findings

Fixed-terminal node-weighted double cut has a tight approximation factor of 2.

Global node-weighted double cut cannot be approximated better than 3/2 under UGC.

Global edge-weighted bicut can be approximated better than factor 2.

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.