Efficient Enumerations for Minimal Multicuts and Multiway Cuts

TL;DR

This paper presents new algorithms for efficiently enumerating all minimal multicuts and multiway cuts in graphs, improving the computational bounds for these problems, which are relevant in network reliability and graph partitioning.

Contribution

It introduces an incremental polynomial delay enumeration algorithm for minimal node multicuts and improves enumeration algorithms for minimal node and edge multiway cuts.

Findings

Provides an incremental polynomial delay enumeration algorithm for minimal node multicuts.

Develops a polynomial delay and exponential space algorithm for minimal node multiway cuts.

Creates a polynomial delay and space enumeration algorithm for minimal edge multiway cuts.

Abstract

Let $G = (V, E)$ be an undirected graph and let $B \subseteq V \times V$ be a set of terminal pairs. A node/edge multicut is a subset of vertices/edges of $G$ whose removal destroys all the paths between every terminal pair in $B$. The problem of computing a {\em minimum} node/edge multicut is NP-hard and extensively studied from several viewpoints. In this paper, we study the problem of enumerating all {\em minimal} node multicuts. We give an incremental polynomial delay enumeration algorithm for minimal node multicuts, which extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008) for minimal edge multicuts. Important special cases of node/edge multicuts are node/edge {\em multiway cuts}, where the set of terminal pairs contains every pair of vertices in some subset $T \subseteq V$, that is, $B = T \times T$. We improve the running time bound for this special case: We devise a polynomial delay and exponential space enumeration algorithm for minimal node multiway cuts and a polynomial delay and space enumeration algorithm for minimal edge multiway cuts.