A polynomial-time algorithm for planar multicuts with few source-sink pairs

TL;DR

This paper presents the first polynomial-time algorithm for solving the minimum multicut problem in planar graphs when the number of source-sink pairs is fixed, expanding the class of tractable instances.

Contribution

It introduces a novel polynomial-time algorithm for planar graphs with a fixed number of source-sink pairs, addressing a previously NP-hard problem.

Findings

First polynomial-time solution for fixed k in planar graphs

Extends tractability beyond outer face source-sink configurations

Addresses a key gap in multicut problem complexity

Abstract

Given an edge-weighted undirected graph and a list of k source-sink pairs of vertices, the well-known minimum multicut problem consists in selecting a minimum-weight set of edges whose removal leaves no path between every source and its corresponding sink. We give the first polynomial-time algorithm to solve this problem in planar graphs, when k is fixed. Previously, this problem was known to remain NP-hard in general graphs with fixed k, and in trees with arbitrary k; the most noticeable tractable case known so far was in planar graphs with fixed k and sources and sinks lying on the outer face.