Odd Multiway Cut in Directed Acyclic Graphs

TL;DR

This paper studies the odd multiway cut problem in directed acyclic graphs, providing fixed-parameter algorithms, approximation bounds, and polyhedral insights, extending previous undirected graph results.

Contribution

It extends the shadow-removal framework to parity problems in DAGs and offers new FPT algorithms, approximation, and polyhedral results for the problem.

Findings

FPT algorithms for odd multiway cut in DAGs

Tight approximability results for 2-terminal DAGs

Inapproximability results for undirected graphs with 2 terminals

Abstract

We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes and the goal is to find a smallest subset of nonterminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our main contribution is a broadening of the shadow-removal framework to address parity problems in DAGs. We complement our FPT results with tight approximability as well as polyhedral results for 2 terminals in DAGs. Additionally, we show inapproximability results for odd multiway edge cut in undirected graphs even for 2 terminals.