A linear time algorithm for a variant of the max cut problem in series parallel graphs

TL;DR

This paper presents a linear time algorithm for solving the maximum connected sides cut problem in series parallel graphs, which is a significant improvement over the NP-hard general case, and also provides a linear time solution for the minimum cut problem in these graphs.

Contribution

The paper introduces the first linear time algorithm for MAX CS CUT in series parallel graphs, and derives a linear time algorithm for the minimum cut problem in the same class.

Findings

Linear time algorithm for MAX CS CUT in series parallel graphs

Linear time algorithm for minimum cut in series parallel graphs

Efficient solutions without maximum flow computations

Abstract

Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $\delta (U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $\Omega$ such that $w(\Omega)$ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.