Combinatorial optimization with 2-joins

TL;DR

This paper develops polynomial-time algorithms for key graph problems in certain perfect graphs decomposable by 2-joins, expanding the understanding of graph classes where these techniques are effective.

Contribution

It introduces new combinatorial algorithms for maximum weighted clique, stable set, and coloring in specific perfect graph classes decomposable by 2-joins, and analyzes their limitations.

Findings

Algorithms work for perfect graphs without balanced skew partition or homogeneous pair.

Extends methods to even-hole-free graphs without star cutset.

Shows NP-hardness in graphs decomposable into bipartite and line graphs.

Abstract

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset. We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.