Finding a Maximum-Weight Convex Set in a Chordal Graph

TL;DR

This paper introduces the first polynomial-time algorithm for finding a maximum-weight convex set in chordal graphs, a problem that generalizes previous cases and relates to closure problems in posets.

Contribution

It presents the first efficient algorithm for the maximum-weight convex set problem in chordal graphs, expanding the scope of solvable instances.

Findings

The problem is solvable in polynomial time for chordal graphs.

The algorithm generalizes solutions for trees and split graphs.

It connects to closure problems in posets and directed graphs.

Abstract

We consider a natural combinatorial optimization problem on chordal graphs, the class of graphs with no induced cycle of length four or more. A subset of vertices of a chordal graph is (monophonically) convex if it contains the vertices of all chordless paths between any two vertices of the set. The problem is to find a maximum-weight convex subset of a given vertex-weighted chordal graph. It generalizes previously studied special cases in trees and split graphs. It also happens to be closely related to the closure problem in partially ordered sets and directed graphs. We give the first polynomial-time algorithm for the problem.