Extended formulations for induced tree and path polytopes of chordal graphs

TL;DR

This paper develops extended formulations for induced tree and path polytopes in chordal graphs, proving their properties and implications for polynomial-time optimization of related problems.

Contribution

It introduces new extended space formulations for these polytopes, proves their Hilbert basis property, and explores their facet-defining inequalities and optimization implications.

Findings

Induced tree polytope has a compact formulation.

Induced path polytope formulation has exponential inequalities.

Linear optimization over these polytopes is polynomial-time for chordal graphs.

Abstract

In this article, we give two extended space formulations, respectively, for the induced tree and path polytopes of chordal graphs with vertex and edge variables. These formulations are obtained by proving that the induced tree and path extended incidence vectors of chordal graphs form Hilbert basis. This also shows that both polytopes have the integer decomposition property in chordal graphs. Whereas the formulation for the induced tree polytope is easily seen to have a compact size, the system we provide for the induced path polytope has an exponential number of inequalities. We show which of these inequalities define facets and exhibit a superset of the facet-defining ones that can be enumerated in polynomial time. We show that for some graphs, the latter superset contains redundant inequalities. As corollaries, we obtain that the problems of finding an induced tree or path maximizing a linear function over the edges and vertices are solvable in polynomial time for the class of chordal graphs.