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Extended formulations for induced tree and path polytopes of chordal graphs

Alexandre Dupont-Bouillard

TL;DR

The paper develops extended space formulations for induced tree and induced path polytopes in chordal graphs by proving that their extended incidence vectors form a Hilbert basis, which implies IDP and enables compact or tractable polyhedral descriptions. It shows a compact linear description for the induced-tree polytope and an exponential-inequality, facet-aware yet polynomially enumerable path description, yielding a polynomial-time solvable optimization for both structures on chordal graphs. A central technical contribution is the Hilbert-basis proof for induced paths in chordal graphs, supported by a refined decomposition and reduction framework; this leads to a polynomial-size extended formulation for the induced path polytope and a complete facet characterization. Overall, the results provide efficient polynomial-time solvability of maximum vertex/edge weighted induced trees and paths on chordal graphs and establish a practical extended formulation for the induced path polytope.

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.

Extended formulations for induced tree and path polytopes of chordal graphs

TL;DR

The paper develops extended space formulations for induced tree and induced path polytopes in chordal graphs by proving that their extended incidence vectors form a Hilbert basis, which implies IDP and enables compact or tractable polyhedral descriptions. It shows a compact linear description for the induced-tree polytope and an exponential-inequality, facet-aware yet polynomially enumerable path description, yielding a polynomial-time solvable optimization for both structures on chordal graphs. A central technical contribution is the Hilbert-basis proof for induced paths in chordal graphs, supported by a refined decomposition and reduction framework; this leads to a polynomial-size extended formulation for the induced path polytope and a complete facet characterization. Overall, the results provide efficient polynomial-time solvability of maximum vertex/edge weighted induced trees and paths on chordal graphs and establish a practical extended formulation for the induced path polytope.

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.

Paper Structure

This paper contains 8 sections, 18 theorems, 11 equations, 5 figures.

Table of Contents

  1. Induced trees
  2. Induced paths
  3. Induced tree polytope of chordal graphs
  4. Induced path polytope of chordal graphs
  5. Induced paths of chordal graphs and Hilbert basis
  6. Assume henceforth that $\Psi_{\overline{K}_\Psi}^u$ is non-empty.
  7. We may now assume that $\Psi_2^u$ is nonempty.
  8. Extended formulation for the induced path polytope

Key Result

Theorem 1.1

The induced tree extended incidence vectors of a chordal graph form a Hilbert basis.

Figures (5)

  • Figure 1: Sets of induced trees considered in the proof of Theorem \ref{['the:inducedPathChordal']}
  • Figure 2: Structure of inequalities \ref{['iq:neighCliquebb']} for a given vertex $w$ and clique $K$ of its neighborhood.
  • Figure 3: Types of induced paths considered in Section \ref{['sec:inducedpath1']}
  • Figure 4: Matrix of points built in the proof of Theorem \ref{['the:facetorbit']} where $G'$ is the graph with vertex set $V$ and edge set $(E\setminus \delta(w) )\cup (\delta(w) \cap \delta(K))$
  • Figure 5: An orbit defining clique whose associated inequalities \ref{['iq:neighCliquebb']} are not facet defining

Theorems & Definitions (19)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1: dupontbouillard2024contractions
  • Claim 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 9 more