Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extended formulations for the multilinear polytope of acyclic hypergraphs

Alberto Del Pia, Aida Khajavirad

TL;DR

This work analyzes binary polynomial optimization through a hypergraph lens, focusing on the multilinear polytope MP(G) and its tightness relative to LP relaxations across acyclicity notions. It establishes size bounds and tractability conditions for polynomial-size extended formulations: α-acyclic hypergraphs admit EPFs of size 2^r min{|V|,|F|} when rank r is bounded, with NP-hardness results otherwise; β-acyclic hypergraphs admit a compact EPF with sparse inequalities and variables using nest points and decomposition. The paper also provides explicit original-space characterizations, showing MP(G) = MP^{LP}(G) for Berge-acyclic graphs and introducing flower and running intersection inequalities to capture γ-acyclic and related structures, with practical implications for solvers such as BARON. It discusses limitations due to potentially dense β-acyclic facets and highlights separation complexities, offering insight into when strong LP relaxations are feasible and how to exploit hypergraph structure in optimization.

Abstract

This article provides an overview of our joint work on binary polynomial optimization over the past decade. We define the multilinear polytope as the convex hull of the feasible region of a linearized binary polynomial optimization problem. By representing the multilinear polytope with hypergraphs, we investigate the connections between hypergraph acyclicity and the complexity of the facial structure of the multilinear polytope. We characterize the acyclic hypergraphs for which a polynomial-size extended formulation for the multilinear polytope can be constructed in polynomial time.

Extended formulations for the multilinear polytope of acyclic hypergraphs

TL;DR

This work analyzes binary polynomial optimization through a hypergraph lens, focusing on the multilinear polytope MP(G) and its tightness relative to LP relaxations across acyclicity notions. It establishes size bounds and tractability conditions for polynomial-size extended formulations: α-acyclic hypergraphs admit EPFs of size 2^r min{|V|,|F|} when rank r is bounded, with NP-hardness results otherwise; β-acyclic hypergraphs admit a compact EPF with sparse inequalities and variables using nest points and decomposition. The paper also provides explicit original-space characterizations, showing MP(G) = MP^{LP}(G) for Berge-acyclic graphs and introducing flower and running intersection inequalities to capture γ-acyclic and related structures, with practical implications for solvers such as BARON. It discusses limitations due to potentially dense β-acyclic facets and highlights separation complexities, offering insight into when strong LP relaxations are feasible and how to exploit hypergraph structure in optimization.

Abstract

This article provides an overview of our joint work on binary polynomial optimization over the past decade. We define the multilinear polytope as the convex hull of the feasible region of a linearized binary polynomial optimization problem. By representing the multilinear polytope with hypergraphs, we investigate the connections between hypergraph acyclicity and the complexity of the facial structure of the multilinear polytope. We characterize the acyclic hypergraphs for which a polynomial-size extended formulation for the multilinear polytope can be constructed in polynomial time.
Paper Structure (5 sections, 8 theorems, 7 equations, 3 figures)

This paper contains 5 sections, 8 theorems, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The multilinear polytope of $\alpha$-acyclic hypergraphs
  3. Polynomial-size extended formulations
  4. Explicit characterizations in the original space
  5. Acknowledgments.

Key Result

Theorem 1

${\rm BQP}(G) = {\rm MP}^{{\rm LP}}(G)$ if and only if the graph $G$ is acyclic.

Figures (3)

  • Figure 1: An illustration of the proof technique for \ref{['padberg']}.
  • Figure 2: An illustration of the proof technique for \ref{['thalpha']}.
  • Figure 3: An illustration of the proof technique for \ref{['thbeta']}.

Theorems & Definitions (8)

  • Theorem 1: Pad89
  • Theorem 2: dPDiG23ALG
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8