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The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

Alberto Del Pia, Aida Khajavirad

TL;DR

This work addresses building strong LP relaxations for binary polynomial optimization by introducing the pseudo-Boolean polytope $\mathcal{P}_{pB}(H)$ represented via signed hypergraphs. It develops the recursive inflate-and-decompose (RID) framework, combining decomposability, pointed/nested-hypergraph hull descriptions, and edge inflation to yield polynomial-size extended formulations for broad hypergraph classes such as $\beta$-acyclic, $\alpha$-acyclic with log-poly rank, and log-poly gap cases. The authors provide constructive extended formulations for pointed and nested hypergraphs and show how inflation extends applicability while preserving polynomial size under suitable bounds. Overall, the paper unifies and extends prior results on polynomial-size extended formulations for higher-degree binary polynomial optimization and enhances practical LP relaxations for complex combinatorial problems.

Abstract

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing pseudo-Boolean functions. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three.

The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

TL;DR

This work addresses building strong LP relaxations for binary polynomial optimization by introducing the pseudo-Boolean polytope represented via signed hypergraphs. It develops the recursive inflate-and-decompose (RID) framework, combining decomposability, pointed/nested-hypergraph hull descriptions, and edge inflation to yield polynomial-size extended formulations for broad hypergraph classes such as -acyclic, -acyclic with log-poly rank, and log-poly gap cases. The authors provide constructive extended formulations for pointed and nested hypergraphs and show how inflation extends applicability while preserving polynomial size under suitable bounds. Overall, the paper unifies and extends prior results on polynomial-size extended formulations for higher-degree binary polynomial optimization and enhances practical LP relaxations for complex combinatorial problems.

Abstract

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing pseudo-Boolean functions. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three.
Paper Structure (26 sections, 11 theorems, 65 equations, 5 figures)

This paper contains 26 sections, 11 theorems, 65 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Multilinear optimization
  3. Pseudo-Boolean optimization
  4. Our contributions
  5. Notations and preliminaries
  6. Hypergraphs.
  7. Signed hypergraphs.
  8. Organization
  9. Multilinear optimization versus pseudo-Boolean optimization
  10. Reformulating \ref{['prob PBO']} as \ref{['prob MO']}.
  11. The pseudo-Boolean set as a projection of the multilinear set.
  12. The pseudo-Boolean polytope as a projection of the multilinear polytope.
  13. Decomposability of pseudo-Boolean polytopes
  14. Pointed signed hypergraphs
  15. The pseudo-Boolean polytope of nested signed hypergraphs
  16. ...and 11 more sections

Key Result

theorem 1

Let $H = (V,S)$ be a signed hypergraph, and assume that the underlying hypergraph of $H$ has a $\beta$-leaf $v$. Let $s_1 \subseteq s_2 \subseteq \cdots \subseteq s_k$ be the signed edges of $H$ containing $v$, and assume that $S$ contains the signed edges $s_i - v$ such that $|s_i - v| \ge 2$, for

Figures (5)

  • Figure 1: Example of signed hypergraphs $H,H_1,H_2$ that follow the construction of \ref{['th decomp']}. The figure depicts only the underlying hypergraphs of $H,H_1,H_2$, denoted by $G,G_1,G_2$, respectively. The decomposition result holds for any possible signing of the edges of $H$. Note that if in the edge of cardinality three containing $v$, all nodes have sign $-1$, then $v$ is not a nest point in the multilinear hypergraph of $H$, and Theorem 4 in dPKha23MPA cannot be applied.
  • Figure 2: The underlying hypergraph of a signed hypergraph $H=(V,S)$ is shown on the left. The underlying hypergraph of the signed hypergraph $H'$ obtained from $H$ by inflating all signed edges in $S$ to $V$ is shown on the right.
  • Figure 3: The hypergraphs of Example \ref{['expl3']} with $n=4$. The underlying hypergraph of the signed hyerpgraph $H$, which is $\beta$-acyclic is shown on the left. The multilinear hypergraph of $H$ that contains a $\beta$-cycle of length four is shown on the right.
  • Figure 4: The hypergraph $G=(V,E)$ of Example \ref{['expl1']}; $G$ contains a long $\beta$-cycle of length $|V|$.
  • Figure 5: The hypergraph $G$ of Example \ref{['expl2']} with $n=12$ is shown on the left. The $\beta$-cyclic hypergraph obtained from $G$ after inflation operations is shown on the right.

Theorems & Definitions (34)

  • proof
  • definition 1: Decomposability
  • theorem 1
  • proof
  • definition 2: Pointed signed hypergraph
  • definition 3: Nested signed hypergraph
  • remark 1
  • proposition 1
  • remark 2
  • remark 3
  • ...and 24 more