Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear Programming Hierarchies Collapse under Symmetry

Yuri Faenza, Víctor Verdugo, José Verschae, Matías Villagra

TL;DR

The paper shows that for a polytope $P\subseteq[0,1]^n$ with $(k+1)$-transitive symmetry, the level-$k$ relaxations SA$^k$, LS$^k$, and LS$_0^k$ collapse in their ability to certify integer-emptiness, being nonempty simultaneously exactly when $P$ intersects every $(n-k)$-dimensional face of the hypercube $[0,1]^n$. This is achieved by a group-theoretic construction using Reynolds operators and symmetric point-averaging, which yields a concrete $\overline{y}\in M^k(P)$ from symmetric $M^0(P)$ points and a reduced linear system in $\gamma,\lambda$ to certify feasibility. The results provide a simple, unifying way to prove lower bounds on integrality gaps for symmetric polytopes and offer practical bounds on the number of LP-hierarchy rounds needed to certify emptiness in problems with strong symmetry, including Steiner triple covers and knapsack-cover instances. Overall, the work explains why LP-based hierarchies struggle on symmetric instances and motivates developing symmetry-aware tightening techniques that remain effective under high degrees of symmetry.

Abstract

The presence of symmetries is one of the central structural features that make some integer programs challenging for state-of-the-art solvers. In this work, we study the efficacy of Linear Programming (LP) hierarchies in the presence of symmetries. Our main theorem unveils a connection between the algebraic structure of these relaxations and the geometry of the initial integer-empty polytope: We show that under $(k+1)$-transitive symmetries--a measure of the underlying symmetry in the problem--the corresponding relaxation at level $k$ of the hierarchy is non-empty if and only if the initial polytope intersects all $(n-k)$-dimensional faces of the hypercube. In particular, the hierarchies of Sherali-Adams, Lovász-Schrijver, and the Lift-and-Project closure are equally effective at detecting integer emptiness. Our result provides a unifying, group-theoretic characterization of the poor performance of LP-based hierarchies, and offers a simple procedure for proving lower bounds on the integrality gaps of symmetric polytopes under these hierarchies.

Linear Programming Hierarchies Collapse under Symmetry

TL;DR

The paper shows that for a polytope with -transitive symmetry, the level- relaxations SA, LS, and LS collapse in their ability to certify integer-emptiness, being nonempty simultaneously exactly when intersects every -dimensional face of the hypercube . This is achieved by a group-theoretic construction using Reynolds operators and symmetric point-averaging, which yields a concrete from symmetric points and a reduced linear system in to certify feasibility. The results provide a simple, unifying way to prove lower bounds on integrality gaps for symmetric polytopes and offer practical bounds on the number of LP-hierarchy rounds needed to certify emptiness in problems with strong symmetry, including Steiner triple covers and knapsack-cover instances. Overall, the work explains why LP-based hierarchies struggle on symmetric instances and motivates developing symmetry-aware tightening techniques that remain effective under high degrees of symmetry.

Abstract

The presence of symmetries is one of the central structural features that make some integer programs challenging for state-of-the-art solvers. In this work, we study the efficacy of Linear Programming (LP) hierarchies in the presence of symmetries. Our main theorem unveils a connection between the algebraic structure of these relaxations and the geometry of the initial integer-empty polytope: We show that under -transitive symmetries--a measure of the underlying symmetry in the problem--the corresponding relaxation at level of the hierarchy is non-empty if and only if the initial polytope intersects all -dimensional faces of the hypercube. In particular, the hierarchies of Sherali-Adams, Lovász-Schrijver, and the Lift-and-Project closure are equally effective at detecting integer emptiness. Our result provides a unifying, group-theoretic characterization of the poor performance of LP-based hierarchies, and offers a simple procedure for proving lower bounds on the integrality gaps of symmetric polytopes under these hierarchies.

Paper Structure

This paper contains 35 sections, 21 theorems, 88 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Examples and applications
  4. Preliminaries
  5. Linear Programming Hierarchies
  6. Symmetries
  7. Proof of Theorem \ref{['main_theorem']}
  8. Proof of \ref{['mainthm-a']} => \ref{['mainthm-b']}
  9. A high-level description of the proof.
  10. 1. Construction of points from ${\normalfont\text{M}}^0(P)$.
  11. 2. System of constraints defining $\overline y$.
  12. Proofs of the remaining statements from Theorem \ref{['main_theorem']}
  13. Conclusion
  14. On the hypotheses of Theorem \ref{['main_theorem']}
  15. Examples and applications
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $k, n \in {\mathbb{N}}$ such that $k < n$, and let $P \subseteq [0,1]^{n}$ be a $(k+1)$-transitive integer-empty polytope. Then the following statements are equivalent:

Figures (3)

  • Figure 1:
  • Figure 2: On the left, the polytope $P=\text{conv}\left(\{ \pi (1,1/2,0), \pi (1,1,1/10) \, : \, \pi \in C_{3}\} \right)$ from Appendix \ref{['appendix:hypothesis-thm']}. On the right, $\normalfont\text{LS}_{0}(P)$.
  • Figure 3: Schematic representation of $A^{4}_{\Delta}$, with the entries in bold corresponding to the surviving $\tilde{A}^{2}$ submatrix after deleting rows 1-2 and columns 1-2.

Theorems & Definitions (43)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 33 more