Linear Programming Hierarchies Collapse under Symmetry

TL;DR

This paper reveals that LP hierarchies like Sherali-Adams and Lovász-Schrijver fail to effectively approximate symmetric integer programs, with their performance collapsing under high symmetry levels, due to a geometric and algebraic connection.

Contribution

It establishes a group-theoretic framework linking symmetry levels to the effectiveness of LP hierarchies, unifying their limitations in symmetric cases.

Findings

LP relaxations are non-empty iff the initial polytope intersects all hypercube faces.

Hierarchies of Sherali-Adams, Lovász-Schrijver, and Lift-and-Project are equally limited by symmetry.

Provides a method to prove lower bounds on integrality gaps for symmetric polytopes.

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, Lov\'asz-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.