Linear Programming Hierarchies Collapse under Symmetry

2025-11-11

AI Summary

This paper investigates the failure mechanisms of linear programming (LP) hierarchies—such as Sherali–Adams, Lovász–Schrijver, and Lift-and-Project—in symmetric integer programs (IPs), focusing on the ((k+1))-transitive symmetry setting. Method: We establish a unified algebraic–geometric framework that bridges LP hierarchy techniques with group-theoretic analysis, introducing a joint geometric–algebraic modeling approach based on symmetry measures and face-intersection conditions between the relaxed polytope and low-dimensional faces of the hypercube. Contribution/Results: We prove that multiple LP hierarchies are equivalent under such symmetry, revealing that performance bottlenecks stem from geometric intersections—not merely algebraic structure. Our framework yields a concise, geometrically grounded criterion for certifying lower bounds on integrality gaps, thereby providing a novel paradigm for analyzing relaxation tightness in symmetric IPs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Studying Linear Programming hierarchies' efficacy under problem symmetries
• Unveiling connection between algebraic relaxations and polytope geometry
• Characterizing poor performance of LP hierarchies on symmetric problems

Innovation

• LP hierarchies collapse under transitive symmetries
• Hierarchies detect emptiness via hypercube face intersections
• Unified group-theoretic characterization of hierarchy limitations