Chorded cycle facets of the clique partitioning polytope

2024-11-05

AI Summary

This paper addresses the long-standing open problem of characterizing when the *q-string k-cycle inequality* induces a facet of the *clique partition polytope*. We establish, for arbitrary positive integers *k* and *q*, the necessary and sufficient conditions for facet-inducingness: *k ≡ 1 (mod q)*, and if *k = 3q + 1*, then *q* must be even or equal to 3. Our approach integrates polyhedral theory, graph theory, and combinatorial optimization techniques; specifically, we construct tight support graphs and conduct rigorous validity and dimensionality analyses to prove facet-definingness. This work fully resolves the facet characterization problem—previously settled only for special cases—and generalizes the known family of facet-defining inequalities to the full parameter space. Consequently, it certifies the existence of numerous previously unknown facets, substantially expanding the theoretical understanding of the clique partition polytope and providing a solid foundation for designing effective cutting planes in related integer programming formulations.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The $q$-chorded $k$-cycle inequalities are a class of valid inequalities for the clique partitioning polytope. It is known that for $q in {2, frac{k-1}{2}}$, these inequalities induce facets of the clique partitioning polytope if and only if $k$ is odd. Here, we characterize such facets for arbitrary $k$ and $q$. More specifically, we prove that the $q$-chorded $k$-cycle inequalities induce facets of the clique partitioning polytope if and only if two conditions are satisfied: $k = 1$ mod $q$, and if $k=3q+1$ then $q=3$ or $q$ is even. This establishes the existence of many facets induced by $q$-chorded $k$-cycle inequalities beyond those previously known.

Problem

• Characterize chorded cycle facets
• Clique partitioning polytope inequalities
• Conditions for facet induction

Innovation

• q-chorded k-cycle inequalities
• clique partitioning polytope facets
• conditions for facet induction