On Circuit Imbalance and 0/1 Circuits for Coloring and Spanning Forest Problems

TL;DR

This paper investigates the role of circuit imbalance in linear programming formulations of graph problems, revealing exponential challenges in general cases but identifying interpretable circuit sets with efficient properties for specific problems.

Contribution

It demonstrates that while general circuit sets can have exponential imbalance, specific interpretable circuit subsets enable efficient circuit-based algorithms for coloring and forest problems.

Findings

General constraint structures can lead to exponential circuit imbalance.

Interpretable circuit sets with imbalance 1 exist for certain graph problems.

Restricted circuit walks on these sets ensure polynomial bounds on reachability.

Abstract

Circuits are fundamental objects in linear programming and oriented matroid theory, representing the elementary difference vectors of a polyhedron between points in its affine space. A recent concept introduced by Ekbatani, Natura, and V\'egh, the circuit imbalance, serves as a complexity measure relevant to iteration bounds for circuit-based augmentation and circuit diameters, as well as the general interpretability of circuits in terms of the underlying application. In this paper, we analyze linear programming formulations of relaxed combinatorial optimization problems to prove two contrasting types of results related to the circuit imbalance. On one hand, we identify simple and common constraint structures, in particular arising in graph-theoretic problems, that inherently lead to an exponential circuit imbalance. These constructions show that, in quite general situations, working with the entire set of circuits poses significant challenges for an application of circuit augmentation or the study of circuit diameters. On the other hand, through a case study of two classic graph-theoretic problems with exponential imbalance, the vertex graph coloring problem and the maximum weight forest problem, we exhibit the existence of sets and subsets of highly interpretable circuits of (best-case) imbalance 1. These sets correspond to the recoloring of vertices or to the addition or removal of edges, respectively, for example generalizing classic concepts of Kempe dynamics in coloring. Their interpretability in terms of the underlying application facilitates a study of circuit walks in the corresponding polytopes. We prove that a restriction of circuit walks to these sets suffices to not only guarantee reachability of the integral extreme-points of the skeleton, but leads to linear and constant circuit diameter bounds, respectively.