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

TL;DR

The paper investigates circuits and the circuit-imbalance measure in LP relaxations of graph-structured problems, revealing that many natural constraint families induce exponential circuit imbalance. It then shows that, despite this, highly interpretable 0/1 circuits exist for two classic problems—vertex coloring and maximum-weight forest—enabling short, Kempe-type circuit walks and strong diameter bounds. By connecting 0/1 circuits to Kempe dynamics and combinatorial reconfiguration, the authors derive both reachability results and explicit limits on walk lengths. They also discuss limitations and open questions about when restricting to 0/1 circuits suffices to guarantee short reaches, and how these ideas interplay with circuit augmentation bounds.

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 Végh, 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.

Figures (14)

• Figure 1: The construction of the vector ${\hbox{\boldmath$\bf g$}}$ in the proof of Theorem \ref{['thm:induce_imbalance']}. Each edge is labeled with its corresponding value in ${\hbox{\boldmath$\bf g$}}$, where dashed lines indicate a value of 0.
• Figure 2: The construction of the vector ${\hbox{\boldmath$\bf g$}}$ in the proof of Theorem \ref{['thm:path_imbalance']}. Each edge is labeled with its corresponding value in ${\hbox{\boldmath$\bf g$}}$.
• Figure 3: The construction of the vector ${\hbox{\boldmath$\bf g$}}$ in the proof of Theorem \ref{['thm:cycle_imbalance']}. Each edge is labeled with it's corresponding value in ${\hbox{\boldmath$\bf g$}}$, where dashed lines indicate a value of 0.
• Figure 4: The construction of the vector ${\hbox{\boldmath$\bf g$}}$ in the proof of Theorem \ref{['thm:star_imbalance']}. Each edge is labeled with its corresponding value in ${\hbox{\boldmath$\bf g$}}$.
• Figure 5: Left: Proper coloring of the cycle on four vertices using colors {1,2,3}. Right: Proper coloring using the minimum number of colors of the cycle on four vertices using colors {1,3}.

Theorems & Definitions (77)

• Definition 1
• Definition 2: Circuits
• Definition 3
• Definition 4
• Definition 5: See e.g. ccps-97
• Definition 6
• Definition 7
• Definition 8
• Definition 9
• Theorem 2.1