On Circuit Diameter Bounds via Circuit Imbalances

TL;DR

This work analyzes circuit diameter as a tractable relaxation of combinatorial diameter for polyhedra, introducing a refined bound that scales with the circuit-imbalance $\kappa_A$ rather than subdeterminants. The authors develop a conformal-circuit decomposition framework and a shoot-towards-the-optimum strategy to prove a bound of $O\big(m\min\{m,n-m\}\log(m+\kappa_A)\big)$ for standard form LPs, and extend it to capacitated cases with an additive $(n-m)\log n$ term. They also introduce practical circuit-augmentation algorithms for feasibility and optimization that run in strongly polynomial time when $\kappa_A$ is not too large, using Ratio-Circuit and Support-Circuit oracles, and they discuss reductions to and from general circuit formulations. Overall, the results improve prior dependence on $\kappa_A$ and offer a rigorous pathway toward efficient circuit-based LP solving and a deeper understanding of circuit diameter in relation to proximity and augmentation techniques.

Abstract

We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq u\}$ for $A \in \mathbb{R}^{m \times n}$ is bounded by $O(m \min\{m, n-m\} \log(m+ κ_A)+n \log n)$, where $κ_A$ is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., all entries of $A$ have polynomially bounded encoding length in $n$. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in $O(mn^2\log(n+κ_A))$ augmentation steps.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: DHNV20
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof