On Circuit Diameter Bounds via Circuit Imbalances (2111.07913v4)

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 $O(m \min{m, n-m} \log(m+ \kappa_A)+n \log n)$, where $\kappa_A$ 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 $O(mn^2\log(n+\kappa_A))$ augmentation steps.