On circuit diameter bounds via circuit imbalances

Abstract

We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIAM J. Discrete Math. 29(1), 113–121 (2015)) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system \(\{x\in \mathbb {R}^n:\, Ax=b,\, \mathbb {0}\le x\le u\}\) 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.