On Circuit Diameter Bounds via Circuit Imbalances

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 is an element of R-n : Ax = b, 0 <= x <= u} for A is an element of R-mxn is bounded by O(m(2) 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(n(3) log(n + kappa(A))) augmentation steps.