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∈Rn:Ax=b,0≤x≤u} for A∈ Rm × n is bounded by O(m2log (m+ κA) + nlog n), where κA 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(n3log (n+ κA) ) augmentation steps.