On Circuit Diameter Bounds via Circuit Imbalances
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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 ∈ℝ^n: Ax=b, 0≤ x≤ u} for A ∈ℝ^m × n is bounded as O(m^2 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 e.g. if 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 optimization LP in O(n^3log(n+κ_A)) augmentation steps.
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