Cut Sparsifiers for Balanced Digraphs - Ikeda, Motoki; Tanigawa, Shin-ichi (Approximation and Online Algorithms, 2018)
In this paper we consider a cut sparsification problem for digraphs parametrized by balancedness. A weighted digraph $$D=(V,E)$$ is said to be $$α $$ -balanced if the total weight of the edges from U to $$V\setminus U$$ is at most $$α $$ times the total weight of the edges from $$V\setminus U$$ to U for any $$U\subseteq V$$ . Based on the combinatorial cut-sparsification framework by Fung et al. (2011), we show that for any $$α $$ -balanced weighted digraph D with n vertices and m edges there is a weighted subdigraph $$D'$$ with $$O(α ε ^{-2} n\log n\log (nW))$$ edges that $$(1+ε )$$ -cut-approximates D, where W is the maximum weight of an edge in D. We also show how to compute such a cut sparsifier in $$O(m\log α +α ^3 n\log W poly(\log n))$$ time with high probability. Applying our sparsifier as a preprocessing, the running time of the minimum cut approximation algorithm by Ene et al. (2016) is improved to $$O(m\log α +α ^3 ε ^{-4} n poly(\log n))$$ for an $$α $$ -balanced digraph with n vertices and m edges.