Cut Sparsifiers for Balanced Digraphs

• Published in: Approximation and Online Algorithms Vol. 11312; pp. 277 - 294
• Main Authors: Ikeda, Motoki, Tanigawa, Shin-ichi
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2018; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Balanced digraph; Cut sparsification; Minimum cut problem
• ISBN: 9783030046927; 3030046923
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-04693-4_17

In this paper we consider a cut sparsification problem for digraphs parametrized by balancedness. A weighted digraph $$D=(V,E)$$ is said to be $$\alpha $$ -balanced if the total weight of the edges from U to $$V\setminus U$$ is at most $$\alpha $$ 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 $$\alpha $$ -balanced weighted digraph D with n vertices and m edges there is a weighted subdigraph $$D'$$ with $$O(\alpha \epsilon ^{-2} n\log n\log (nW))$$ edges that $$(1+\epsilon )$$ -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 $$O(m\log \alpha +\alpha ^3 n\log W \mathrm{poly}(\log n))$$ 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 $$O(m\log \alpha +\alpha ^3 \epsilon ^{-4} n \mathrm{poly}(\log n))$$ for an $$\alpha $$ -balanced digraph with n vertices and m edges.