On Mimicking Networks Representing Minimum Terminal Cuts

Given a capacitated undirected graph G = (V,E) with a set of terminals K ⊂ V , a mimicking network is a smaller graph H = (VH , EH) that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier VH contains the set of terminals K and for every bipartition U,K − U of the terminals K, the size of the minimum cut separating U from K − U in G is exactly equal to the size of the minimum cut separating U from K − U in H. This notion of a mimicking network was introduced by Hagerup, Katajainen, Nishimura and Ragde [HKNR95] who also exhibited a mimicking network of size 2 k for every graph with k terminals. The best known lower bound on the size of a mimicking network is linear in the number of terminals. More precisely, the best known lower bound is k + 1 for graphs with k terminals [CSWZ00]. In this work, we improve both the upper and lower bounds reducing the doubly-exponential gap between them to a single-exponential gap. Specifically, we obtain the following upper and lower bounds on mimicking networks: • Given a graph G, we exhibit a construction of mimicking network with at most (|K|−1)’th Dedekind number (≈ 2( (k−1) b(k−1)/2c)) of vertices (independent of size of V ). Furthermore, we show that the construction is optimal among all restricted mimicking networks – a natural class of mimicking networks that are obtained by clustering vertices together. • There exists graphs with k terminals that have no mimicking network of size smaller than 2 k−1 2 . We also exhibit improved constructions of mimicking networks for trees and graphs of bounded tree-width. keywords: Approximation algorithms, Graph algorithms, Vertex sparsification, Cut sparsifier, Mimicking networks, Terminal cuts, Realizable external flow, Network flow. ∗School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332-0765. Email: [email protected], [email protected] †EECS, Univ of California, Berkeley, CA. Email: [email protected] ‡Dept of Management, London School of Economics. Email: [email protected] ar X iv :1 20 7. 63 71 v1 [ cs .D S] 2 6 Ju l 2 01 2