On Mimicking Networks Representing Minimum Terminal Cuts

TL;DR

This paper advances the understanding of mimicking networks by establishing tighter bounds on their size, providing optimal constructions, and demonstrating limitations for general graphs and special cases like trees.

Contribution

It introduces new upper and lower bounds on mimicking network sizes, showing optimality within restricted classes and improving previous exponential gap results.

Findings

Constructed mimicking networks with size bounded by the Dedekind number.

Proved existence of graphs requiring exponentially large mimicking networks.

Provided improved constructions for trees and bounded tree-width graphs.

Abstract

Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K \subset V$, a mimicking network is a smaller graph $H=(V_H,E_H)$ that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier $V_H$ 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 (1995) who also exhibited a mimicking network of size $2^{2^{k}}$ 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 (Chaudhuri et al. 2000). 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: 1) Given a graph $G$, we exhibit a construction of mimicking network with at most $(|K|-1)$'th Dedekind number ($\approx 2^{{(k-1)} \choose {\lfloor {{(k-1)}/2} \rfloor}}$) of vertices (independent of size of $V$). Furthermore, we show that the construction is optimal among all {\it restricted mimicking networks} -- a natural class of mimicking networks that are obtained by clustering vertices together. 2) There exists graphs with $k$ terminals that have no mimicking network of size smaller than $2^{\frac{k-1}{2}}$. We also exhibit improved constructions of mimicking networks for trees and graphs of bounded tree-width.