Vertex Sparsification for Edge Connectivity

TL;DR

This paper develops new graph sparsification techniques called connectivity-c mimicking networks that preserve terminal connectivity up to a threshold, enabling efficient dynamic connectivity queries and survivable network design.

Contribution

It introduces the first constructions of connectivity-c mimicking networks with size bounds and algorithms for their efficient creation, advancing graph sparsification and network algorithms.

Findings

Existence of connectivity-c mimicking networks with O(kc^4) edges.

Algorithms to construct these networks in time m(c log n)^{O(c)}.

Applications to dynamic c-edge connectivity queries and survivable network design.

Abstract

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+\epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call connectivity-$c$ mimicking network, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity-$c$ mimicking networks with $O(kc^4)$ edges exist and can be found in time $m(c\log n)^{O(c)}$. We also give a separate algorithm that constructs such graphs with $k \cdot O(c)^{2c}$ edges in time $mc^{O(c)}\log^{O(1)}n$. These results lead to the first data structures for answering fully dynamic offline $c$-edge-connectivity queries for $c \ge 4$ in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.