Vertex Sparsification for Edge Connectivity

Abstract

Graph compression or sparsification is a basic informationtheoretic and computational question. A major open problem in this research area is whether (1 + )-approximate cutpreserving 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 kThese results lead to the first data structures for answering fully dynamic offline c-edge-connectivity queries for c ≥ 4 in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

Additive Sparsification of CSPs

“…There are also vertex sparsifiers, in which one reduces the number of vertices. Vertex sparsifiers have been studied for cut sparsification (between special vertices called terminals) [22,26,25,15] as well as for spectral sparsification [24].…”

“…where the first line follows from (15) for the different hypergraphs, the second line from the triangle inequality, the third from Lemma 21, the fourth is due to λ r ∈ {0, 1} for all r, and the last is the definition of ε 0 . Again,…”