Log In Sign Up

Vertex Sparsification for Edge Connectivity

by   Parinya Chalermsook, et al.

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether (1+ϵ)-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(clog n)^O(c). We also give a separate algorithm that constructs such graphs with k · 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 ≥ 4 in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.


page 1

page 2

page 3

page 4


Vertex Sparsification for Edge Connectivity in Polynomial Time

An important open question in the area of vertex sparsification is wheth...

Graph Connectivity with Noisy Queries

Graph connectivity is a fundamental combinatorial optimization problem t...

Vertex Sparsifiers for Hyperedge Connectivity

Recently, Chalermsook et al. [SODA'21(arXiv:2007.07862)] introduces a no...

Mimicking Networks Parameterized by Connectivity

Given a graph G=(V,E), capacities w(e) on edges, and a subset of termina...

A Faster Local Algorithm for Detecting Bounded-Size Cuts with Applications to Higher-Connectivity Problems

Consider the following "local" cut-detection problem in a directed graph...

Deterministic Fault-Tolerant Connectivity Labeling Scheme with Adaptive Query Processing Time

The f-fault-toleratant connectivity labeling (f-FTC labeling) is a schem...

Hedge Connectivity without Hedge Overlaps

Connectivity is a central notion of graph theory and plays an important ...