DeepAI
Log In Sign Up

Deterministic Graph Cuts in Subquadratic Time: Sparse, Balanced, and k-Vertex

10/17/2019
by   Yu Gao, et al.
0

We study deterministic algorithms for computing graph cuts, with focus on two fundamental problems: balanced sparse cut and k-vertex connectivity for small k (k=O( n)). Both problems can be solved in near-linear time with randomized algorithms, but their previous deterministic counterparts take at least quadratic time. In this paper, we break this bound for both problems. Interestingly, achieving this for one problem crucially relies on doing so for the other. In particular, via a divide-and-conquer argument, a variant of the cut-matching game by [Khandekar et al.`07], and the local vertex connectivity algorithm of [Nanongkai et al. STOC'19], we give a subquadratic time algorithm for k-vertex connectivity using a subquadratic time algorithm for computing balanced sparse cuts on sparse graphs. To achieve the latter, we improve the previously best mn bound for approximating balanced sparse cut for the whole range of m. This starts from (1) breaking the n^3 barrier on dense graphs to n^ω + o(1) (where ω < 2.372) using the the PageRank matrix, but without explicitly sweeping to find sparse cuts; to (2) getting the Õ(m^1.58) bound by combining the J-trees by [Madry FOCS `10] with the n^ω + o(1) bound above, and finally; to (3) getting the m^1.5 + o(1) bound by recursively invoking the second bound in conjunction with expander-based graph sparsification. Interestingly, our final m^1.5 + o(1) bound lands at a natural stopping point in the sense that polynomially breaking it would lead to a breakthrough for the dynamic connectivity problem.

READ FULL TEXT

page 1

page 2

page 3

page 4

10/25/2022

Deterministic Small Vertex Connectivity in Almost Linear Time

In the vertex connectivity problem, given an undirected n-vertex m-edge ...
10/31/2019

Computing and Testing Small Connectivity in Near-Linear Time and Queries via Fast Local Cut Algorithms

Consider the following "local" cut-detection problem in a directed graph...
05/14/2019

Computing and Testing Small Vertex Connectivity in Near-Linear Time and Queries

We present a new, simple, algorithm for the local vertex connectivity pr...
10/17/2019

A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

We consider the classical Minimum Balanced Cut problem: given a graph G,...
03/29/2021

Engineering Nearly Linear-Time Algorithms for Small Vertex Connectivity

Vertex connectivity is a well-studied concept in graph theory with numer...
09/19/2022

A Simple Framework for Finding Balanced Sparse Cuts via APSP

We present a very simple and intuitive algorithm to find balanced sparse...
05/20/2022

Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player

A (ϕ,ϵ)-Expander-decomposition of a graph G is a partition of V into clu...