Deterministic Edge Connectivity and Max Flow using Subquadratic Cut Queries

TL;DR

This paper introduces the first deterministic sub-quadratic query algorithm for finding global min-cuts in simple graphs, along with an efficient method for s-t max-flows, advancing graph cut computation techniques.

Contribution

It presents a novel deterministic algorithm with sub-quadratic query complexity for global min-cut and s-t max-flow problems in the cut query model.

Findings

Achieves $ ilde{O}(n^{5/3})$ query complexity for global min-cut.

Develops an algorithm for s-t max-flows of size $ ilde{O}(n)$ with similar query complexity.

Provides efficient cut-query implementations for expander decomposition and isolating cuts.

Abstract

We give the first deterministic algorithm that makes sub-quadratic queries to find the global min-cut of a simple graph in the cut query model. Given an $n$-vertex graph $G$, our algorithm makes $\widetilde{O}(n^{5/3})$ queries to compute the global min-cut in $G$. As a key ingredient, we also show an algorithm for finding $s$-$t$ max-flows of size $\widetilde{O}(n)$ in $\widetilde{O}(n^{5/3})$ queries. We also show efficient cut-query implementations of versions of expander decomposition and isolating cuts, which may be of independent interest.