Faster All-Pairs Minimum Cut: Bypassing Exact Max-Flow

TL;DR

This paper introduces a new sparsifier that preserves all minimum s,t-cuts in unweighted graphs, enabling faster algorithms for All-Pairs Minimum Cut across various computational models without relying on exact max-flow computations.

Contribution

The authors develop a sparsifier constructed via approximate max-flow computations, leading to improved algorithms for APMC in multiple models.

Findings

Achieved a randomized cut-query algorithm with  O(n^{3/2}) queries.

Designed a deterministic fully-dynamic algorithm with n^{3/2+o(1)} worst-case update time.

Created a two-pass streaming algorithm with  O(n^{3/2}) space.

Abstract

All-Pairs Minimum Cut (APMC) is a fundamental graph problem that asks to find a minimum $s,t$-cut for every pair of vertices $s,t$. A recent line of work on fast algorithms for APMC has culminated with a reduction of APMC to $\mathrm{polylog}(n)$-many max-flow computations. But unfortunately, no fast algorithms are currently known for exact max-flow in several standard models of computation, such as the cut-query model and the fully-dynamic model. Our main technical contribution is a sparsifier that preserves all minimum $s,t$-cuts in an unweighted graph, and can be constructed using only approximate max-flow computations. We then use this sparsifier to devise new algorithms for APMC in unweighted graphs in several computational models: (i) a randomized algorithm that makes $\tilde{O}(n^{3/2})$ cut queries to the input graph; (ii) a deterministic fully-dynamic algorithm with $n^{3/2+o(1)}$ worst-case update time; and (iii) a randomized two-pass streaming algorithm with space requirement $\tilde{O}(n^{3/2})$. These results improve over the known bounds, even for (single pair) minimum $s,t$-cut in the respective models.