A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs

TL;DR

This paper introduces a quantum algorithm that efficiently computes the minimum s-t cut in dense undirected graphs, achieving sublinear query complexity and outperforming classical algorithms in certain dense graph scenarios.

Contribution

It presents a novel quantum algorithm for s-t minimum cut with sublinear query complexity, especially effective for dense simple graphs.

Findings

Quantum algorithm computes min s-t cut with high probability

Query complexity is  O(n^{11/6}) for dense graphs

Classical algorithms require  ull m queries to decide connectivity

Abstract

An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$ to $t$. In this work we describe a quantum algorithm for the minimum $s{\operatorname{-}}t$ cut problem on undirected graphs. For an undirected graph with $n$ vertices, $m$ edges, and integral edge weights bounded by $W$, the algorithm computes with high probability the weight of a minimum $s{\operatorname{-}}t$ cut after $\widetilde O(\sqrt{m} n^{5/6} W^{1/3})$ queries to the adjacency list of $G$. For simple graphs this bound is always $\widetilde O(n^{11/6})$, even in the dense case when $m = \Omega(n^2)$. In contrast, a randomized algorithm must make $\Omega(m)$ queries to the adjacency list of a simple graph $G$ even to decide whether $s$ and $t$ are connected.