Query Complexity of Global Minimum Cut

TL;DR

This paper develops a randomized algorithm to approximate the minimum cut size in a graph using local queries, matching known lower bounds and resolving the query complexity for this problem.

Contribution

It introduces an optimal query complexity algorithm for estimating minimum cut size and establishes tight lower bounds for deciding and finding minimum cuts with local queries.

Findings

The algorithm approximates minimum cut size with high probability using minimal queries.

Lower bounds show that certain decision and exact minimum cut problems require linear queries in the number of edges.

Results hold even with additional random edge queries, confirming the bounds' robustness.

Abstract

In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like {\sc Degree}, {\sc Neighbor}, and {\sc Adjacency} queries. Given $\epsilon \in (0,1)$, the algorithm with high probability outputs an estimate $\hat{t}$ satisfying the following $(1-\epsilon) t \leq \hat{t} \leq (1+\epsilon) t$, where $m$ is the number of edges in the graph and $t$ is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is $\min\left\{m+n,\frac{m}{t}\right\}\mbox{poly}\left(\log n,\frac{1}{\epsilon}\right)$ where $n$ is the number of vertices in the graph. Eden and Rosenbaum showed that $\Omega(m/t)$ many local queries are required for approximating the size of minimum cut in graphs. These two results together resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries. Building on the lower bound of Eden and Rosenbaum, we show that, for all $t \in \mathbb{N}$, $\Omega(m)$ local queries are required to decide if the size of the minimum cut in the graph is $t$ or $t-2$. Also, we show that, for any $t \in \mathbb{N}$, $\Omega(m)$ local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size $t$. Both of our lower bound results are randomized, and hold even if we can make {\sc Random Edge} query apart from local queries.