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Deterministic Edge Connectivity and Max Flow using Subquadratic Cut Queries

Aditya Anand, Thatchaphol Saranurak, Yunfan Wang

TL;DR

This work gives the first deterministic algorithm that makes sub-quadratic queries to find the global min-cut of a simple graph in the cut query model, and shows efficient cut-query implementations of versions of expander decomposition and isolating cuts, which may be of independent interest.

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.

Deterministic Edge Connectivity and Max Flow using Subquadratic Cut Queries

TL;DR

This work gives the first deterministic algorithm that makes sub-quadratic queries to find the global min-cut of a simple graph in the cut query model, and shows efficient cut-query implementations of versions of expander decomposition and isolating cuts, which may be of independent interest.

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 -vertex graph , our algorithm makes queries to compute the global min-cut in . As a key ingredient, we also show an algorithm for finding - max-flows of size in queries. We also show efficient cut-query implementations of versions of expander decomposition and isolating cuts, which may be of independent interest.

Paper Structure

This paper contains 25 sections, 24 theorems, 13 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Other Query Models.
  3. Technical Overview
  4. Maximum $s$-$t$ flow:
  5. Global minimum cut:
  6. Preliminaries
  7. Graphs:
  8. Flows and residual graphs:
  9. Types of queries:
  10. Flow Algorithm
  11. Simple Primitives using BIS Queries
  12. Reminder of Dinitz's Algorithm
  13. Implementing Dinitz's Algorithm via BIS Queries
  14. Global Min-cut Algorithm
  15. Dominating Sets are Separated by Small Cuts
  16. ...and 10 more sections

Key Result

theorem 1.1

Given a simple undirected graph $G$, there is a deterministic algorithm that computes the global minimum cut in $G$ using $\widetilde{O}(n^{5/3})$ cut queries.

Theorems & Definitions (64)

  • theorem 1.1
  • theorem 1.2
  • definition 3.1: Cut Query
  • definition 3.2: Bipartite Independent Set (BIS) Query
  • lemma 3.3
  • proof
  • lemma 4.1
  • proof
  • corollary 4.2
  • proof
  • ...and 54 more