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A Short Note on Relevant Cuts

Nico Domschke, Thomas Gatter, Richard Golnik, Peter F. Stadler

TL;DR

This note investigates relevant cuts in weighted undirected graphs, showing that a cut is relevant exactly when it is a weight-minimal $u,v$-cut for some pair $u,v$ of distinct vertices, and that the collection of such cuts can be exponential in size. It leverages Gomory-Hu trees and PQ-DAG representations to develop polynomial-time enumeration strategies, notably restricting to PQ-DAGs on the edges of a Gomory-Hu tree for efficiency. The authors implement and compare several approaches (GUS-P, GUS-T, YEH) on large chemical-graph benchmarks, finding that GUS-T generally offers the best performance while maintaining versatility. The work has practical implications for graph-based genetic algorithms and chemical-space explorations, and provides open-source tooling for researchers to enumerate relevant cuts.

Abstract

The set of relevant cuts in a graph is the union of all minimum weight bases of the cut space. A cut is relevant if and only if it is the a minimum weight cut between two distinct vertices.

A Short Note on Relevant Cuts

TL;DR

This note investigates relevant cuts in weighted undirected graphs, showing that a cut is relevant exactly when it is a weight-minimal -cut for some pair of distinct vertices, and that the collection of such cuts can be exponential in size. It leverages Gomory-Hu trees and PQ-DAG representations to develop polynomial-time enumeration strategies, notably restricting to PQ-DAGs on the edges of a Gomory-Hu tree for efficiency. The authors implement and compare several approaches (GUS-P, GUS-T, YEH) on large chemical-graph benchmarks, finding that GUS-T generally offers the best performance while maintaining versatility. The work has practical implications for graph-based genetic algorithms and chemical-space explorations, and provides open-source tooling for researchers to enumerate relevant cuts.

Abstract

The set of relevant cuts in a graph is the union of all minimum weight bases of the cut space. A cut is relevant if and only if it is the a minimum weight cut between two distinct vertices.

Paper Structure

This paper contains 6 sections, 9 theorems, 2 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relevant Cuts
  4. Algorithmic Considerations
  5. Computational Experiments
  6. Concluding Remarks

Key Result

Lemma 2.1

Let $G$ be a connected graph, $s,t\in V(G)$ distinct vertices, and $\mathcal{D}$ a cut basis. Then $\mathcal{D}$ contains an $s,t$-cut.

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 8 more