Budget-constrained cut problems

TL;DR

This paper introduces a budget-constrained extension of classic min-cut and max-cut problems, proving NP-completeness and developing exact and heuristic algorithms, with extensive computational evaluation.

Contribution

It extends traditional cut problems by incorporating a cost constraint, proving NP-completeness, and proposing algorithms including a Lagrangian relaxation method.

Findings

NP-completeness of the budget-constrained problems

Lagrangian relaxation algorithm often yields optimal solutions

Extensive computational experiments demonstrate algorithm effectiveness

Abstract

The minimum and maximum cuts of an undirected edge-weighted graph are classic problems in graph theory. While the Min-Cut Problem can be solved in P, the Max-Cut Problem is NP-Complete. Exact and heuristic methods have been developed for solving them. For both problems, we introduce a natural extension in which cutting an edge induces a cost. Our goal is to find a cut that minimizes the sum of the cut weights but, at the same time, restricts its total cut cost to a given budget. We prove that both restricted problems are NPComplete and we also study some of its properties. Finally, we develop exact algorithms to solve both as well as a non-exact algorithm for the min-cut case based on a Lagreangean relaxation that generally provides optimal solutions. Their performance is reported by an extensive computational experience.