Faster Algorithms for Parametric Global Minimum Cut Problems

TL;DR

This paper introduces faster, strongly polynomial algorithms for solving parametric global minimum cut problems, including finding breakpoints and maximum cut values, improving over traditional methods.

Contribution

The authors develop new algorithms that outperform naive Megiddo's parametric search for parametric min cut problems, showing the next breakpoint problem is easier than the max value problem.

Findings

Faster algorithms for parametric min cut problems

Strongly polynomial complexity achieved

Next breakpoint problem is easier than max value problem

Abstract

The parametric global minimum cut problem concerns a graph $G = (V,E)$ where the cost of each edge is an affine function of a parameter $\mu \in \mathbb{R}^d$ for some fixed dimension $d$. We consider the problems of finding the next breakpoint in a given direction, and finding a parameter value with maximum minimum cut value. We develop strongly polynomial algorithms for these problems that are faster than a naive application of Megiddo's parametric search technique. Our results indicate that the next breakpoint problem is easier than the max value problem.