New algorithms for Steiner tree reoptimization

TL;DR

This paper introduces polynomial-time approximation schemes for Steiner tree reoptimization, significantly improving previous constant-factor algorithms, especially for cases where edge costs decrease, and establishes their optimality under P≠NP.

Contribution

The paper develops novel techniques to create polynomial-time approximation schemes for Steiner tree reoptimization, surpassing all prior constant-factor algorithms and addressing previously unresolved cases.

Findings

Established polynomial-time approximation schemes for Steiner tree reoptimization.

Achieved the first approximation schemes for scenarios with decreasing edge costs.

Proved the optimality of these schemes under P≠NP.

Abstract

{\em Reoptimization} is a setting in which we are given an (near) optimal solution of a problem instance and a local modification that slightly changes the instance. The main goal is that of finding an (near) optimal solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as {\em Steiner tree reoptimization}. Steiner tree reoptimization is a collection of strongly NP-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decade. In this paper we improve upon all these results by developing a novel technique that allows us to design {\em polynomial-time approximation schemes}. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP.