Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More

TL;DR

This paper presents a novel meta-algorithm that efficiently adapts unweighted algorithms for weighted problems like TSP and Max-Cut when input weights have small doubling, avoiding pseudo-polynomial complexity issues.

Contribution

It introduces a new approach using Freiman's theorem to convert weights into polynomially bounded integers, enabling faster algorithms for weighted problems with small doubling weights.

Findings

Complexity proportional to unweighted versions for small doubling weights

Avoids pseudo-polynomial complexity in weighted problems

Applicable to problems like TSP, Max-Cut, and k-Clique

Abstract

Despite much research, hard weighted problems still resist super-polynomial improvements over their textbook solution. On the other hand, the unweighted versions of these problems have recently witnessed the sought-after speedups. Currently, the only way to repurpose the algorithm of the unweighted version for the weighted version is to employ a polynomial embedding of the input weights. This, however, introduces a pseudo-polynomial factor into the running time, which becomes impractical for arbitrarily weighted instances. In this paper, we introduce a new way to repurpose the algorithm of the unweighted problem. Specifically, we show that the time complexity of several well-known NP-hard problems operating over the $(\min, +)$ and $(\max, +)$ semirings, such as TSP, Weighted Max-Cut, and Edge-Weighted $k$-Clique, is proportional to that of their unweighted versions when the set of input weights has small doubling. We achieve this by a meta-algorithm that converts the input weights into polynomially bounded integers using the recent constructive Freiman's theorem by Randolph and W\k{e}grzycki [ESA 2024] before applying the polynomial embedding.