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arXiv · 2601.00768

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

Mihail Stoian
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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.

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Mihail Stoian. 2026-01-02. Mind the Gap. Doubling Constant Parametrization of Weighted Problems: TSP, Max-Cut, and More. https://doi.org/10.4230/lipics.stacs.2026.29

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