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 Węgrzycki [ESA 2024] before applying the polynomial embedding.

翻译：尽管已有大量研究，困难的加权问题仍难以在教科书解法基础上实现超多项式改进。另一方面，这些问题的无权版本近期已取得突破性加速进展。目前，将无权版本算法应用于加权版本的唯一途径是对输入权重进行多项式嵌入。然而，这会为运行时间引入伪多项式因子，导致算法在面对任意权重实例时失去实用性。本文提出一种重新利用无权问题算法的新方法。具体而言，我们证明在$(\min, +)$与$(\max, +)$半环上运行的若干经典NP难问题（如TSP、加权最大割、边加权$k$-团）的时间复杂度，当输入权重集合具有较小倍增常数时，与其无权版本的时间复杂度成正比。我们通过元算法实现这一目标：首先利用Randolph与Węgrzycki在[ESA 2024]中提出的构造性Freiman定理将输入权重转换为多项式有界整数，再实施多项式嵌入。