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

TL;DR

The paper addresses the persistent gap between weighted NP-hard problems and their unweighted speedups, which is often exacerbated by pseudo-polynomial dependence on the largest weight. It introduces a doubling-constant, weight-parametrized framework grounded in constructive Freiman’s theorem to convert weights into polynomially bounded integers, enabling time that matches the unweighted algorithms for problems on the min-plus and max-plus semirings. The core contribution is a meta-algorithm that, for problems with additive objective and small-doubling weight sets, yields $O_{\mathcal{C}}^*(T(n))$ time whenever the unweighted version can be solved in time $T(n)$ by an algebraic algorithm; this is instantiated for TSP, Weighted Max-Cut, edge-weighted $k$-clique, and Minimum Steiner Tree, among others. The results highlight a new avenue where additive combinatorics tools can dramatically broaden the scope of fast exact algorithms beyond unweighted cases, with potential extensions to polynomial-time problems and faster min-plus-type computations.

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 Węgrzycki [ESA 2024] before applying the polynomial embedding.

Theorems & Definitions (26)

• Corollary 1.0
• Theorem 1.1: Meta-algorithm
• Corollary 1.1
• Corollary 1.1
• Corollary 1.1
• Definition 2.1: Solution polynomial
• Theorem 2.2: Constructive Freiman's Theorem esa-paper
• Lemma 3.0
• Theorem 3.1: Meta-algorithm
• proof