Approximate Min-Sum Subset Convolution

TL;DR

This work tackles the NP-hard problem of min-sum subset convolution by introducing a $(1+\varepsilon)$-approximate variant that supports exponential-time approximation for a broad class of hard optimization problems. It bridges sequence convolution techniques and subset convolutions, delivering both a weakly- and a strongly-polynomial approximation framework, with runtimes of $\widetilde{O}(2^n \log M / \varepsilon)$ and $\widetilde{O}(2^{3n/2} / \sqrt{\varepsilon})$ respectively, and an exact $\widetilde{O}(2^{3n/2})$-time min-max subset convolution. The results enable out-of-the-box $(1+\varepsilon)$-approximation schemes for problems such as minimum-cost $k$-coloring and prize-collecting Steiner tree, independent of the input range $M$, by leveraging a strongly-polynomial approximate min-sum subset convolution. This advances tropical subset convolutions after a two-decade gap and expands the algorithmic toolkit for convolution-based hardness in both theory and practical contexts.

Abstract

Exponential-time approximation has recently gained attention as a practical way to deal with the bitter NP-hardness of well-known optimization problems. We study for the first time the $(1 + \varepsilon)$-approximate min-sum subset convolution. This enables exponential-time $(1 + \varepsilon)$-approximation schemes for problems such as minimum-cost $k$-coloring, the prize-collecting Steiner tree, and many others in computational biology. Technically, we present both a weakly- and strongly-polynomial approximation algorithm for this convolution, running in time $\widetilde O(2^n \log M / \varepsilon)$ and $\widetilde O(2^\frac{3n}{2} / \sqrt{\varepsilon})$, respectively. Our work revives research on tropical subset convolutions after nearly two decades.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: Extension of approx_min_plus
• Theorem 5
• Theorem 6
• Theorem 6
• Theorem 6
• Lemma 7: Sum-to-Max Covering approx_min_plus
• Lemma 7