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Approximate Min-Sum Subset Convolution (2404.11364v3)

Published 17 Apr 2024 in cs.DS

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 $\widetilde O(2n \log M / \varepsilon)$ and $\widetilde O(2\frac{3n}{2} / \sqrt{\varepsilon})$, respectively. Our work revives research on tropical subset convolutions after nearly two decades.

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