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- W2198078872 abstract "The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a faster, $O^*(2^{(0.5-delta)n})$-time algorithm, with some constant $delta > 0$. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size $beta$, defined as the largest number of subsets of the $n$ input integers that yield the same sum. For every $epsilon > 0$ we give a truly faster algorithm for instances with $beta leq 2^{(0.5-epsilon)n}$, as well as instances with $beta geq 2^{0.661n}$. Consequently, we also obtain a characterization in terms of the popular density parameter $n/log_2 t$: if all instances of density at least $1.003$ admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from novel combinations of earlier algorithms for Subset Sum and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory." @default.
- W2198078872 created "2016-06-24" @default.
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- W2198078872 date "2015-08-25" @default.
- W2198078872 modified "2023-09-27" @default.
- W2198078872 title "Dense Subset Sum may be the hardest" @default.
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