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- W4226508266 abstract "Given $(a_1, dots, a_n, t) in mathbb{Z}_{geq 0}^{n + 1}$, the Subset Sum problem ($mathsf{SSUM}$) is to decide whether there exists $S subseteq [n]$ such that $sum_{i in S} a_i = t$. There is a close variant of the $mathsf{SSUM}$, called $mathsf{Subset~Product}$. Given positive integers $a_1, ..., a_n$ and a target integer $t$, the $mathsf{Subset~Product}$ problem asks to determine whether there exists a subset $S subseteq [n]$ such that $prod_{i in S} a_i=t$. There is a pseudopolynomial time dynamic programming algorithm, due to Bellman (1957) which solves the $mathsf{SSUM}$ and $mathsf{Subset~Product}$ in $O(nt)$ time and $O(t)$ space. In the first part, we present {em search} algorithms for variants of the Subset Sum problem. Our algorithms are parameterized by $k$, which is a given upper bound on the number of realisable sets (i.e.,~number of solutions, summing exactly $t$). We show that $mathsf{SSUM}$ with a unique solution is already NP-hard, under randomized reduction. This makes the regime of parametrized algorithms, in terms of $k$, very interesting. Subsequently, we present an $tilde{O}(kcdot (n+t))$ time deterministic algorithm, which finds the hamming weight of all the realisable sets for a subset sum instance. We also give a poly$(knt)$-time and $O(log(knt))$-space deterministic algorithm that finds all the realisable sets for a subset sum instance. In the latter part, we present a simple and elegant randomized $tilde{O}(n + t)$ time algorithm for $mathsf{Subset~Product}$. Moreover, we also present a poly$(nt)$ time and $O(log^2 (nt))$ space deterministic algorithm for the same. We study these problems in the unbounded setting as well. Our algorithms use multivariate FFT, power series and number-theoretic techniques, introduced by Jin and Wu (SOSA'19) and Kane (2010)." @default.
- W4226508266 created "2022-05-05" @default.
- W4226508266 creator A5055791894 @default.
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- W4226508266 date "2021-12-21" @default.
- W4226508266 modified "2023-09-24" @default.
- W4226508266 title "Efficient reductions and algorithms for variants of Subset Sum" @default.
- W4226508266 doi "https://doi.org/10.48550/arxiv.2112.11020" @default.
- W4226508266 hasPublicationYear "2021" @default.
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