Abstract
Given positive integers \(a_1, \ldots , a_n\) and a target integer t, the \(\mathsf {Subset~Product}\) problem asks to determine whether there exists a subset \(S \subseteq [n]\) such that \(\prod _{i \in S} a_i=t\). It differs from the Subset Sum problem where the multiplication operation is replaced by addition. There is a pseudopolynomial-time dynamic programming algorithm which solves the \(\mathsf {Subset~Product}\) in O(nt) time and \(\varOmega (t)\) space.
In this paper, we present a simple and elegant randomized algorithm for \(\mathsf {Subset~Product}\) in \(\tilde{O}(n + t^{o(1)})\) expected-time. Moreover, we also present a \(\textsf{poly}(nt)\) time and \(O(\log ^2 (nt))\) space deterministic algorithm.
In fact, we solve a more general problem called the \(\textsf{SimulSubsetSum}\). This problem was introduced by Kane 2010. Given k instances of Subset Sum, it asks to decide whether there is a ‘common’ solution to all the instances. Kane gave a logspace algorithm for this problem. We show a polynomial-time reduction from \(\mathsf {Subset~Product}\) to \(\textsf{SimulSubsetSum}\) and also give efficient algorithm for the latter. Our algorithms use multivariate FFT, power series and number-theoretic techniques, introduced by Jin and Wu (SOSA 2019) and Kane (2010).
The full version is available at https://drive.google.com/file/d/1xUX29eVZ_2J1zv062dH4V8Yc8NhqwV1X/view?usp=share_link.
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Notes
- 1.
Expected time complexity is \(\exp (O(\sqrt{\log t \log \log t}))\), which is smaller than \(t^{O(1/\sqrt{\log \log t})} =t^{o(1)}\), which will be the time taken in the next step. Moreover, we are interested in randomized algorithms, hence expected run-time is.
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Dutta, P., Rajasree, M.S. (2023). Efficient Reductions and Algorithms for Subset Product. In: Bagchi, A., Muthu, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2023. Lecture Notes in Computer Science, vol 13947. Springer, Cham. https://doi.org/10.1007/978-3-031-25211-2_1
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