Algebraic Algorithms for Variants of Subset Sum

Abstract

Given $(a_1, \dots , a_n, t) \in \mathbb {Z}_{\ge 0}^{n + 1}$, the Subset Sum problem ($\mathsf {SSUM}$) is to decide whether there exists $S \subseteq [n]$ such that $\sum _{i \in S} a_i = t$. Bellman (1957) gave a pseudopolynomial time dynamic programming algorithm which solves the Subset Sum in O(nt) time and O(t) space.

In this work, we present 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 $\mathsf {NP}$-hard, under randomized reduction. This makes the regime of parametrized algorithms, in terms of k, very interesting.

Subsequently, we present an $\tilde{O}(k\cdot (n+t))$ time deterministic algorithm, which finds the hamming weight of all the realisable sets for a subset sum instance. We also give a ${\mathsf {poly}}(knt)$-time and $O(\log (knt))$-space deterministic algorithm that finds all the realisable sets for a subset sum instance. Our algorithms use analytic and number-theoretic techniques.

The full version is available at this link.

P. Dutta—Supported by Google PhD Fellowship.

M. S. Rajasree—Supported by Prime Minister’s Research Fellowship.