arXiv subjects

Mahesh Sreekumar Rajasree

Publications and source records attributed to Mahesh Sreekumar Rajasree.

4 recordsLinked to original sources

arXiv2026

Pigeonhole Equal Subset Sum: Subexponential Algorithm, Tight Lower Bounds, and Average-Case Analysis

The Equal Subset Sum problem asks, given n positive integers bounded by M, to find two distinct non-empty subsets with the same sum. We study this problem in the regime $M = 2^{o(n)}$, where the pigeonhole principle guarantees that a solution exists, and the main challenge is to find one efficiently. A straightforward dynamic-programming algorithm runs in time poly(n).$M$. After a sequence of improvements, the current best algorithm, due to Jin, Williams, and Zhang (ESA 2025), achieves running time poly(n).$M^{1/3}$. In this work, we give the first poly(n).$M^{o(1)}$-time algorithm for Equal Subset Sum in the regime $M = 2^{o(n)}$. We complement this algorithmic result with a tight conditional lower bound showing that any substantial improvement would require a major breakthrough in approximation algorithms for lattice problems, building on recent techniques of Vafa and Vaikuntanathan (STOC 2025). Finally, we show that our techniques yield an asymptotically faster algorithm for the regime when $M = Θ(2^{n})$: a poly(n)$\cdot 2^{n/4}$-time algorithm in the average case.

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arXiv2026

On the Largest Cardinality Subset Sum

We study Largest Cardinality Subset Sum (LCSS): given $n$ nonnegative integers and a target $t$, compute a subset summing to $t$ with maximum cardinality. We also consider its all-targets version (ATLCSS), which asks for this maximum for every sum $s\in\{1,\ldots,t\}$. The LCSS problem sits squarely between the subset sum and the knapsack problem. A straightforward dynamic-programming algorithm runs in time $O(nt)$ and solves ATLCSS, and using Jin and Bringmann (both in STOC 2024, independent work) $0$-$1$ knapsack results we achieve a $\widetilde{O}(n + a_{\max}^2)$-time deterministic algorithm for LCSS. Our main result is a randomized $\widetilde{O}(n+\sqrt{n}\,t)$-time algorithm for LCSS. Then, we complement this result with a randomized $\widetilde{O}(n+t^{3/2})$-time algorithm for ATLCSS. Together, our two algorithms give a $\widetilde{O}(n+t\min\{\sqrt{n},\sqrt{t}\})$-time algorithm for LCSS. As an application of ATLCSS, we solve the search version of Projection Subset Sum, introduced by Dutta and Rajasree (SYNSAC 2023), in $\widetilde{O}(n+t^{3/2})$ time for every fixed $\ell_p$ norm with $p\geq 1$. For $\ell_1$, we obtain $\widetilde{O}(n+t)$ time, removing the unique-representation assumption required by their algorithm. Both algorithms output an optimal modified vector and a witnessing subset.

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arXiv2024

On the Maximum Distance Sublattice Problem and Closest Vector Problem

In this paper, we introduce the Maximum Distance Sublattice Problem (MDSP). We observed that the problem of solving an instance of the Closest Vector Problem (CVP) in a lattice $\mathcal{L}$ is the same as solving an instance of MDSP in the dual lattice of $\mathcal{L}$. We give an alternate reduction between the CVP and MDSP. This alternate reduction does not use the concept of dual lattice.

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arXiv2022

Efficient reductions and algorithms for variants of Subset Sum

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 $\tilde{O}(k\cdot (n+t))$ time deterministic algorithm, which finds the hamming weight of all the realisable sets for a subset sum instance. We also give a 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).

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