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.