Derandomizing Pseudopolynomial Algorithms for Subset Sum
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909981061677056 |
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| author | Chan, Timothy M. |
| author_facet | Chan, Timothy M. |
| contents | We reexamine the classical subset sum problem: given a set $X$ of $n$ positive integers and a number $t$, decide whether there exists a subset of $X$ that sums to $t$; or more generally, compute the set $\mbox{out}$ of all numbers $y\in\{0,\ldots,t\}$ for which there exists a subset of $X$ that sums to $y$. Standard dynamic programming solves the problem in $O(tn)$ time. In SODA'17, two papers appeared giving the current best deterministic and randomized algorithms, ignoring polylogarithmic factors: Koiliaris and Xu's deterministic algorithm runs in $\widetilde{O}(t\sqrt{n})$ time, while Bringmann's randomized algorithm runs in $\widetilde{O}(t)$ time. We present the first deterministic algorithm running in $\widetilde{O}(t)$ time.
Our technique has a number of other applications: for example, we can also derandomize the more recent output-sensitive algorithms by Bringmann and Nakos [STOC'20] and Bringmann, Fischer, and Nakos [SODA'25] running in $\widetilde{O}(|\mbox{out}|^{4/3})$ and $\widetilde{O}(|\mbox{out}|\sqrt{n})$ time, and we can derandomize a previous fine-grained reduction from 0-1 knapsack to min-plus convolution by Cygan et al. [ICALP'17]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01390 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Derandomizing Pseudopolynomial Algorithms for Subset Sum Chan, Timothy M. Data Structures and Algorithms We reexamine the classical subset sum problem: given a set $X$ of $n$ positive integers and a number $t$, decide whether there exists a subset of $X$ that sums to $t$; or more generally, compute the set $\mbox{out}$ of all numbers $y\in\{0,\ldots,t\}$ for which there exists a subset of $X$ that sums to $y$. Standard dynamic programming solves the problem in $O(tn)$ time. In SODA'17, two papers appeared giving the current best deterministic and randomized algorithms, ignoring polylogarithmic factors: Koiliaris and Xu's deterministic algorithm runs in $\widetilde{O}(t\sqrt{n})$ time, while Bringmann's randomized algorithm runs in $\widetilde{O}(t)$ time. We present the first deterministic algorithm running in $\widetilde{O}(t)$ time. Our technique has a number of other applications: for example, we can also derandomize the more recent output-sensitive algorithms by Bringmann and Nakos [STOC'20] and Bringmann, Fischer, and Nakos [SODA'25] running in $\widetilde{O}(|\mbox{out}|^{4/3})$ and $\widetilde{O}(|\mbox{out}|\sqrt{n})$ time, and we can derandomize a previous fine-grained reduction from 0-1 knapsack to min-plus convolution by Cygan et al. [ICALP'17]. |
| title | Derandomizing Pseudopolynomial Algorithms for Subset Sum |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2601.01390 |