More than 40 years ago, Schroeppel and Shamir presented an algorithm that solves the Subset Sum problem for $n$ integers in time $O^*(2^{0.5n})$ and space $O^*(2^{0.25n})$. The time upper bound remains unbeaten, but the space upper bound has been improved to $O^*(2^{0.249999n})$ in a recent breakthrough paper by Nederlof and W\k{e}grzycki (STOC 2021). Their algorithm is a clever combination of a number of previously known techniques with a new reduction and a new algorithm for the Orthogonal Vectors problem. In this paper, we give two new algorithms for Subset Sum. We start by presenting an Arthur--Merlin algorithm: upon receiving the verifier's randomness, the prover sends an $n/4$-bit long proof to the verifier who checks it in (deterministic) time and space $O^*(2^{n/4})$. The simplicity of this algorithm has a number of interesting consequences: it can be parallelized easily; also, by enumerating all possible proofs, one recovers upper bounds on time and space for Subset Sum proved by Schroeppel and Shamir in 1979. As it is the case with the previously known algorithms for Subset Sum, our algorithm follows from an algorithm for $4$-SUM: we prove that, using verifier's coin tosses, the prover can prepare a $\log_2 n$-bit long proof verifiable in time $\tilde{O}(n)$. Another interesting consequence of this result is the following fine-grained lower bound: assuming that $4$-SUM cannot be solved in time $O(n^{2-\varepsilon})$ for all $\varepsilon>0$, Circuit SAT cannot be solved in time $O(g2^{(1-\varepsilon)n})$, for all $\varepsilon>0$. Then, we improve the space bound by Nederlof and W\k{e}grzycki to $O^*(2^{0.246n})$ and also simplify their algorithm and its analysis. We achieve this space bound by further filtering sets of subsets using a random prime number. This allows us to reduce an instance of Subset Sum to a larger number of instances of smaller size.

翻译：四十多年前，Schroeppel与Shamir提出了一种算法，能在时间$O^*(2^{0.5n})$和空间$O^*(2^{0.25n})$内解决包含$n$个整数的子集和问题。迄今为止，该时间上界尚未被超越，然而在Nederlof和Węgrzycki发表于STOC 2021的最新突破性论文中，空间上界已被改进至$O^*(2^{0.249999n})$。他们的算法巧妙结合了多项已知技术，并针对正交向量问题提出了一种新归约和新算法。本文提出了两种新的子集和问题算法。首先，我们给出一种Arthur-Merlin算法：验证方接收随机数后，证明方发送长度为$n/4$比特的证明，验证方可在（确定性）时间与空间$O^*(2^{n/4})$内完成验证。该算法的简洁性产生了若干有趣推论：它易于并行化；此外，通过枚举所有可能证明，可恢复Schroeppel与Shamir于1979年证明的子集和问题时间与空间上界。与现有子集和问题算法类似，我们的算法源于对$4$-SUM问题的求解：我们证明，利用验证方的掷币结果，证明方可以准备长度为$\log_2 n$比特的证明，并在时间$\tilde{O}(n)$内完成验证。该结果的另一个有趣推论是如下细粒度下界：若$4$-SUM无法在时间$O(n^{2-\varepsilon})$（对所有$\varepsilon>0$）内求解，则电路SAT也无法在时间$O(g2^{(1-\varepsilon)n})$（对所有$\varepsilon>0$）内求解。接着，我们将Nederlof与Węgrzycki的空间界改进至$O^*(2^{0.246n})$，同时简化了其算法及分析过程。我们通过使用随机素数对子集进行进一步筛选来实现这一空间界。这使我们能够将子集和问题的实例归约为更多数量、更小规模的子实例。