In the classical Subset Sum problem we are given a set $X$ and a target $t$, and the task is to decide whether there exists a subset of $X$ which sums to $t$. A recent line of research has resulted in $\tilde{O}(t)$-time algorithms, which are (near-)optimal under popular complexity-theoretic assumptions. On the other hand, the standard dynamic programming algorithm runs in time $O(n \cdot |\mathcal{S}(X,t)|)$, where $\mathcal{S}(X,t)$ is the set of all subset sums of $X$ that are smaller than $t$. Furthermore, all known pseudopolynomial algorithms actually solve a stronger task, since they actually compute the whole set $\mathcal{S}(X,t)$. As the aforementioned two running times are incomparable, in this paper we ask whether one can achieve the best of both worlds: running time $\tilde{O}(|\mathcal{S}(X,t)|)$. In particular, we ask whether $\mathcal{S}(X,t)$ can be computed in near-linear time in the output-size. Using a diverse toolkit containing techniques such as color coding, sparse recovery, and sumset estimates, we make considerable progress towards this question and design an algorithm running in time $\tilde{O}(|\mathcal{S}(X,t)|^{4/3})$. Central to our approach is the study of top-$k$-convolution, a natural problem of independent interest: given sparse polynomials with non-negative coefficients, compute the lowest $k$ non-zero monomials of their product. We design an algorithm running in time $\tilde{O}(k^{4/3})$, by a combination of sparse convolution and sumset estimates considered in Additive Combinatorics. Moreover, we provide evidence that going beyond some of the barriers we have faced requires either an algorithmic breakthrough or possibly new techniques from Additive Combinatorics on how to pass from information on restricted sumsets to information on unrestricted sumsets.

中文翻译：

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Top-k-Convolution 和对近线性输出敏感子集和的探索

在经典的子集和问题中，我们给定了一个集合 $X$ 和一个目标 $t$，任务是确定是否存在一个 $X$ 的子集，其总和为 $t$。最近的一系列研究产生了 $\tilde{O}(t)$-time 算法，这些算法在流行的复杂性理论假设下是（接近）最优的。另一方面，标准动态规划算法在 $O(n \cdot |\mathcal{S}(X,t)|)$ 中运行，其中 $\mathcal{S}(X,t)$ 是集合小于 $t$ 的 $X$ 的所有子集和。此外，所有已知的伪多项式算法实际上解决了更强大的任务，因为它们实际上计算了整个集合 $\mathcal{S}(X,t)$。由于前面提到的两个运行时间是不可比的，本文我们问是否可以做到两全其美：运行时间 $\tilde{O}(|\mathcal{S}(X,t)|)$。特别是，我们问 $\mathcal{S}(X,t)$ 是否可以在输出大小的近线性时间内计算。使用包含颜色编码、稀疏恢复和总集估计等技术的各种工具包，我们在这个问题上取得了相当大的进展，并设计了一个及时运行的算法 $\tilde{O}(|\mathcal{S}(X,t) |^{4/3})$。我们的方法的核心是研究 top-$k$-卷积，这是一个具有独立兴趣的自然问题：给定具有非负系数的稀疏多项式，计算其乘积的最低 $k$ 非零单项式。我们设计了一个在时间 $\tilde{O}(k^{4/3})$ 中运行的算法，结合了 Additive Combinatorics 中考虑的稀疏卷积和 sumset 估计。而且，