AI 中文总结
研究多维密集子集和的加法结构，通过证明\(\mathcal{S}(A)\)的相关性质降低密度阈值、扩大整点区域，且与一维情况匹配，进而开发出\(\tilde{O}(n)\)时间算法解决多维密集子集和问题。
AI 中文摘要
我们研究多维中密集子集和的加法结构，并利用该结构为密集子集和问题开发高效算法。给定在\(d\)维超矩形\([N_1]\times [N_2]\times\cdots\times [N_d]\)中的\(n\)个向量的集合\(A\)，研究其所有子集和的集合\(\mathcal{S}(A)\)的结构。聚焦于\(n \gg \sqrt{\Phi}\)且\(\Phi = N_1 \times \cdots \times N_d\)的密集情况。表明对于任何常数\(d\geq 1\)，若\(n \gg \sqrt{\Phi}\)，则\(\mathcal{S}(A)\)包含多维中的长广义级数。若进一步没有非平凡格能包含多数\(A\)，则\(\mathcal{S}(A)\)包含带状体中的所有整点。与之前\(d \geq 2\)的结果相比，显著降低了密度阈值并扩大了整点属于\(\mathcal{S}(A)\)的区域，还与一维情况的界匹配。利用组合结果，还开发了多维密集子集和问题的\(\tilde{O}(n)\)时间算法。
英文摘要
We study the additive structure of dense subset sum in multi-dimension, and use the structure to develop efficient algorithms for the dense subset sum problem. More precisely, given a set $A$ of $n$ vectors in the $d$-dimensional hyperrectangle $[N_1]\times [N_2]\times\cdots\times [N_d]$, we study the structure of $\mathcal{S}(A)$, which is the set of all subset sums of $A$. We focus on the dense regime of the problem where $n \gg \sqrtΦ$ and $Φ= N_1 \times \cdots \times N_d$. We show that for any constant $d\geq 1$, if $n \gg \sqrtΦ$, then $\mathcal{S}(A)$ contains a long generalized progression in multi-dimension. If we further have that no non-trivial lattice can contain the majority of $A$, then $\mathcal{S}(A)$ contains all the integer points in the zonotope $\{x_1\vec{a}_1 + \cdots + x_n\vec{a}_n: o(1)\leq x_j \leq 1-o(1), x_j \in \mathbb{R}\}$. Compared to the previous results for $d \geq 2$, our result significantly reduces the density threshold and enlarges the region inside which all the integer points belong to $\mathcal{S}(A)$. Also, it matches the bound for the 1-dimensional case. Using our combinatorics result, we also develop an $\tilde{O}(n)$-time algorithm for the dense subset sum problem in multi-dimension.
CommentsA preliminary version to appear in FOCS'26
