arXiv subjects

Tingwei Hu

Publications and source records attributed to Tingwei Hu.

3 recordsLinked to original sources

arXiv2026

A Sharp Upper Bound for the Frobenius Number

Let $0<a_1<\cdots<a_n$ be integers with $n\ge 2$ and $\gcd(a_1,\ldots,a_n)=1$. Their Frobenius number $F(a_1,\ldots,a_n)$ is the largest integer that cannot be expressed as a nonnegative integer combination of $a_1,\ldots,a_n$. Erdős and Graham proved the classical bound $F(a_1,\ldots,a_n)\le 2a_{n-1}\lfloor a_n/n\rfloor-a_n$, which depends on the two largest generators. Dixmier subsequently proved their conjecture $F(a_1,\ldots,a_n)\le a_n^2/(n-1)$, a bound depending only on $a_n$ and $n$ that removes the factor $2$ from the corresponding quadratic estimate. Following the Erdős--Graham approach, we remove the factor $2$ while retaining the dependence on $a_{n-1}$, and prove $F(a_1,\ldots,a_n)\le a_{n-1}\lceil a_n/(n-1)\rceil-a_n-1$. This bound can be substantially smaller than Dixmier's when $a_{n-1}$ is small relative to $a_n$. For every $n\ge3$, we also construct infinitely many instances satisfying the reverse inequality, which shows that the bound is sharp.

math.NT↗

arXiv2026

Dense Subset Sum in Multi-Dimension

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.

cs.DS↗

arXiv2026

Near-Tight Approximation Algorithms for Bottleneck Multiple Knapsack Problems

In the bottleneck multiple knapsack problem, we are given a set of items and a set of knapsacks, where each item has a profit and a weight, and each knapsack has a capacity. Our goal is to assign items to knapsacks so as to maximize the minimum profit received by any knapsack subject to the capacity constraint. When all knapsacks have identical capacity, we give a $(\frac{2}{3} - \varepsilon)$-approximation algorithm for any constant $\varepsilon > 0$. This result almost matches the $(\frac{2}{3} + \varepsilon)$ inapproximability bound for the bottleneck multiple subset sum problem (Caprara et al., 2000). When the knapsacks can have arbitrary capacities, we propose a $(\frac{1}{2} - \varepsilon)$-approximation algorithm for any constant $\varepsilon > 0$. We also prove a hardness bound of $(\frac{1}{2} + \varepsilon)$ for any constant $\varepsilon > 0$.

cs.DS↗