arXiv math/01122572001-12-22math.NTcs.CCmath.CO

The computational complexity of the local postage stamp problem

Jeffrey Shallit

英文摘要：

The well-studied local postage stamp problem (LPSP) is the following: given a positive integer k, a set of postive integers 1 = a1 < a2 < ... < ak and an integer h >= 1, what is the smallest positive integer which cannot be represented as a linear combination x1 a1 + ... + xk ak where x1 + ... + xk <= h and each xi is a non-negative integer? In this note we prove that LPSP is NP-hard under Turing reductions, but can be solved in polynomial time if k is fixed.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The Postage Stamp Problem - Formulae and proof for the case of 3 denominations

A_k = {1, a_2, ..., a_k} is an h-basis for n if every positive integer not exceeding n can be expressed as the sum of no more than h values a_i; an extremal h-basis A_k is one for which n is as large as possible. Computing such extremal bases has become known as the Postage Stamp Problem. In 1968 Gerd Hofmeister published formulae for solutions to the Postage Stamp Problem with three denominations (h = 3). This paper presents an alternative approach developed independently by the author 25 years later.

2013-11-05math.COmath.NT

Addition Chains Meet Postage Stamps: Reducing the Number of Multiplications

We introduce stamp chains. A stamp chain is a finite integer sequence that is both an addition chain and an additive 2-basis, i.e., a solution to the postage stamp problem. We provide a simple method for converting known postage stamp solutions of length k into stamp chains of length k+1. Using stamp chains, we construct an algorithm that computes u(x^i) for i=1,...,n in less than n-1 multiplications, if u is a function that can be computed at zero cost, and if there exists another zero-cost function v such that v(a,b) = u(ab). This can substantially reduce the computational cost of repeated m…

2014-02-03math.COmath.NT

The Frobenius postage stamp problem, and beyond

Let $A$ be a finite subset of $\mathbb{Z}^n$, which generates $\mathbb{Z}^n$ additively. We provide a precise description of the $N$-fold sumsets $NA$ for $N$ sufficiently large, with some explicit bounds on "sufficiently large."

2020-04-27math.COmath.NT

The Postage Stamp Problem and Essential Subsets in Integer Bases

Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h,k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that E(h,k) = Θ_{k} ([h^{k}/\log h]^{1/(k+1)}). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known "postage stamp problem" in fini…

2008-07-02math.COmath.NT

Compact symmetric solutions to the postage stamp problem

We derive lower and upper bounds on possible growth rates of certain sets of positive integers $A_k=\{1= a_1 < a_2 < ... < a_{k}\}$ such that all integers $n\in \{0, 1, 2, ..., ka_{k}\}$ can be represented as a sum of no more than $k$ elements of $A_k$, with repetition.

2008-10-14math.NT

Early Pruning in the Restricted Postage Stamp Problem

A set of non-negative integers is an additive basis with range $n$, if its sumset covers all consecutive integers from 0 to $n$, but not $n+1$. If the range is exactly twice the largest element of the basis, the basis is restricted. Restricted bases have important special properties that facilitate efficient searching. With the help of these properties, we have previously listed the extremal restricted bases up to length $k = 41$. Here, with a more prudent use of the properties, we present an improved search algorithm and list all extremal restricted bases up to $k = 47$.

2015-03-11math.NT

更多相关论文收起更多论文

大邮票数量下邮票问题的改进界

Improved bounds on the postage stamp problem for large numbers of stamps

针对集合$\{1,2,\cdots,n\}$的$h$-折叠基最小基数$F_h(n)$，本文通过概率方法结合Berry-Esseen定理改进下界，利用Jia-Shen构造确立非平凡上界，大幅优化了$h$足够大时的渐近界。

2025-11-11math.COmath.NT

The continuous postage stamp problem

For a real set $A$ consider the semigroup $S(A)$, additively generated by $A$; that is, the set of all real numbers representable as a (finite) sum of elements of $A$. If $A \subset (0,1)$ is open and non-empty, then $S(A)$ is easily seen to contain all sufficiently large real numbers, and we let $G(A) := \sup \{u \in R \colon u \notin S(A) \}$. Thus, $G(A)$ is the smallest number with the property that any $u>G(A)$ is representable as indicated above. We show that if the measure of $A$ is large, then $G(A)$ is small; more precisely, writing for brevity $α:= \mes A$ we have $$ G(A) \le (1-α) \…

2009-11-27math.NT