Related papers: On the Hardness of Almost All Subset Sum Problems …

The Subset Sum problem asks whether a given set of $n$ positive integers contains a subset of elements that sum up to a given target $t$. It is an outstanding open question whether the $O^*(2^{n/2})$-time algorithm for Subset Sum by…

We prove that the subset sum problem has a polynomial time computable certificate of infeasibility for all $a$ weight vectors with density at most $1/(2n)$ and for almost all integer right hand sides. The certificate is branching on a…

We study the computational complexity of one of the particular cases of the knapsack problem: the subset sum problem. For solving this problem we consider one of the basic variants of the Branch-and-Bound method in which any sub-problem is…

The subset sum problem is known to be an NP-hard problem in the field of computer science with the fastest known approach having a run-time complexity of $O(2^{0.3113n})$. A modified version of this problem is known as the perfect sum…

We introduce and study four optimization problems that generalize the well-known subset sum problem. Given a node-weighted digraph, select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints…

Given a set (or multiset) S of n numbers and a target number t, the subset sum problem is to decide if there is a subset of S that sums up to t. There are several methods for solving this problem, including exhaustive search,…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the…

For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. We prove that if a sequence $B=\{11\leq b_1<b_2<\cdots\}$ satisfies $b_2=3b_1+5$, $b_3=3b_2+2$ and $b_{n+1}=3b_n+4b_{n-1}$ for all $n\geq 3$, then there is a sequence of…

This paper presents a deterministic algorithmic approach of exploring the solution space of the Subset Sum Problem. The algorithm presented is input-robust and structurally adaptive. Exploration is guided and narrows into areas in the…

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known…

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erd\H{o}s asked whether this implies…

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high…

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number $\beta$. We prove that if $\beta$ is real quadratic, then the number of partitions is always finite if and only if some conjugate of $\beta$…

Pilz's conjecture states that for any finite set $A=\{a_1,a_2,\dots,a_k\}$ of positive integers and positive integer $n$ in the union of the sets $\{a_1,2a_1,\dots,na_1\},\dots, \{a_k,2a_k,\dots,na_k\}$ (considered as a multiset) at least…

This work is about a partition problem which is an instance of the distance magic graph labeling problem. Given positive integers $n,k$ and $p_1\le p_2\le \cdots\le p_k$ such that $p_1+\cdots+p_k=n$ and $k$ divides $\sum_{i=1}^ni$, we study…

In the field of algorithmic analysis, one of the more well-known exercises is the subset sum problem. That is, given a set of integers, determine whether one or more integers in the set can sum to a target value. Aside from the brute-force…

Given $(a_1, \dots, a_n, t) \in \mathbb{Z}_{\geq 0}^{n + 1}$, the Subset Sum problem ($\mathsf{SSUM}$) is to decide whether there exists $S \subseteq [n]$ such that $\sum_{i \in S} a_i = t$. There is a close variant of the $\mathsf{SSUM}$,…

The modular subset sum problem consists of deciding, given a modulus $m$, a multiset $S$ of $n$ integers in $0..m-1$, and a target integer $t$, whether there exists a subset of $S$ with elements summing to $t \mod m $, and to report such a…