Long Arithmetic Progressions in Sumsets and Subset Sums: Constructive Proofs and Efficient Witnesses

2025-03-25

AI Summary

This paper addresses the non-constructive nature of classical results in additive combinatorics—such as Sárközy’s theorem—concerning the existence of long arithmetic progressions (APs) in sumsets and subset sums. We present the first near-linear-time constructive algorithm: given a set $A subseteq [n]$, it explicitly outputs, in $ ilde{O}(n)$ time, an arithmetic progression of length $Omega(n)$ along with an explicit representation of each term as a sum of elements from $A$. Our approach integrates combinatorial number-theoretic constructions, greedy compression encoding, modular arithmetic optimization, and dynamic representation tracking. As a consequence, we obtain the first efficient search algorithm for Subset Sum: for dense instances, it solves the problem in $ ilde{O}(n)$ time; for the unbounded case, it achieves $O(n log a_{max})$ time when the target $t geq c a_{max}^2 / n$, thereby resolving—constructively and efficiently—the long-standing search-version hardness of Subset Sum.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Applications of cryptography

Abstract

Existence of long arithmetic progression in sumsets and subset sums has been studied extensively in the field of additive combinatorics. These additive combinatorics results play a central role in the recent progress of fundamental problems in theoretical computer science including Knapsack and Subset Sum. The non-constructiveness of relevant additive combinatorics results affects their application in algorithms. In particular, several additive combinatorics-based algorithms for Subset Sum work only for the decision version of the problem, but not for the search version. We provide constructive proofs for finite addition theorems [S'arkH{o}zy'89 '94], which are fundamental results in additive combinatorics concerning the existence of long arithmetic progression in sumsets and subset sums. Our constructive proofs yield a near-linear time algorithm that returns an arithmetic progression explicitly, and moreover, for each term in the arithmetic progression, it also returns its representation as the sum of elements in the base set. As an application, we obtain an $ ilde{O}(n)$-time algorithm for the search version of dense subset sum now. Another application of our result is Unbounded Subset Sum, where each input integer can be used an infinite number of times. A classic result on the Frobenius problem [ErdH{o}s and Graham '72] implies that for all $t geq 2a^2_{max}/n$, the decision version can be solved trivially in linear time. It remains unknown whether the search version can be solved in the same time. Our result implies that for all $t geq ca^2_{max}/n$ for some constant $c$, a solution for Unbounded Subset Sum can be obtained in $O(n log a_{max})$ time.

Problem

• Constructive proofs for arithmetic progressions in sumsets
• Efficient algorithm for search version of Subset Sum
• Solution for Unbounded Subset Sum in near-linear time

Innovation

• Constructive proofs for finite addition theorems
• Near-linear time algorithm for arithmetic progression
• Efficient solution for Unbounded Subset Sum