A Faster Algorithm for Pigeonhole Equal Sums

An important area of research in exact algorithms is to solve Subset-Sum-type problems faster than meet-in-middle. In this paper we study Pigeonhole Equal Sums, a total search problem proposed by Papadimitriou (1994): given \(n\) positive integers \(w_1,\dots,w_n\) of total sum \(\sum_{i=1}^n w_i < 2^n-1\), the task is to find two distinct subsets \(A, B \subseteq [n]\) such that \(\sum_{i\in A}w_i=\sum_{i\in B}w_i\). Similar to the status of the Subset Sum problem, the best known algorithm for Pigeonhole Equal Sums runs in \(O^*(2^{n/2})\) time, via either meet-in-middle or dynamic programming (Allcock, Hamoudi, Joux, Klingelh\"{o}fer, and Santha, 2022). Our main result is an improved algorithm for Pigeonhole Equal Sums in \(O^*(2^{0.4n})\) time. We also give a polynomial-space algorithm in \(O^*(2^{0.75n})\) time. Unlike many previous works in this area, our approach does not use the representation method, but rather exploits a simple structural characterization of input instances with few solutions.