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.
Categories: cs.DS