  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.
