A Faster Algorithm for Pigeonhole Equal Sums
Ce Jin, Hongxun Wu
TL;DR
The paper studies Pigeonhole Equal Sums, a total-search variant of Subset Sum with promise $w([n])<2^n-1$, and presents faster-than-meet-in-the-middle algorithms. It achieves a randomized $O^*(2^{0.4n})$-time algorithm and a polynomial-space $O^*(2^{0.75n})$-time algorithm by exploiting a simple structural characterization of inputs with few solutions, avoiding the representation method. The approach splits into two cases based on the defect parameter $d$: a small-$d$ regime using structural constraints to obtain $O^*(\sqrt{\Delta})$ time, and a large-$d$ regime using subsampling with modular DP to obtain $O^*( (2^{2n}/\Delta)^{1/3})$ time, balanced at $\Delta=2^{0.8n}$. A polynomial-space variant combines these ideas with low-space element-distinctness techniques to reach $O^*(2^{0.75n})$ time, preserving poly$(n)$ space. Together, these results advance exact, sub-exponential-time algorithms for Pigeonhole Equal Sums and highlight new techniques that bypass the representation method in this problem class.
Abstract
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, Klingelhöfer, 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.