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 &l...
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description | 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. |
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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.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Algorithms ; Dynamic programming ; Polynomials ; Structural analysis ; Sums</subject><ispartof>arXiv.org, 2024-03</ispartof><rights>2024. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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subjects | Algorithms Dynamic programming Polynomials Structural analysis Sums |
title | A Faster Algorithm for Pigeonhole Equal Sums |
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