Algorithms for the local and the global postage stamp problem
Léo Colisson Palais, Jean-Guillaume Dumas, Alexis Galan, Bruno Grenet, Aude Maignan
TL;DR
The paper tackles the local and global postage stamp problems, proving NP-hardness for the local variant and introducing algorithms that simultaneously improve efficiency for the LPSP and provide polynomial-time approximations for the GPSP. It analyzes multiple approximation strategies, including Fibonacci-based, greedy block, and recursive divide-and-conquer constructions, and establishes that balanced recursive partitioning yields superior asymptotics in many regimes. It also connects these combinatorial methods to practical cryptographic applications by enabling more efficient homomorphic evaluation on encrypted data. Collectively, the work delivers tighter algorithmic bounds, practical baselines, and cryptographic relevance for stamp-based optimization problems.
Abstract
We consider stamps with different values (denominations) and same dimensions, and an envelope with a fixed maximum number of stamp positions. The local postage stamp problem is to find the smallest value that cannot be realized by the sum of the stamps on the envelope. The global postage stamp problem is to find the set of denominations that maximize that smallest value for a fixed number of distinct denominations. The local problem is NP-hard and we propose here a novel algorithm that improves on both the time complexity bound and the amount of required memory. We also propose a polynomial approximation algorithm for the global problem together with its complexity analysis. Finally we show that our algorithms allow to improve secure multi-party computations on sets via a more efficient homomorphic evaluation of polynomials on ciphered values.