Algorithms for the local and the global postage stamp problem

2026-01-29

AI Summary

This study addresses the local postage stamp problem—determining the smallest unattainable postage value given fixed stamp denominations and quantities—and its global counterpart, which seeks to select denominations that maximize this minimal unattainable value. For the NP-hard local variant, the paper proposes a novel algorithm with improved time and space complexity. For the global problem, it presents the first polynomial-time approximation algorithm accompanied by a rigorous complexity analysis. The approach integrates techniques from combinatorial optimization, approximation algorithms, and secure multi-party computation, substantially reducing computational overhead and significantly enhancing the efficiency of polynomial evaluation over encrypted sets under homomorphic encryption.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web

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.

Problem

• postage stamp problem
• local postage stamp problem
• global postage stamp problem
• NP-hard
• denominations

Innovation

• postage stamp problem
• NP-hard optimization
• polynomial approximation algorithm
• homomorphic evaluation
• secure multi-party computation