Algorithms for the local and the global postage stamp problem

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.

💡 Research Summary

The paper tackles two classic combinatorial optimization problems that arise from the “postage stamp problem”: the Local Postage Stamp Problem (LPSP) and the Global Postage Stamp Problem (GPSP). In LPSP, a fixed set of k distinct stamp denominations Aₖ and a maximum of s stamp positions are given; the goal is to compute the s‑range nₛ(Aₖ), i.e., the largest integer N such that every value 1…N can be expressed as a sum of at most s stamps from Aₖ. It is known that LPSP is NP‑hard, and only for constant k does a polynomial‑time algorithm exist. The authors improve upon Mossige’s classic s‑range algorithm by eliminating the linear dependence on s in both time and memory. Their new dynamic‑programming scheme compresses the reachability table and updates reachable values in O(k·s·log aₖ) operations, a substantial reduction from the previous O(k·s²·aₖ) bound. Memory usage drops from O(s·aₖ) to O(log aₖ), making the algorithm practical for large s and large denominations.

GPSP asks, for given parameters k (number of denominations) and s (maximum stamps), to find a basis Aₖ that maximizes nₛ(Aₖ). Exact solutions are unknown; prior work relied on specific constructions such as the Fibonacci‑based basis (when k = s) and the Alter‑Barnett block construction (for k ≥ s). The Fibonacci basis yields nₛ(Fₖ) = f_{2k+1}−1 but performs poorly when k and s differ. The Alter‑Barnett method partitions the k denominations into s blocks of arithmetic progressions, but its analysis only provided a lower bound.

The authors first generalize the Alter‑Barnett construction, balancing block sizes more evenly (using (s−r) blocks of size q and r blocks of size q+1) and proving a tight upper bound, which improves the asymptotic constant factor. They then adopt a recursive divide‑and‑conquer (D&C) strategy originally suggested by Mrose. Given two sub‑bases A_{k₁} and B_{k₂} with s₁‑range and s₂‑range, they construct a combined basis C_{k₁+k₂} for s₁ + s₂ stamps and prove the inequality
n_{s₁+s₂}(C) ≥ (n_{s₁}(A)+1)(n_{s₂}(B)+1)−1.
Using this relation, they design a midpoint cut algorithm that repeatedly halves both k and s, yielding the lower bound nₛ(k) ≥ (1 + k·s)^{s−1}. They show that this bound is asymptotically optimal among all possible cuts, and that a dynamic‑programming search for the best first cut can improve constants further. Comparative analysis demonstrates that the D&C approach dominates the Fibonacci and greedy (Alter‑Barnett) constructions for most regimes, especially when k ≤ s. Empirical results, obtained with their open‑source GStamps library, confirm the theoretical predictions: the midpoint D&C algorithm consistently achieves larger s‑ranges than the Fibonacci basis, and the “best‑cut” variant (which uses DP to select optimal split points) yields the highest values at the cost of super‑polynomial time.

Beyond combinatorial optimization, the authors explore cryptographic applications. In fully homomorphic encryption (FHE) settings, evaluating a polynomial on encrypted inputs requires many homomorphic multiplications and additions. By interpreting polynomial coefficients as stamp denominations and using the high‑range bases produced by their algorithms, the number of required homomorphic operations can be reduced from O(s·k) to O(log s·log k). This reduction translates into tangible speed‑ups for secure multi‑party computation protocols that involve set operations or polynomial evaluations on private data.

In summary, the paper makes three major contributions: (1) a faster, memory‑efficient exact algorithm for the local postage stamp problem; (2) a suite of polynomial‑time approximation algorithms for the global problem, with rigorous analysis showing the recursive midpoint D&C method is asymptotically optimal; and (3) a demonstration that these combinatorial improvements directly benefit homomorphic encryption‑based secure computation. The work bridges number theory, algorithm design, and cryptographic engineering, providing both theoretical insights and practical tools.