Global Postage Stamp Problem

• The Global Postage Stamp Problem is a combinatorial optimization challenge that determines optimal stamp denominations for representing consecutive integers with a limited number of stamps.
• It employs recursive divide-and-conquer methods, asymptotic analysis, and geometric techniques to derive bounds and construct extremal bases.
• Recent research highlights NP-hardness, polynomial-time approximations, and connections to additive number theory, underscoring its significance in discrete optimization.

The Global Postage Stamp Problem (GPSP) concerns the combinatorial optimization of integer bases for additive representation under bounded sum constraints. Given a fixed number $k$ of stamp denominations and a maximum stamp count %%%%1%%%%, the GPSP seeks to select denominations to maximize the largest consecutive integer $h$ representable as a sum of at most $m$ stamps (with repetitions). The GPSP has deep ties to additive number theory, computational complexity, discrete optimization, and has motivated advances in asymptotic analysis, explicit extremal constructions, and algorithmic methodology.

1. Formal Definition and Problem Statement

Fix integers $m\in\mathbb{N}$ (maximum allowed stamps, also called the range parameter) and $k\in\mathbb{N}$ (number of denominations). Consider a basis $A = \{ a_1, \ldots, a_k \} \subset \mathbb{N}$ with $a_1=1< a_2 < \cdots < a_k$. For this $A$, define

$$h(A) = \max \left\{ h \in \mathbb{N} : \forall n \leq h,\ \exists\ (\lambda_1, \ldots, \lambda_k) \in \mathbb{N}_0^k,\ \sum_{j=1}^{k} \lambda_j \leq m,\ n = \sum_{j=1}^{k} \lambda_j a_j \right\}.$$

The Global Postage Stamp Problem asks: For given $m$ and $k$, to find a $k$-element set $A$ maximizing $h(A)$. Define the optimum

$h^*(m, k) = \max_{|A|=k,\, a_1=1} h(A),$

and call any $A$ achieving equality an extremal basis. This formulation subsumes the classical "h-basis" concept and encompasses both structural and computational complexity.

2. Algorithmic Techniques and Approximation

The GPSP is known to be NP-hard in the local (decision) version for $m$ as parameter, but admits efficient polynomial-time approximation algorithms for the global maximization variant. Colisson Palais et al. (Palais et al., 29 Jan 2026) introduced a recursive divide-and-conquer approach, building upon a combinatorial lemma attributed to Mrose (1974):

• If $A_1$ has size $k_1$ and $m_1$-range $h_1$, and $A_2$ has size $k_2$ and $m_2$-range $h_2$, then the union $A = A_1 \cup \{ (h_1+1) a : a \in A_2 \}$ forms a basis of size $k_1 + k_2$ with $(m_1 + m_2)$-range at least $(h_1+1)(h_2+1) - 1$.

The resulting recursive algorithm, outlined as follows, achieves the best known polynomial-time guarantee:

def BuildBasis(k, m): if k == 1: return [1] if m == 1: return range(1, k+1) k1, k2 = k//2, k - k//2 m1, m2 = m//2, m - m//2 A1 = BuildBasis(k1, m1) A2 = BuildBasis(k2, m2) h1 = compute_range(A1, m1) A2_scaled = [(h1+1) * a for a in A2] return A1 + A2_scaled

This construction ensures

$h(A) \geq (1 + m/k)^k - 1 \quad \text{(for $k \leq m$)},$

and analogous bounds in the $m<k$ regime.

Time complexity is $O(k \log k)$, space is $O(k)$. Empirically, combining this method with brute-force for small subbase regimes further tightens performance to near-optimal (Palais et al., 29 Jan 2026).

3. Asymptotic Bounds for Large Parameters

Asymptotic analysis for GPSP at large $m$ or $k$ falls under additive basis theory. The minimal size $F_m(n)$ of an $m$-fold basis for $[n]$ exhibits sharp exponential transitions. Faust and Tait (Faust et al., 31 Jul 2025) provide improved bounds:

• For every $\varepsilon > 0$ and sufficiently large $m$:

$$(1/2 - \varepsilon) m! \sqrt{2\pi e} n \leq F_m(n)^m \leq \left((\sqrt{3}/2 + \varepsilon) m\right)^m n$$

Thus,

$$(1/2)\sqrt{2\pi e}\; m! \cdot n \lesssim F_m(n)^m \lesssim (\sqrt{3}/2\, m)^m\, n$$

A probabilistic method using Berry–Esseen and a modular digit-expansion construction for bases in $\mathbb{Z}/b\mathbb{Z}$ underlie these improvements. These results essentially halve prior leading constants and, for the first time, establish nontrivial (subexponential-in-$m$) upper bounds for all $m>2$.

The full asymptotic regime remains open; the gap between lower and upper constants is substantial. This suggests further refinements—possibly via advanced local limit theorems or more optimal group-theoretic constructions—are plausible future directions (Faust et al., 31 Jul 2025).

4. Explicit Extremal Bases for Small $k$

For $k=3$, closed-form characterization is established. Challis (Challis, 2013) provides parametric formulae (for $m \geq 81$) expressing

$M(3, s) = X_{\text{opt}}(s)$

in terms of $s = 9t + r$, for $0 \leq r < 9$, with corresponding constructive formulas for the extremal set $\{1, a_2, a_3\}$. The solution employs the concept of stride generators and thread diagrams, leading to finite combinatorial enumeration valid for all large $s$.

For $k=4$, Challis (Challis, 2014) describes two universal families—Hofmeister and Braunschädel bases—parameterized by period-12 cycles in $h$. All known extremal bases for $k=4$ (and large $h$) belong to one of these two forms:

• Hofmeister: $$(k_{21}, k_{31}, k_{32}, k_{41}, k_{42}, k_{43}) = (9,4,3,7,2,2)$$
• Braunschädel: $(9,2,3,7,2,2)$

Coverage (the maximal representable $n$) is expressed as quartic polynomials in $t$ with explicit offsets. No new symbolic forms have been observed despite extensive computational exploration for $h \leq 700$. Selmer’s correspondence and Mossige’s admissibility criteria further support this observed rigidity.

A plausible implication is that for fixed $k$, the space of extremal bases may be exhausted by finitely many periodic patterns up to slowly drifting offset vectors, though a rigorous proof is still lacking (Challis, 2014).

5. Structural and Geometric Analysis

Granville and Shakan (Granville et al., 2020) provide a general geometric solution for the $N$-fold sumset $N A$ in all dimensions once $N$ is large. For $A \subset \mathbb{Z}_{\geq 0}$ with $\gcd(A)=1$, and $b=\max A$,

$$N A = [0, bN] \;\setminus\; (E(A) \cup (bN-E(b-A)))$$

for $N \geq 2 \lfloor b/2 \rfloor$, where $E(A)$ is a finite exceptional set. The result generalizes to higher dimensions via convex geometry and additive combinatorics, yielding asymptotic Ehrhart-like polynomiality for the full sumset cardinality.

This analysis tightly characterizes the regions of guaranteed coverage and identifies precise endpoints where representation can fail, reducing the global problem to the structure of the “exceptional set,” which has cardinality $O(N^{n-1})$ in $n$ dimensions.

6. Open Problems and Current Research Questions

Despite extensive progress, several fundamental questions remain:

• Can the exponent or base in polynomial-time approximation bounds (e.g., those of Colisson Palais et al.) be further improved via new combinatorial constructions (Palais et al., 29 Jan 2026)?
• Is there a fixed-parameter tractable (FPT) exact algorithm in $k$ or $m$ for finding extremal bases, despite the NP-hardness in $m$ (Palais et al., 29 Jan 2026)?
• What is the precise asymptotic behavior of $h^*(m,k)$ as $m,k \to \infty$? The gap between known lower and upper bounds is exponentially wide in the regime of large $m$ (Faust et al., 31 Jul 2025).
• For $k=4$, will one of Hofmeister or Braunschädel bases strictly dominate for all but finitely many $h$, or will optimal forms oscillate indefinitely (Challis, 2014)?
• Can the drift of offset vectors $c_{ij}(r)$ in the $k=4$ case be fully characterized, and is it possible to prove that no new structural types exist for fixed $k$ (Challis, 2014)?
• For arbitrary $k$, does the geometric/exceptional set analysis of (Granville et al., 2020) offer improved explicit bounds for moderately sized parameters?

7. Connections and Applications

The GPSP is linked to additive basis theory, computational number theory, and the algorithmics of integer representations. Recent algorithmic advances facilitate more efficient protocols for secure multi-party computation via improved homomorphic polynomial evaluation (notably, from the work of Colisson Palais et al. (Palais et al., 29 Jan 2026)). The interplay between explicit constructions, combinatorial optimization, and analytic number theory continues to stimulate new technical developments and cross-disciplinary applications.

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Key References:

• (Palais et al., 29 Jan 2026) for algorithms and polynomial-time approximations
• (Faust et al., 31 Jul 2025) for new exponential-in-$h$ asymptotic bounds
• (Granville et al., 2020) for geometric and structural results on sumsets
• (Challis, 2014, Challis, 2013) for explicit constructions in small $k$ cases