Optimal Assembly Addition Chains

• Optimal assembly addition chains are combinatorial structures that generalize integer addition chains to non-numeric settings by minimizing assembly steps.
• They leverage binary and two-piece decompositions to establish structural bounds and guide efficient algorithm design across various domains.
• Despite NP-completeness in exact computation, heuristic methods like m-ary approaches and tree searches yield tight approximations in practice.

An optimal assembly addition chain is a combinatorial structure that generalizes the classical notion of addition chains from the set of positive integers ℕ to arbitrary algebraic or combinatorial spaces, with the goal of minimizing the number of “assembly” steps required to build a target object from a prescribed set of elementary building blocks. This generalization retains the core recursive property of addition chains but extends it to non-numeric settings, providing a formalism that incorporates objects such as strings, graphs, and polyominoes. The theory closely parallels classical bounds and methods in integer addition chains and reveals deep connections with algebraic complexity, combinatorial optimization, and efficient algorithm design (Cronin et al., 19 Dec 2025).

1. Foundational Concepts: Multi-Magma and Assembly Addition Chains

Let $S$ be a set of objects and $BB \subset S$ a specified finite set of elementary “building blocks.” An assembly multi-magma is defined as a triple $(S, \odot, BB)$, where $\odot: 2^S \times 2^S \to 2^S$ is a binary operation on subsets of $S$ that models object assembly.

Given $O \in S \setminus BB$, an assembly addition chain (AAC) for $O$ is a sequence $(O_1, O_2, \ldots, O_r = O)$ such that:

• $O_1 \in BB \odot BB$,
• For each $i>1$, $O_i \in \{X\} \odot \{Y\}$ for some $X, Y$ in $\{O_1, ..., O_{i-1}\} \cup BB$.

The assembly index $a(O)$ is defined as the minimal possible length $r$ of any such chain for $O$. If $a(O) = \min \{ L(C) \mid C \in AAC(O) \}$, then $C$ is called an optimal assembly addition chain (OAAC) for $O$ (Cronin et al., 19 Dec 2025).

The size function $s: S \to \mathbb{N}$ assigns $s(B) = 1$ for $B \in BB$, and $s(O) = r + 1$ when $O$ is constructed from $r$ assembly steps. In an assembly space, the operation $\odot$ is required to satisfy $s(O) = s(P) + s(Q)$ for $O \in \{P\} \odot \{Q\}$.

2. Classical Addition Chains and the Integer Setting

For $S = \mathbb{N}$ and $BB = \{1\}$, standard addition chains are sequences $1 = a_0 < a_1 < \cdots < a_k = n$ where each $a_j = a_i + a_r$ for $0 \leq i, r < j$. The minimal chain length is $\ell(n)$. Classical results include:

• Lower bound: $\ell(n) \ge \lceil \log_2 n \rceil$
• Trivial upper bound (binary method): $\ell(n) \le \lfloor \log_2 n \rfloor + v_2(n) - 1$, with $v_2(n)$ the binary weight
• Schönhage’s lower bound: $\ell(n) \ge \log_2 n + \log_2 v_2(n) - 2.13$
• For almost all $n$, $$\ell(n) = \frac{\log n}{\log 2} + (1 + o(1)) \frac{\log n}{\log\log n}$$ (Koninck et al., 9 Apr 2025).

In the generalized setting, it is proved that

$a(O) \geq \ell(s(O)),$

with equality when $S = \mathbb{N}$, reflecting the fact that the combinatorial structure imposes no greater complexity than the brute numeric total (Cronin et al., 19 Dec 2025).

3. Structural Bounds in Assembly Spaces

Upper and lower bounds for $a(O)$ in general assembly spaces (with $s = s(O)$) follow the integer case but adapt to structural features:

• Coarse universal bounds: $\log_2 s \leq a(O) \leq s-1$.
• Binary-decomposable objects (BD(S)): If $s$ has binary expansion $s = 2^{n_1} + \cdots + 2^{n_H}$ and there exist appropriate decompositions in $S$, then

$$a(O) \leq (H - 1) + \sum_{i=1}^{n_1} \min \left( \sum_{j=1}^H \lfloor 2^{n_j - i} \rfloor, |S(2^i)| \right ),$$

where $|S(2^i)|$ is the number of objects of size $2^i$ (Cronin et al., 19 Dec 2025).

• Two-piece-decomposable objects (2PD(S)): If $O$ can be constructed by pairwise assembly, then

$$a(O) \leq \min(\lfloor s/2 \rfloor, |S(2)|) + \lceil s/2 \rceil - 1.$$

These structural theorems are specialized in concrete combinatorial domains.

4. Algorithmic Construction and Computational Complexity

Determining $a(O)$ exactly is NP-complete, matching the intractability of finding minimal integer addition chains (Cronin et al., 19 Dec 2025). However, class-specific heuristics and greedy/m-ary-type methods enable tight approximations:

• m-ary methods: For $g$-ary assembly, the m-ary method provides a construction of length at most $\lceil \log_g s \rceil + \mu_g(s)$, where $\mu_g(s)$ is the base-$g$ digit weight (Elias et al., 2016).
• Factor methods and tree search: Factorization-based (for integer-like $S$) and tree-expansion algorithms yield exact chains for moderate-sized objects but have exponential complexity.
• Stamp chains and additive 2-bases: In the integer case, stamp chains—structures that satisfy both addition chain and additive basis properties—admit efficient O($\sqrt{n}$)-multiplication assembly for computing all $u(x^i)$, $i=1,\ldots,n$ in “zero-cost” composite domains (Kohonen et al., 2013).

For strings, graphs, and polyominoes, these strategies are adapted by exploiting the space’s decomposition properties.

5. Representative Examples and Explicit Constructions

The theory encompasses a range of combinatorial objects:

• j-Strings: All strings over an alphabet of size $j$ ($S$: all strings, $BB$: $j$ letters). Every string of length $L$ is binary-decomposable, and $|S(2^i)| = j^{2^i}$. Bounds are thus given by the binary weight $H(L)$ and the structure of the j-ary tree (Cronin et al., 19 Dec 2025).
• Colored Connected Graphs: The assembly is over gluing monochromatic edges; Kotzig’s theorem ensures any connected graph is two-piece-decomposable. For a graph with $e$ edges and $C$ colors,

$$a(G) \leq \min(\lfloor e/2 \rfloor, C) + \lceil e/2 \rceil - 1.$$

• Colored Polyominoes: Using the skeleton graph reduction, polyomino assembling inherits the two-piece decomposability; the explicit bound involves both the number of unit squares and coloring parameters.

For each of these, the universal lower bound $a(O) \ge \ell(s(O))$ applies.

6. Connections with Classical and Generalized Addition Chains

The assembly addition chain formalism subsumes classical addition chain phenomena, including:

• Generalized g-addition chains: For integers and $g$-ary operations, the minimum length $l_g(d)$ satisfies $$\lceil \log_g d \rceil \leq l_g(d) \leq \lceil \log_g d \rceil + \mu_g(d)$$, with typical “gap” proportional to $\mu_g(\lceil \log_g d \rceil)$ (Elias et al., 2016).
• Scholz–Brauer–Schönhage bounds: The universal lower and upper bounds in assembly spaces are direct analogues of these foundational results: $$\log_2 s(O) + \log_2 H(s(O)) - 2.13 \leq a(O) \leq \log_2 s(O) + H(s(O)) - 1$$, where $H(s(O))$ is the binary digit weight (Cronin et al., 19 Dec 2025).
• Carry-based inequalities for $2^n-1$ (Scholz conjecture): Recent progress constructs OAACs for objects parameterized by $2^n-1$ by controlling carry propagations within the assembly steps, yielding sharper upper bounds than the classical “worst-case” (Agama, 2021).

7. Implications, Extensions, and Open Problems

The extension to arbitrary assembly spaces has multiple consequences:

• The optimality criteria and associated bounds provide generic design principles for constructing explicit OAACs in applied settings (e.g., DNA assembly, combinatorial enumeration, algorithmic group theory).
• The analysis of structural decomposability (binary/2-piece) predicts when substantial reductions in chain length are possible compared to naive assembly.
• Universal lower bounds enforce fundamental complexity-theoretic barriers independent of object symmetries.

Open problems include:

• Tightening the gaps between upper and lower bounds in the generalized setting, especially for complex combinatorial objects.
• Full characterization of OAACs for spaces with rich internal symmetries or restricted glue operations.
• The computational complexity of near-optimal heuristic algorithms in combinatorially rich spaces (Cronin et al., 19 Dec 2025).

A plausible implication is that assembly addition chains—by encoding both algebraic and combinatorial structure—could inform the design of efficient algorithms in domains as diverse as symbolic computation, structural chemistry, and network topology generation.

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References:

• "Assembly Addition Chains" (Cronin et al., 19 Dec 2025)
• "On the minimal length of addition chains" (Koninck et al., 9 Apr 2025)
• "On Generalized Addition Chains" (Elias et al., 2016)
• "Addition Chains Meet Postage Stamps: Reducing the Number of Multiplications" (Kohonen et al., 2013)
• "On addition chains and progress on the Scholz conjecture" (Agama, 2021)