Structure-Aware Informed Tree Search (IQTS)

• IQTS is a structure-aware tree search algorithm that strategically expands partial solutions using dependency and grammar rules to efficiently navigate constrained combinatorial spaces.
• It integrates classical Monte Carlo Tree Search with quantum sub-solvers, employing heuristics, branch pruning, and solution repair to optimize engineering designs and supply chain logistics.
• IQTS improves solution quality and reduces computational cost, as evidenced by significant finite-element analysis reductions and competitive performance against reinforcement learning methods.

Structure-Aware Informed Tree Search (IQTS) is a decision procedure for solving large-scale combinatorial optimization problems whose feasible sets or objective landscapes are largely determined by underlying dependency, assembly, or grammar-induced structural relations. IQTS strategically expands partial solutions in a tree-structured search, using domain‐specific structural information to limit and inform the exploration of branches. It has seen application in both classical engineering design, notably truss optimization via Monte Carlo Tree Search (MCTS), and quantum-classical hybrid logistics optimization via tree-structured division of QUBOs with quantum sub-solvers. IQTS yields improvements in both feasible solution quality and computational cost relative to reinforcement learning and uninformed tree search, particularly in settings where the solution space is sharply constrained by sequential dependencies or construction grammars (Heese et al., 5 Feb 2026, Garayalde et al., 2024).

1. Structural Foundations and Problem Formulation

IQTS is predicated on the exploitation of combinatorial structure inherent to a given optimization domain. In supply chain QUBO applications, the structure derives from a part-dependency tree (PBS), representing how parts, suppliers, and sites relate through hierarchical assemblies and dependency chains. In truss optimization, the structure emerges from a generative graph grammar, in which every move is regulated by rules that ensure mechanical validity (e.g., statically determinate additions via prescribed operators) (Heese et al., 5 Feb 2026, Garayalde et al., 2024).

In the logistics QUBO, the scalarized objective is

$$\min_{x \in \{0,1\}^{N_x}} Q(x) = \sum_{n=1}^4 \overline C_n(w,\bar y) + \sum_{m=1}^6 \lambda_m P_m(x)$$

with $x$ capturing assignment and penalty-ancilla variables, and extensive combinatorial penalties $P_m$ encoding constraint satisfaction (Heese et al., 5 Feb 2026). For truss optimization, partial states are graphs $(V,E)$, with expansion by grammar rules guaranteeing all child states remain physically admissible (Garayalde et al., 2024).

2. Search Tree Construction and Expansion Policies

A core feature of IQTS is tree expansion guided by structural information. In the logistics QUBO setting, the part-dependency tree (PBS) defines nodes as local variable subsets ($n$ variables on subtree $T \subset I$), with high PBS levels (deeper dependencies) prioritized for exploration. For trusses, the only children permitted at each search node are those produced by grammar rules (direct addition $\mathcal{D}$ and replacement $\mathcal{T}$) parameterized by active/inactive nodes and edges, ensuring all expanded states are statically determinate (Heese et al., 5 Feb 2026, Garayalde et al., 2024).

The algorithm proceeds as follows:

• Select a "hot" element (e.g., a part or truss element) with the highest unexplored structural priority.
• Generate a connected subtree or permissible expansion using the relevant grammar.
• Restrict search to a small number of variables or design actions ($n$); all other variables remain fixed.
• Expansion (and subsequent roll-outs, in the truss case) only follows structure-legal transitions, directly enforcing domain feasibility during search (Heese et al., 5 Feb 2026, Garayalde et al., 2024).

3. Heuristic Guidance, Pruning, and Repair Mechanisms

The informed nature of IQTS is realized through multiple mechanisms:

• Heuristic hot-spotting: Priority is given to unexplored nodes with maximal structural depth, reflecting the influence of high-level dependencies on solution genesis.
• Branch and pruning strategies: In logistics QUBO, each subtree root is only revisited after all elements have been cycled (round-robin), maintaining coverage and diversity. In truss MCTS, expansion is pruned strictly by grammar legality.
• Solution repair: After local subproblem solves in logistics, informed solution fixer (ISF) and improver (ISI) subroutines are invoked to restore feasibility and enhance solution quality through local greedy swaps (Heese et al., 5 Feb 2026). For trusses, using only grammar-legal expansions obviates the need for post-hoc repairs.
• Termination: The algorithm halts after a prescribed number of subproblem solves or MCTS episodes, returning the best feasible solution encountered (Heese et al., 5 Feb 2026, Garayalde et al., 2024).

4. Algorithmic Realizations and Quantum-Classical Interfaces

IQTS instantiates as a classical MCTS or as a hybrid quantum-classical solver depending on application domain.

Truss MCTS pseudocode:

Algorithm IQTS-MCTS(s₀, Nₑ, α, β)
  for each episode:
    // Selection: traverse tree, selecting child maximizing UCT
    // Expansion: expand via legal grammar rules only
    // Simulation: random grammar-legal rollout to terminal state, compute reward
    // Backpropagation: propagate terminal reward to all nodes along path

The UCT score is given by

$$UCT_j = (1-\alpha) \frac{v_j^\Sigma}{n_j} + \alpha \sqrt{ \frac{2 \ln \sum_\ell n_\ell}{n_j} }$$

with possible augmentation for best return $v^{best}$ and variance $\text{var}(v)$ via an additional parameter $\beta$ (Garayalde et al., 2024).

Logistics QUBO IQTS:

At each iteration:

• Select subtree $T$ of size $m$, free $n$ binary variables.
• Restrict the QUBO to those variables; solve the sub-QUBO via QAOA of depth $p$ on quantum hardware or simulator.
• Integrate assignment, run ISF/ISI to enforce constraint satisfaction/local improvement.
• Terminate after $\kappa$ iterations (Heese et al., 5 Feb 2026).

Hybrid quantum-classical implementations leverage current NISQ hardware (IonQ Aria-1, 25 qubits) for quantum subproblem solves; all orchestration and repair is classical, implemented in Python with QAOA through PennyLane/AWS Braket (Heese et al., 5 Feb 2026).

5. Performance Analysis and Empirical Results

IQTS demonstrates robustness and efficiency across diverse problem settings:

• Truss optimization: Across six benchmark designs, IQTS/MCTS consistently achieved optimal or near-optimal solutions with 30–76% fewer finite-element analyses compared to deep Q-learning. In extremely large state spaces (≥10$^7$ states), the method consistently reached global optima with minimal simulative effort (mean 507 FE runs), attributed to the exhaustive yet structure-pruned search (Garayalde et al., 2024).

| Case | Objective Ratio | Percentile | FE Runs | % Reduction vs. DQN | |------|----------------|------------|---------|---------------------| | 1 | 100% | 100% | 106 | –74.7% | | 2 | 100% | 100% | 517 | –76.3% | | 3 | 100% | 100% | 966 | –56.5% | | 4 | 91.9% | 99.9% | 1672 | –70.9% | | 5 | 95.2% | 99.99% | 9739 | –70.7% | | 6 | 90.4% | 99.98% | 7931 | –31.3% |

• Supply chain QUBO: On IonQ Aria-1 hardware, IQTS+QAOA achieved a Pareto-approximate front with hypervolume 55.20, nearly matching classical simulated annealing's 54.44 at small subproblem size ($n \leq 25$). The hybrid bilevel solver (HBS) achieves higher hypervolumes in aggregate. The primary utility of QAOA-based IQTS may be realized as quantum hardware scales (Heese et al., 5 Feb 2026).

Key performance metrics include objective quality (hypervolume, displacement), convergence speed (number of tree expansions or QUBO subproblems solved), and comparative efficiency to RL and classical baselines.

6. Advantages, Limitations, and Outlook

Advantages:

• Enforced structure ensures each search node corresponds to a feasible or near-feasible design, preventing wasted computation on inadmissible candidates.
• Heuristic and grammar-based expansion yields fast convergence and scalability to large, highly constrained state spaces.
• Monte Carlo back-propagation propagates credit to all ancestors instantly, unlike step-wise RL.

Limitations:

• Parallelization is limited because dependency chains force sequential updating (logistics QUBO); in principle, distinct subtrees can be scheduled in parallel if propagation of constraints is strictly local (Heese et al., 5 Feb 2026).
• Quantum subproblems remain small due to hardware constraints; thus, current quantum advantage is not yet visible (Heese et al., 5 Feb 2026).
• Hyperparameter sensitivity (e.g., subtree size $m$, number of free variables $n$, number of steps $\kappa$, UCT weights $\alpha$, $\beta$) necessitates empirical tuning for best results (Heese et al., 5 Feb 2026, Garayalde et al., 2024).

Scaling quantum hardware may enable IQTS subproblem sizes to exceed classical capabilities, providing quantum advantage in hybrid optimization workflows, especially in industrial logistics and design applications (Heese et al., 5 Feb 2026).

7. Comparative Context and Research Directions

IQTS generalizes well beyond the initial domains (supply chain assembly, truss design) to optimization problems where legal expansion paths are induced by dependency, assembly, or generative grammars. It addresses the limitations of Q-learning and deep Q-learning: avoiding state-action table explosion, and improving the timeliness and granularity of credit assignment for early design decisions (Garayalde et al., 2024).

A plausible implication is that IQTS provides a scalable template for integration with both quantum and classical sub-solvers in domains with combinatorial assembly, especially as quantum hardware capability increases. Depth-adaptive exploration and further meta-optimization of hyperparameters remain open research fronts (Heese et al., 5 Feb 2026, Garayalde et al., 2024).