Cross-Domain Heuristic Search

• Cross-domain heuristic search is a framework that leverages domain-independent principles such as relaxation and solution statistics to guide search without domain-specific tuning.
• It applies robust methodologies across classical planning, constraint programming, and combinatorial optimization, using techniques like delete-relaxation and counting-based heuristics.
• Emerging models including deep RL and LLM-based planners enhance cross-domain adaptability, enabling effective zero-shot generalization in diverse, evolving problem spaces.

Cross-domain heuristic search is the paper and engineering of general-purpose heuristic search methods that exhibit strong empirical performance and adaptability across multiple distinct problem domains. These approaches distill and automate domain-independent search principles, operator selection, and learning methods, operating in heterogeneous spaces such as classical planning, combinatorial optimisation, and constraint programming. The central challenge is to design heuristics, control policies, and architectures that function robustly with only high-level task information, variable structural semantics, and non-uniform search operators, minimizing reliance on domain-specific tuning or expert knowledge.

1. Foundational Principles and Domain-Independence

A broad foundation for cross-domain heuristic search is the abstraction of domain-independent problem structures and the generalization of heuristic guidance. Classical planners like FF (Fast Forward) formalize this using the STRIPS syntax $(O,I,G)$, where the delete-relaxation heuristic $h_\mathrm{FF}(S)$—computed by extracting a relaxed plan that ignores transient (negative) action effects—yields search guidance that is fully domain-independent (Hoffmann et al., 2011). The delete-relaxation technique involves constructing a relaxed planning graph, extracting a level-wise plan, and using the plan length as a heuristic. Because this process operates solely on formal action encodings and goals, it applies uniformly across disparate domains (e.g., logistics, scheduling, Blocksworld), capturing positive interactions (shared precondition reuse) without explicit domain engineering.

In constraint programming, domain-independence is achieved through abstraction over global, structured constraints. For instance, counting-based search exploits solution densities across constraints like alldifferent, regular, and knapsack, guiding search by maximizing the preservation of feasible global solutions rather than relying solely on variable-local statistics (Pesant et al., 2014). For combinatorial optimization, frameworks like HyFlex abstract the solution representation and heuristic pool, exposing a domain-agnostic interface through which hyper-heuristic controllers operate (Burke et al., 2011).

2. Key Architectural Approaches

Domain-Independent Heuristics via Relaxation and Solution Statistics

• Planning with Relaxations: FF's heuristic search operates by computing relaxed solutions that ignore delete lists and extracting helpful actions and goal agendas automatically. Enforced hill-climbing (EHC) augments greedy search with systematic breadth-first exploration to escape plateaus.
• Counting-Based Branching in CSPs: The maxSD heuristic in constraint programming selects variable-value assignments maximizing the standard deviation of solution densities, leveraging combinatorial counts of global constraint satisfaction (Pesant et al., 2014).
• Portfolio and Hyper-Heuristic Controllers: HyFlex and related hyper-heuristics decouple the execution of low-level, domain-provided heuristics from high-level policy, enabling the latter to select, combine, and adapt operators using only performance feedback—no domain semantics required (Burke et al., 2011, Sobotka et al., 2 Sep 2025).

Learning and Foundation Models

Recent work explores data-driven generalization by learning domain-independent heuristic functions from experience:

• Residual Heuristic Learning: Models trained to predict corrections to classical heuristics (e.g., FF, CEA) across diverse planning domains can generate heuristics that transfer between domains even as state structures and action spaces change (Gomoluch et al., 2017).
• Foundation Models for Heuristics: Deep RL-trained value iteration networks, when provided with domain transition encodings (e.g., one-hot action availability), generalize heuristic cost-to-go estimates to unseen domains (e.g., n-puzzle action-space variants) without retraining (Khandelwal et al., 1 Jun 2024).
• LLM-Based Action Search: SayCanPay composes a fixed LLM proposal module ("Say") with learnable feasibility and payoff modules ("Can," "Pay"), implementing beam-based search over open-ended natural language plans with empirically robust cross-domain generalization (Hazra et al., 2023).

3. Cross-Domain Generalization Mechanisms

Generalization is achieved through several mechanisms:

• Abstracted Feature Spaces: Hand-crafted or learned features (plan properties, relaxed plan lengths, goal distances, constraint densities) are engineered to describe state-space properties agnostic to domain details (Hoffmann et al., 2011, Gomoluch et al., 2017, Pesant et al., 2014).
• Operator and Solution Diversity: Frameworks like HyFlex encapsulate domain-specific solution representations and operators while exposing them via unified APIs, allowing controller algorithms to remain domain-agnostic (Burke et al., 2011).
• Text-Only Interfaces and Embeddings: By using natural language representations of state histories, goals, and actions (as in SayCanPay), pipeline components can seamlessly adapt to new task formulations without retraining architectural machinery (Hazra et al., 2023).
• Transition-Function Encoding: Neural heuristics ingest explicit representations of transition dynamics (e.g., state-action adjacency vectors), enabling adaptation to variations in permissible moves and problem dynamics (Khandelwal et al., 1 Jun 2024).

4. Empirical Performance and Benchmarking

A commitment to robust benchmarking underpins progress in this area:

• Competition and Frameworks: The CHeSC competition (built atop HyFlex) establishes multi-domain evaluation with unified protocols, Borda count aggregation, and domain-agnostic solution APIs (Burke et al., 2011).
• Quantitative Results: FF's heuristic consistently outperforms planner-specific or SAT-based systems across logistics, scheduling, and permutation flow shop domains, with search-space reductions attributable to helpful action pruning, goal agendas, and enforced hill-climbing (Hoffmann et al., 2011).
• Machine-Learned Heuristics: NN-FF (feed-forward neural net correction to FF) achieves up to $3.5\times$ fewer state expansions on Parking problems than base FF, with coverage approaching baseline domain-independent heuristics in several benchmarks (Gomoluch et al., 2017).
• Pathfinding Foundation Models: The inclusion of explicit domain transition info allows the trained heuristic to yield $R^2=0.98$, CCC=0.99 on unseen 15-puzzle domains, outperforming generic domain-independent methods like Fast Downward while closely matching domain-optimal DeepCubeA (Khandelwal et al., 1 Jun 2024).
• LLM-Guided Search: SayCanPay exhibits planning success rates ranging from $\approx70\%$ (Ravens-Hanoi, beam search) to $77\%$ (Ravens-Blocks), and demonstrates significant uplifts from beam search and oracle LLM proposal, isolating bottlenecks to action proposal quality (Hazra et al., 2023).
• Hyper-Heuristic Performance: Intensified LLH composition and solution acceptance/repetition strategies enable trivial random selectors (NHH*) to attain or exceed state-of-the-art results on HyFlex domains and in challenging scheduling and routing instances, confirming the primacy of LLH design over sophisticated selection (Sobotka et al., 2 Sep 2025).

5. Algorithmic Patterns and Control Policies

Table: Representative Cross-Domain Heuristic Search Approaches

Approach	Domain Coverage	Heuristic/Control Principle
FF (Fast Forward)	Planning/Scheduling	Delete-relaxation w/ relaxed plan extraction
Counting-based (maxSD)	Constraint Programming	Maximize SD of global constraint densities
SayCanPay (LLM planning)	NL-Plan, RL Benchmarks	LLM proposal + learned affordance/payoff eval
HyFlex hyper-heuristics	Combinatorial/Optimization	Domain-independent heuristic/operator control
RL Foundation Model	Pathfinding/Puzzle	Deep RL w/ transition-encoded cost prediction

Common patterns include best-first or beam search with inadmissible but informative heuristics, variable-depth operator application conditioned on solution improvement, dynamic acceptance/annealing, and exploitation of global problem structure for guidance (Hoffmann et al., 2011, Pesant et al., 2014, Sobotka et al., 2 Sep 2025).

6. Limitations and Future Directions

Documented limitations and research avenues include:

• Negative Interaction Blindness: Delete-relaxation implicitly ignores necessary negative interactions, rendering search less effective in domains requiring precise resource management or deep action chaining (Hoffmann et al., 2011).
• Oracle Dependence: LLM-based planners bottleneck on quality of action proposal; use of more expressive or instruction-tuned affordance/payoff modules is hypothesized to enhance generalization (Hazra et al., 2023).
• Label and Model Scalability: Learned heuristics require optimally labeled plans; for large domains, reliance on exhaustive solution labels becomes impractical—suggesting the need for semi-supervised or reinforcement learning augmentation (Gomoluch et al., 2017, Khandelwal et al., 1 Jun 2024).
• Operator Set Composition: Strong evidence suggests that the breadth and strategy for composing the LLH set is a limiting factor in cross-domain performance, often surpassing the benefit obtainable from more intricate selection or control policies (Sobotka et al., 2 Sep 2025).
• Structural Generalization: There remains an open challenge in jointly leveraging symbolic representations, graph neural architectures, and dynamic embedding of structural knowledge for unbounded classes of new domains (Khandelwal et al., 1 Jun 2024, Gomoluch et al., 2017).

A plausible implication is that future cross-domain heuristic search will converge to frameworks that (i) systematically engineer or learn powerful operator portfolios, (ii) automate abstraction and feature generation, and (iii) utilize foundation models capable of explicit adaptation to variable transition and reward structures via modular state and transition encodings—thus bridging the classical “domain barrier” that has historically separated problem formulations.

7. Broader Impact and Outlook

Cross-domain heuristic search has reshaped the landscape of automated planning, combinatorial optimization, and constraint satisfaction by providing robust, adaptable search mechanisms. Benchmarking frameworks such as HyFlex and CHeSC have standardized evaluation and spurred rapid methodological progress. The field is currently witnessing an overview of symbolic relaxations, statistical operator analysis, learning-based heuristics, and natural language processing, advancing towards "zero-shot" generalization and ever lower requirement for expert domain engineering (Hazra et al., 2023, Hoffmann et al., 2011, Sobotka et al., 2 Sep 2025, Pesant et al., 2014, Burke et al., 2011, Gomoluch et al., 2017, Khandelwal et al., 1 Jun 2024). This suggests cross-domain heuristic search will remain a central theme as AI systems are pressured to reason and act across increasingly diverse, dynamically evolving environments.