Figure 1: Problem instance where perfect heuristic is not strictly optimally efficient with GBFS. However, the path (A, C,D, E) has cost 10 instead of 11 . Then h is a perfect ranking for GBFS on Γ. Proof. We carry the proof by induction with respect to the number of expanded states. Let's now make the induction step and assume the theorem holds for the first A 0 B 1 C 1 D 2 A 1 1 9 9 1 Figure 2: Problem instance where optimally efficient heuristic does not exists for GBFS.

The overall network architecture is shown in Figure 1. This work was done when the author was with Rutgers University. The overall network architecture is shown in Figure 1. We also apply the ReLU activation after its first and second layers. Empirical evaluations show that NHE exhibits admissibility and consistency.

This work was done when the author was with Rutgers University.

While most heuristics studied in heuristic search depend only on the state, some accumulate information during search and thus also depend on the search history. Various existing approaches use such dynamic heuristics in $\mathrm{A}^*$-like algorithms and appeal to classic results for $\mathrm{A}^*$ to show optimality. However, doing so ignores the complexities of searching with a mutable heuristic. In this paper we formalize the idea of dynamic heuristics and use them in a generic algorithm framework. We study a particular instantiation that models $\mathrm{A}^*$ with dynamic heuristics and show general optimality results. Finally we show how existing approaches from classical planning can be viewed as special cases of this instantiation, making it possible to directly apply our optimality results.

Many sequential decision-making problems can be formulated as shortest-path problems, where the objective is to reach a goal state from a given starting state. Heuristic search is a standard approach for solving such problems, relying on a heuristic function to estimate the cost to the goal from any given state. Recent approaches leverage reinforcement learning to learn heuristics by applying deep approximate value iteration. These methods typically rely on single-step Bellman updates, where the heuristic of a state is updated based on its best neighbor and the corresponding edge cost. This work proposes a generalized approach that enhances both state sampling and heuristic updates by performing limited-horizon searches and updating each state's heuristic based on the shortest path to the search frontier, incorporating both edge costs and the heuristic values of frontier states.

In recent years, large language models (LLMs) have shown remarkable capabilities in various artificial intelligence problems. However, they fail to plan reliably, even when prompted with a detailed definition of the planning task. Attempts to improve their planning capabilities, such as chain-of-thought prompting, fine-tuning, and explicit "reasoning" still yield incorrect plans and usually fail to generalize to larger tasks. In this paper, we show how to use LLMs to generate correct plans, even for out-of-distribution tasks of increasing size. For a given planning domain, we ask an LLM to generate several domain-dependent heuristic functions in the form of Python code, evaluate them on a set of training tasks within a greedy best-first search, and choose the strongest one. The resulting LLM-generated heuristics solve many more unseen test tasks than state-of-the-art domain-independent heuristics for classical planning. They are even competitive with the strongest learning algorithm for domain-dependent planning. These findings are especially remarkable given that our proof-of-concept implementation is based on an unoptimized Python planner and the baselines all build upon highly optimized C++ code. In some domains, the LLM-generated heuristics expand fewer states than the baselines, revealing that they are not only efficiently computable, but sometimes even more informative than the state-of-the-art heuristics. Overall, our results show that sampling a set of planning heuristic function programs can significantly improve the planning capabilities of LLMs.

Figure 1: Problem instance where perfect heuristic is not strictly optimally efficient with GBFS. However, the path (A, C,D, E) has cost 10 instead of 11 . Then h is a perfect ranking for GBFS on Γ. Proof. We carry the proof by induction with respect to the number of expanded states. Let's now make the induction step and assume the theorem holds for the first A 0 B 1 C 1 D 2 A 1 1 9 9 1 Figure 2: Problem instance where optimally efficient heuristic does not exists for GBFS.

The overall network architecture is shown in Figure 1. This work was done when the author was with Rutgers University. The overall network architecture is shown in Figure 1. We also apply the ReLU activation after its first and second layers. Empirical evaluations show that NHE exhibits admissibility and consistency.