Admissible heuristic

About: Admissible heuristic is a research topic. Over the lifetime, 197 publications have been published within this topic receiving 15329 citations. The topic is also known as: admissible heuristics.

Scheduling FMS problems with heuristic search function and transition-timed Petri nets

Abstract: In flexible manufacturing system (FMS) scheduling problems, the production process and constraints can be modeled by Petri nets concisely. However, the search space for an optimal or suboptimal schedule will exponentially grow with the increase of problem size. In this paper, we consider the scheduling problems of FMS in the framework of timed-transition Petri nets. In the framework, each token owns a individual timestamp which faciliates the analysis of the concurrency characteristic of modeled system. To save search effort, we provide an admissible heuristic function which also considers the available time of shared machine resources and subparts during calculating the lower bound of the remaining time for unprocessed operations. We prove the heuristic function is more effective than prior ones. Thus, an optimal scheduling strategy can be obtained at much less effort. Several numerical experiments are provided to demonstrate the effect of the improved heuristic function.

Heuristics for numeric planning via subgoaling

Abstract: The paper presents a new relaxation for hybrid planning with continuous numeric and propositional state variables based on subgoaling, generalising the subgoaling principle underlying the hmax and hadd heuristics to such problems. Our relaxation improves on existing interval-based relaxations by taking into account some negative interactions between effects when achieving a subgoal, resulting in better estimates. We show conditions on the planning model ensuring that this new relaxation leads to tractable, and, for the hmax version, admissible, heuristics. The new relaxation can be combined with the interval-based relaxation, to derive heuristics applicable to general numeric planning, while still providing more informed estimates for the subgoals that meet these conditions. Experiments show the effectiveness of its inadmissible and admissible version on satisficing and optimal numeric planning, respectively. As far as we know, this is the first admissible heuristic enabling cost-optimal numeric planning.

Lookahead Pathologies and Meta-level Control in Real-Time Heuristic Search

Abstract: Admissible and consistent heuristic functions are usually preferred in single-agent heuristic search as they guarantee optimal solutions with complete search methods such as A* and IDA*. Real-time problems, however, frequently make a complete search intractable due to space and/or time limitations. For instance, a path-planning agent in a real-time strategy game may need to take an action before its complete search has the time to finish. In such cases, incomplete search techniques (such as RTA*, SRTA*, RTDP, DTA*) can be used. Such algorithms conduct a limited ply lookahead and then evaluate the states envisioned using a heuristic function. The action selected on the basis of such evaluations can be suboptimal due to the incompleteness of search and inaccuracies in the heuristic. It is usually believed that deeper lookahead increases the chances of taking the optimal action. We demonstrate that it is not necessarily the case and that selecting lookahead depth dynamically can significantly improve the performance. 1 Real-time Heuristic Search The basic framework of real-time heuristic search is as follows. The agent traverses a state space by taking an action in each state. Its goal is to reach one of the predetermined “goal” states. The standard assumption is that the state space, the action set, and the set of goal states are fixed. Thus, each problem instance can be described by the agent’s initial state. Note that such search framework can be easily extended to more general decision-making. One way of doing so is via Markov Decision Processes as it is usually done in the field of Reinforcement Learning [1]. Throughout the paper, we will assume that the agent has a perfect domain model of the environment but cannot always tell which states are better than others. Thus, it is forced to use a heuristic estimate (henceforth heuristic function) of the state quality or value. Often such heuristic function is an estimate on the distance between the state in question and the closest goal state. Complete search methods such as A* [8] and IDA* [10] produce optimal solutions when based on an admissible heuristic function. The primary drawbacks are the exponential running time and the necessity to wait until the search completes before the first action can be taken [11]. This limits the applicability of complete search in practice as the deliberation time per action can be severely limited [9], the domain model can be expensive, the goal states can be difficult to recognize [13]. Consequently, despite numerous advances in improving heuristic functions [12, 21], incomplete real-time/on-line search methods remain the practical choice for complex real-life problems. Various real-time search methods have been proposed including: RTA* [11], RTDP [1], SRTA* and DTA* [22]. Such algorithms base their decisions on heuristic information collected from a partial tree expansion (lookahead) prior to reaching the goal state. Since the heuristic function is generally inaccurate and the search is incomplete, suboptimal solutions can be produced even with admissible and consistent heuristics. It is widely believed that looking ahead deeper improves the solution quality (e.g., [11]). Consequently, a considerable amount of effort has been put into increasing the lookahead depth by using selective search (search extensions) and hardware/software optimizations. In this paper we demonstrate that looking ahead deeper can actually decrease the chances of taking the optimal action as well as the overall solution quality. This phenomenon is known as lookahead pathology. Additionally, we show that selecting the lookahead depth dynamically can greatly improve the solution quality. 2 Related Work & Our Novel Contributions Lookahead pathologies within the mini-max search in two-player games have been investigated extensively in the past. In [15, 16, 2, 3, 4, 5, 17], the primary cause of pathologies was deemed to be the independence of heuristic values of the leaf nodes. Such games were called non-incremental. Large branching factors were also considered contributing to a pathology. Later, [14] added non-inertness (i.e., a constant branching factor) to the list of suspects. More recent work considered single-agent heuristic search and demonstrated that pathologies are possible even with admissible and consistent heuristic functions [6]. In this paper we extend the previous efforts in the following ways: (i) several performance metrics (e.g., overall solution quality, total running time, etc.) are introduced for single-agent heuristic search and lookahead pathologies are shown for all of them; (ii) the demonstration is carried out in the standard testbed of the 8 puzzle as opposed to more contrived and artificial environments; (iii) finally we show that using meta-level control of the search via selecting the lookahead depth dynamically has a significant potential. 3 Lookahead Pathologies Throughout the paper we will assume the standard RTA* real-time heuristic search model [11] as it is a general and modular technique allowing for further extensions (e.g., SRTA* and DTA* [22] or RTDP [1]). Thus, we consider a single-agent heuristic search in a discrete state domain with a finite number of deterministic actions. The states (set S) and actions (set A) form a directed graph with certain specified vertices representing the goal states. The edges (actions) are weighed with action costs: c : A→ R. The agent is provided with a perfect domain model: δ : S ×A→ S. We define the true distance-to-goal function h∗(s) as the sum of action costs along the shortest path from state s to the closest goal state. Generally speaking, the agent uses an approximation h to the unavailable h∗. The approximation is typically inaccurate insomuch as: ∃s ∈ S [h∗(s) 6= h(s)] . For a fixed starting state s, function g(s′) is defined as the sum of action costs along the shortest path from s to s′. Finally, the sum of h or h∗ and g is typically denoted by f or f∗. It easy to see that f∗ remains constant along any optimal path from a fixed state s to the closest goal. Also note that, for any state s′, action a1 is inferior to action a2 iff f∗(δ(s′, a1)) > f∗(δ(s′, a2)). Located in the state s, the agent can use its perfect model δ to predict the states it will get to upon taking various sequences of actions. Two depth 2 lookahead trees are illustrated in Figure 1. RTA* defines policy π(s, p) as follows: (i) consider s’s immediate children {ci}; (ii) for each child ci construct the lookahead search trees of p plies deep by envisioning terminal states of all action sequences of p actions (whenever possible); (iii) evaluate the leaf nodes of each lookahead tree rooted in ci using the f function and select the minimum-valued state which becomes the lookaheadaugmented value of the ci; (iv) output the single action leading to the child ci with the minimum lookahead-augmented f -value (resolve ties randomly). Additionally, a hash table of all previously visited states is maintained. The f -value of a state x is drawn from the table whenever x has been previously visited. After each move, the hash-table value of the state s just left is updated with the second-best f -value among its children ci. This mechanism prevents infinite looping in RTA* under certain s

Searching without a heuristic: efficient use of abstraction

Abstract: In problem domains where an informative heuristic evaluation function is not known or not easily computed, abstraction can be used to derive admissible heuristic values. Optimal path lengths in the abstracted problem are consistent heuristic estimates for the original problem. Pattern databases are the traditional method of creating such heuristics, but they exhaustively compute costs for all abstract states and are thus usually appropriate only when all instances share the same single goal state. Hierarchical heuristic search algorithms address these shortcomings by searching for paths in the abstract space on an as-needed basis. However, existing hierarchical algorithms search less efficiently than pattern database constructors: abstract nodes may be expanded many times during the course of a base-level search. We present a novel hierarchical heuristic search algorithm, called Switchback, that uses an alternating direction of search to avoid abstract node re-expansions. This algorithm is simple to implement and demonstrates superior performance to existing hierarchical heuristic search algorithms on several standard benchmarks.

Probabilistic planning with non-linear utility functions and worst-case guarantees

Abstract: Markov Decision Processes are one of the most widely used frameworks to formulate probabilistic planning problems. Since planners are often risk-sensitive in high-stake situations, non-linear utility functions are often introduced to describe their preferences among all possible outcomes. Alternatively, risk-sensitive decision makers often require their plans to satisfy certain worst-case guarantees.We show how to combine these two approaches by considering problems where we maximize the expected utility of the total reward subject to worst-case constraints. We generalize several existing results on the structure of optimal policies to the constrained case, both for finite and infinite horizon problems. We provide a Dynamic Programming algorithm to compute the optimal policy, and we introduce an admissible heuristic to effectively prune the search space. Finally, we use a stochastic shortest path problem on large real-world road networks to demonstrate the practical applicability of our method.