Optimal planning and heuristic search systems solve state-space search problems by finding a least-cost path from start to goal. As a byproduct of having an optimal path they also determine the optimal solution cost. In this paper we focus on the problem of determining the optimal solution cost for a state-space search problem directly, i.e., without actually finding a solution path of that cost. We present an efficient algorithm, BiSS, based on ideas of bidirectional search and stratified sampling that produces accurate estimates of the optimal solution cost. Our method is guaranteed to return the optimal solution cost in the limit as the sample size goes to infinity. We show empirically that our method makes accurate predictions in several domains. In addition, we show that our method scales to state spaces much larger than can be solved optimally. In particular, we estimate the average solution cost for the 6x6, 7x7, and 8x8 Sliding-Tile Puzzle and provide indirect evidence that these estimates are accurate.

ABSTRACT Traditional heuristic search algorithms use the ranking of states that a heuristic function provides to guide the search. In this paper-with the objective of improving suboptimality and runtime of search algorithms when only weak heuristics are available-we present Stratified Tree Search (STS), a suboptimal heuristic search algorithm that uses a heuristic to partition the state space to guide the search. We call this partition a type system. STS assumes that nodes of the same type will lead to solutions of the same cost. Thus, STS expands only one node of each type in every level of search. We show that in general STS offers a good tradeoff between solution quality and search speed by varying the size of the type system. However, in some cases, STS might not provide a fine adjustment of this tradeoff. We present a variant of STS, Beam STS (BSTS), that allows one to make fine adjustments of this tradeoff. BSTS combines the ideas of STS with those of Beam Search. Our empirical results in benchmark domains show that both STS and BSTS can find solutions of lower suboptimality in less time than standard heuristic search algorithms for finding suboptimal solutions.

Heuristic search is a general problem solving technique. While most evaluations of heuristic search focus on the speed of search, there are relatively few techniques for predicting when search will end. This paper provides a study of progress estimating techniques for optimal, suboptimal, and bounded suboptimal heuristic search algorithms. We examine two previously proposed techniques, search velocity and search vacillation, as well as two new approaches, path-based estimation and distribution-based estimation. We find that both new approaches are better at estimating the remaining amount of search effort than previous work in all three varieties of search, occasionally erring by less than 5%.

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In the last International Planning Competition (IPC 2011), the most efficient planners in the satisficing track were planners that used unit-cost heuristics. These heuristics ignore the real cost of the actions and return instead an estimate of the plan length to the goal. The main advantage of these heuristics compared with real-cost heuristics is that they solve a greater number of problems (also known as coverage), which has a high impact on the IPC score. However, a priori heuristics that predict the real cost should find solutions of better quality. To increase the effectiveness of real-cost heuristics and reduce the impact of their drawbacks without losing quality, we study the use of machine learning techniques to automatically obtain good combinations of those heuristics per domain. In particular, regression techniques are used to predict the real cost from any state to the goal. We use the heuristic estimations and the real costs obtained from solving easy problems as attributes. Later, we feed those instances to several machine learning techniques to obtain prediction models. All learned models approximate the real value with high correlation. Then, we implemented the most suitable model in a planner and evaluated it on harder problems. With this new planner we can solve 56 more problems than using the best real-cost heuristics for each domain separately. Our approach is also better regarding solution quality.

Most bounded suboptimal algorithms in the search literature have been developed so as to be -admissible. This means that the solutions found by these algorithms are guaranteed to be no more than a factor of (1 + ) greater than optimal. However, this is not the only possible form of suboptimality bounding. For example, another possible suboptimality guar-antee is that of additive bounding, which requires that the cost of the solution found is no more than the cost of the optimal solution plus a constant γ. In this work, we consider the problem of developing algo-rithms so as to satisfy a given, and arbitrary, suboptimality requirement. To do so, we develop a theoretical framework which can be used to construct algorithms for a large class of possible suboptimality paradigms. We then use the frame-work to develop additively bounded algorithms, and show that in practice these new algorithms effectively trade-off ad-ditive solution suboptimality for runtime. 1

ruml at cs.unh.edu Many important problems are too difficult to solve optimally. A traditional approach to such problems is bounded suboptimal search, which guarantees solution costs within a userspecified factor of optimal. Recently, a complementary approach has been proposed: bounded-cost search, where solution cost is required to be below a user-specified absolute bound. In this paper, we show how bounded-cost search can incorporate inadmissible estimates of solution cost and solution length. This information has previously been shown to improve bounded suboptimal search and, in an empirical evaluation over five benchmark domains, we find that our new algorithms surpass the state-of-the-art in bounded-cost search as well, particularly for domains where action costs differ.

The notion of finding a solution that is approximately optimal with high probability was recently introduced to the field of heuristic search, formalized as Probably Approximately Correct Heuristic Search, or PAC search in short. A big challenge when constructing a PAC search algorithm is to identify when a given solution achieves the desired sub-optimality with the required confidence, allowing the search to halt and return the incumbent solution. In this paper we propose two novel methods for identifying when a PAC search can halt. Unlike previous work, the new methods provided in this paper become more knowledgeable as the search progresses. This can speedup the search, since the search can halt earlier with the proposed methods and still keeping the desired PAC solution quality guarantees. Experimental results indeed show a substantial speedup of the search in comparison to the previous approach for PAC search. 1

Jabbari Arfaee, Zilles, and Holte presented the bootstrap learning system, a system that learns strong heuristic functions for state-space problems. They showed that IDA * with a bootstrap heuristic is able to quickly find near-optimal solutions in several problem domains. However, the process the bootstrap method uses to learn heuristic functions is time-consuming: it is on the order of days. In this paper we present a learning system that uses an approximation method instead of an exact one to generate the training set required to learn heuristics. We showed recently that solution costs can often be quickly and accurately predicted without having to actually find a solution. In this paper we apply this idea to speedup the process of learning heuristics. In contrast with other learning approaches that use search algorithms to solve problem instances to generate the training set, our system uses a solution cost predictor. We reduce the time required to learn strong heuristics from days to minutes on the domains tested.