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We show how recently-defined abstract models of the Branch-and-Bound algorithm can be used to obtain information on how the nodes are distributed in B&B search trees. This can be directly exploited in the form of probabilities in a sampling algorithm given by Knuth that estimates the size of a search tree. This method reduces the offline estimation error by a factor of two on search trees from Mixed-Integer Programming instances.
Hierarchical task network (HTN) planning is well-known for being an efficient planning approach. This is mainly due to the success of the HTN planning system SHOP2. However, its performance depends on hand-designed search control knowledge. At the time being, there are only very few domain-independent heuristics, which are designed for differing hierarchical planning formalisms. Here, we propose an admissible heuristic for standard HTN planning, which allows to find optimal solutions heuristically. It bases upon the so-called task decomposition graph (TDG), a data structure reflecting reachable parts of the task hierarchy. We show (both in theory and empirically) that rebuilding it during planning can improve heuristic accuracy thereby decreasing the explored search space. The evaluation further studies the heuristic both in terms of plan quality and coverage.
It is well-known that any admissible unidirectional heuristic search algorithm must expand all states whose f-value is smaller than the optimal solution cost when using a consistent heuristic. Such states are called “surely expanded” (s.e.). A recent study characterized s.e. pairs of states for bidirectional search with consistent heuristics: if a pair of states is s.e. then at least one of the two states must be expanded. This paper derives a lower bound, VC, on the minimum number of expansions required to cover all s.e. pairs, and present a new admissible front-to-end bidirectional heuristic search algorithm, Near-Optimal Bidirectional Search (NBS), that is guaranteed to do no more than 2VC expansions. We further prove that no admissible front-to-end algorithm has a worst case better than 2VC. Experimental results show that NBS competes with or outperforms existing bidirectional search algorithms, and often outperforms A* as well.
We want to compute geometric shortest paths in a collection of convex traversable polygons, also known as a navigation mesh. Simple to compute and easy to update, navigation meshes are widely used for pathfinding in computer games. When the mesh is static, shortest path problems can be solved exactly and very fast but only after a costly preprocessing step. When the mesh is dynamic, practitioners turn to online methods which typically compute only approximately shortest paths. In this work we present a new pathfinding algorithm which is compromise-free; i.e. it is simultaneously fast, online and optimal. Our method, Polyanya, extends and generalises Anya; a recent and related interval-based search technique developed for computing geometric shortest paths in grids. We show how that algorithm can be modified to support search over arbitrary sets of convex polygons and then evaluate its performance on a range of realistic and synthetic benchmark problems.
We propose and study, theoretically and empirically, a new random model for the abstract argumentation framework (AF). Our model overcomes some intrinsic difficulties of the only random model of directed graphs in the literature that is relevant to AFs, and makes it possible to study the typical-case complexity of AF instances in terms of threshold behaviours and phase transitions. We proved that the probability for a random AF instance to have a stable/preferred extension goes through a sudden change (from 1 to 0) at the threshold of the parameters of the new model D(n, p, q), satisfying the equation 4q/((1 + q)(1+q)) = p. We showed, empirically, that in this new model, there is a clear easy-hard-easy pattern of hardness (for a typical backtracking-style exact solvers) associated with the phase transition. Our empirical studies indicated that instances from the new model at phase transitions are much harder than those from an Erdos-Renyi-style model with equal edge density. In addition to being an analytically tractable model for understanding the interplay between problems structures and effectiveness of (branching) heuristics used in practical argumentation solvers, the model can also be used to generate, in a systematic way, non-trivial AF instances with controlled features to evaluate the performance of other AF solvers.
Real-time search algorithms are relevant to time-sensitive decision-making domains such as video games and robotics. In such settings, the agent is required to decide on each action under a constant time bound, regardless of the search space size. Despite recent progress, poor-quality solutions can be produced mainly due to state re-visitation. Different techniques have been developed to reduce such a re-visitation with state pruning showing promise. In this paper, we propose a novel pruning approach applicable to the wide class of real-time search algorithms. Given a local search space of arbitrary size, our technique aggressively prunes away all states in its interior, possibly adding new edges to maintain the connectivity of the search space frontier. An experimental evaluation shows that our pruning often improves the performance of a base real-time search algorithm by over an order of magnitude. This allows our implemented system to outperform state-of-the-art real-time search algorithms used in the evaluation.
The graph coloring problem (GCP) is one of the most studied NP hard problems and has numerous applications. Despite the practical importance of GCP, there are limited works in solving GCP for very large graphs. This paper explores techniques for solving GCP on very large real world graphs.We first propose a reduction rule for GCP, which is based on a novel concept called degree bounded independent set.The rule is iteratively executed by interleaving between lower bound computation and graph reduction. Based on this rule, we develop a novel method called FastColor, which also exploits fast clique and coloring heuristics. We carry out experiments to compare our method FastColor with two best algorithms for coloring large graphs we could find. Experiments on a broad range of real world large graphs show the superiority of our method. Additionally, our method maintains both upper bound and lower bound on the optimal solution, and thus it proves an optimal solution when the upper bound meets the lower bound. In our experiments, it proves the optimal solution for 97 out of 142 instances.
We investigate almost uniform sampling from the set of linear extensions of a given partial order. The most efficient schemes stem from Markov chains whose mixing time bounds are polynomial, yet impractically large. We show that, on instances one encounters in practice, the actual mixing times can be much smaller than the worst-case bounds, and particularly so for a novel Markov chain we put forward. We circumvent the inherent hardness of estimating standard mixing times by introducing a refined notion, which admits estimation for moderate-size partial orders. Our empirical results suggest that the Markov chain approach to sample linear extensions can be made to scale well in practice, provided that the actual mixing times can be realized by instance-sensitive upper bounds or termination rules. Examples of the latter include existing perfect simulation algorithms, whose running times in our experiments follow the actual mixing times of certain chains, albeit with significant overhead.