In the multi agent path finding problem (MAPF) paths should be found for several agents, each with a different start and goal position such that agents do not collide. Previous optimal solvers applied global A*-based searches. We present a new search algorithm called Conflict Based Search (CBS). CBS is a two-level algorithm. At the high level, a search is performed on a tree based on conflicts between agents. At the low level, a search is performed only for a single agent at a time. In many cases this reformulation enables CBS to examine fewer states than A* while still maintaining optimality. We analyze CBS and show its benefits and drawbacks. Experimental results on various problems shows a speedup of up to a full order of magnitude over previous approaches.

Conflict-based search for optimal multi-agent path finding

Noncooperative game theory provides a normative framework for analyzing strategic interactions. However, for the toolbox to be operational, the solutions it defines will have to be computed. In this paper, we provide a single reduction that 1) demonstrates NP-hardness of determining whether Nash equilibria with certain natural properties exist, and 2) demonstrates the #P-hardness of counting Nash equilibria (or connected sets of Nash equilibria). We also show that 3) determining whether a pure-strategy Bayes-Nash equilibrium exists is NP-hard, and that 4) determining whether a pure-strategy Nash equilibrium exists in a stochastic (Markov) game is PSPACE-hard even if the game is invisible (this remains NP-hard if the game is finite). All of our hardness results hold even if there are only two players and the game is symmetric. 
Keywords: Nash equilibrium; game theory; computational complexity; noncooperative game theory; normal form game; stochastic game; Markov game; Bayes-Nash equilibrium; multiagent systems.

Complexity Results about Nash Equilibria

In order to promote computer Go and stimulate further development and research in the field, the event activities, Computational Intelligence Forum and World 9times9 Computer Go Championship, were held in Taiwan. This study focuses on the invited games played in the tournament Taiwanese Go players versus the computer program MoGo held at the National University of Tainan (NUTN), Tainan, Taiwan. Several Taiwanese Go players, including one 9-Dan (9D) professional Go player and eight amateur Go players, were invited by NUTN to play against MoGo from August 26 to October 4, 2008. The MoGo program combines all-moves-as-first (AMAF)/rapid action value estimation (RAVE) values, online "upper confidence tree (UCT)-like" values, offline values extracted from databases, and expert rules. Additionally, four properties of MoGo are analyzed including: (1) the weakness in corners, (2) the scaling over time, (3) the behavior in handicap games, and (4) the main strength of MoGo in contact fights. The results reveal that MoGo can reach the level of 3 Dan (3D) with: (1) good skills for fights, (2) weaknesses in corners, in particular, for "semeai" situations, and (3) weaknesses in favorable situations such as handicap games. It is hoped that the advances in AI and computational power will enable considerable progress in the field of computer Go, with the aim of achieving the same levels as computer chess or Chinese chess in the future.

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The Computational Intelligence of MoGo Revealed in Taiwan's Computer Go Tournaments

When solving instances of problem domains that feature a large branching factor, A* may generate a large number of nodes whose cost is greater than the cost of the optimal solution. We designate such nodes as surplus. Generating surplus nodes and adding them to the OPEN list may dominate both time and memory of the search. A recently introduced variant of A* called Partial Expansion A* (PEA*) deals with the memory aspect of this problem. When expanding a node n, PEA* generates all of its children and puts into OPEN only the children with f = f(n). n is reinserted in the OPEN list with the f-cost of the best discarded child. This guarantees that surplus nodes are not inserted into OPEN.

In this paper, we present a novel variant of A* called Enhanced Partial Expansion A* (EPEA*) that advances the idea of PEA* to address the time aspect. Given a priori domain-and heuristic-specific knowledge, EPEA* generates only the nodes with f = f(n). Although EPEA* is not always applicable or practical, we study several variants of EPEA*, which make it applicable to a large number of domains and heuristics. In particular, the ideas of EPEA* are applicable to IDA* and to the domains where pattern databases are traditionally used. Experimental studies show significant improvements in run-time and memory performance for several standard benchmark applications. We provide several theoretical studies to facilitate an understanding of the new algorithm.

/pdf/enhanced-partial-expansion-a-404hiigrhe.pdf

Enhanced partial expansion A

Classical methods such as A* and IDA* are a popular and successful choice for one-player games. However, they fail without an accurate admissible evaluation function. In this paper we investigate whether Monte-Carlo Tree Search (MCTS) is an interesting alternative for one-player games where A* and IDA* methods do not perform well. Therefore, we propose a new MCTS variant, called Single-Player Monte-Carlo Tree Search (SP-MCTS). The selection and backpropagation strategy in SP-MCTS are different from standard MCTS. Moreover, SP-MCTS makes use of a straightforward Meta-Search extension. We tested the method on the puzzle SameGame. It turned out that our SP-MCTS program gained the highest score so far on the standardized test set.

/pdf/single-player-monte-carlo-tree-search-390a4ae3hs.pdf

Single-Player Monte-Carlo Tree Search

Informally, a set of abstractions of a state space S is additive if the distance between any two states in S is always greater than or equal to the sum of the corresponding distances in the abstract spaces. The first known additive abstractions, called disjoint pattern databases, were experimentally demonstrated to produce state of the art performance on certain state spaces. However, previous applications were restricted to state spaces with special properties, which precludes disjoint pattern databases from being defined for several commonly used testbeds, such as Rubik's Cube, TopSpin and the Pancake puzzle. In this paper we give a general definition of additive abstractions that can be applied to any state space and prove that heuristics based on additive abstractions are consistent as well as admissible. We use this new definition to create additive abstractions for these testbeds and show experimentally that well chosen additive abstractions can reduce search time substantially for the (18,4)-TopSpin puzzle and by three orders of magnitude over state of the art methods for the 17-Pancake puzzle. We also derive a way of testing if the heuristic value returned by additive abstractions is provably too low and show that the use of this test can reduce search time for the 15-puzzle and TopSpin by roughly a factor of two.

/pdf/a-general-theory-of-additive-state-space-abstractions-100lw8f8qw.pdf

A general theory of additive state space abstractions

A pattern database (PDB) is a heuristic function stored as a lookup table. Symmetries of a state space are often used to enable multiple values to be looked up in a PDB for a given state. This paper introduces an additional PDB lookup, called the dual PDB lookup. A dual PDB lookup is always admissible but can return inconsistent values. The paper also presents an extension of the well-known pathmax method so that inconsistencies in heuristic values are propagated in both directions (child-to-parent, and parent-to-child) in the search tree. Experiments show that the addition of dual lookups and bidirectional pathmax propagation can reduce the number of nodes generated by IDA* by over one order of magnitude in the TopSpin puzzle and Rubik's Cube, and by about a factor of two for the sliding tile puzzles.

/pdf/dual-lookups-in-pattern-databases-b40vrirh7m.pdf

Dual lookups in pattern databases

https://web.cs.du.edu/~sturtevant/papers/incnew.pdf

Inconsistent heuristics in theory and practice

Korf, Reid, and Edelkamp introduced a formula to predict the number of nodes IDA* will expand on a single iteration for a given consistent heuristic, and experimentally demonstrated that it could make very accurate predictions. In this paper we show that, in addition to requiring the heuristic to be consistent, their formula's predictions are accurate only at levels of the brute-force search tree where the heuristic values obey the unconditional distribution that they defined and then used in their formula. We then propose a new formula that works well without these requirements, i.e., it can make accurate predictions of IDA*'s performance for inconsistent heuristics and if the heuristic values in any level do not obey the unconditional distribution. In order to achieve this we introduce the conditional distribution of heuristic values which is a generalization of their unconditional heuristic distribution. We also provide extensions of our formula that handle individual start states and the augmentation of IDA* with bidirectional pathmax (BPMX), a tech nique for propagating heuristic values when inconsistent heuristics are used. Experimental results demonstrate the accuracy of our new method and all its variations.

/pdf/predicting-the-performance-of-ida-using-conditional-45x486dz2l.pdf

Predicting the performance of IDA* using conditional distributions

In the field of heuristic search it is well-known that improving the quality of an admissible heuristic can significantly decrease the search effort required to find an optimal solution. Existing literature often assumes that admissible heuristics are consistent, implying that consistency is a desirable attribute. To the contrary, this paper shows that an inconsistent heuristic can be preferable to a consistent heuristic. Theoretical and empirical results show that, in many cases, inconsistency can be used to achieve large performance improvements.

Uzi Zahavi

Papers

A general theory of additive state space abstractions

Dual lookups in pattern databases

Inconsistent heuristics in theory and practice

Predicting the performance of IDA* using conditional distributions

Inconsistent heuristics