The cost-optimal track of the international planning competition in 2014 has seen an unexpected outcome. Different to the precursing competition in 2011, where explicit-state heuristic search planning scored best, advances in the state-set exploration with BDDs showed a significant lead. In this paper we review the outcome of the competition, briefly looking into the internals of the competing systems.

Merge-and-shrink (M&S) is a framework to generate abstraction heuristics for cost-optimal planning. A recent approach computes simulation relations on a set of M&S abstractions in order to identify states that are better than others. This relation is then used for pruning states in the search when a "better" state is already known. We propose the usage of simulation relations inside the M&S framework in order to detect irrelevant transitions in abstract state spaces. This potentially simplifies the abstraction allowing M&S to derive more informed heuristics. We also tailor M&S to remove irrelevant operators from the planning task. Experimental results show the potential of our approach to construct well-informed heuristics and simplify the planning tasks prior to the search.

Research on heuristic functions is all about estimating the length (or cost) of solution paths. But what if there is no such path? Many known heuristics have the ability to detect (some) unsolvable states, but that ability has always been treated as a by-product. No attempt has been made to design heuristics specifically for that purpose, where there is no need to preserve distances. As a case study towards leveraging that advantage, we investigate merge-and-shrink abstractions in classical planning. We identify safe abstraction steps (no information loss regarding solvability) that would not be safe for traditional heuristics. We design practical algorithm configurations, and run extensive experiments showing that our heuristics outperform the state of the art for proving planning tasks unsolvable.

Symbolic search using binary decision diagrams (BDDs) can often save large amounts of memory due to its concise representation of state sets. A decisive factor for this method's success is the chosen variable ordering. Generally speaking, it is plausible that dependent variables should be brought close together in order to reduce BDD sizes. In planning, variable dependencies are typically captured by means of causal graphs, and in preceding work these were taken as the basis for finding BDD variable orderings. Starting from the observation that the two concepts of "dependency" are actually quite different, we introduce a framework for assessing the strength of variable ordering heuristics in sub-classes of planning. It turns out that, even for extremely simple planning tasks, causal graph based variable orders may be exponentially worse than optimal. Experimental results on a wide range of variable ordering variants corroborate our theoretical findings. Furthermore, we show that dynamic reordering is much more effective at reducing BDD size, but it is not cost-effective due to a prohibitive runtime overhead. We exhibit the potential of middle-ground techniques, running dynamic reordering until simple stopping criteria hold.

Symbolic search using BDDs usually saves huge amounts of memory, while in some domains its savings are moderate at best. It is an open problem to determine if BDDs work well for a certain domain. Motivated by finding evidences for BDD growths for state space search, in this paper we are concerned with symbolic search in the domain of Connect Four. We prove that there is a variable ordering for which the set of all possible states – when continuing after a terminal state has been reached – can be represented by polynomial sized BDDs, whereas the termination criterion leads to an exponential number of nodes in the BDD given any variable ordering.

Symbolic search with BDDs has shown remarkable performance for cost-optimal deterministic planning by exploiting a succinct representation and exploration of state sets. In this paper we enhance BDD-based planning by applying a combination of domain-independent search techniques: the optimization of the variable ordering in the BDD by approximating the linear arrangement problem, pattern selection for improved construction of search heuristics in form of symbolic partial pattern databases, and a decision procedure for the amount of bidirection in the symbolic search process.

In this paper we propose a new algorithm for solving general two-player turn-taking games that performs symbolic search utilizing binary decision diagrams (BDDs). It consists of two stages: First, it determines all breadth-first search (BFS) layers using forward search and omitting duplicate detection, next, the solving process operates in backward direction only within these BFS layers thereby partitioning all BDDs according to the layers the states reside in. We provide experimental results for selected games and compare to a previous approach. This comparison shows that in most cases the new algorithm outperforms the existing one in terms of runtime and used memory so that it can solve games that could not be solved before with a general approach.

For general two-player turn-taking games, first solvers have been contributed. Algorithms for multi-player games like Maxn, however, cannot classify general games robustly, and its extension Soft-Maxn, which can play optimally against unknown and weak opponents, demands large amounts of memory. As RAM is a scarce resource, this paper proposes a memory-efficient implementation of the Soft-Maxnalgorithm, by exploiting the functional representation of state and evaluation sets with BDDs.