Top- K queries provide a ranked answer using a score that can either be given explicitly or computed from tuple values. Recommender systems use scores, based on user feedback on items with which they interact, to answer top- K queries. Such scores pose the challenge of correctly ranking elements using scores that are more often than not, uncertain. In this work, we address top- K queries based on uncertain scores. We propose to explicitly model the inherent uncertainty in the provided data and to consider a distribution of scores instead of a single score. Rooted in works of database probabilistic ranking, we offer the use of probabilistic ranking as a tool of choice for generating recommendation in the presence of uncertainty. We argue that the ranking approach should be chosen in a manner that maximizes user satisfaction, extending state-of-the-art on quality aspect of top- K answers over uncertain data, their relationship to top- K semantics, and improve ranking with uncertain scores in recommender systems. Towards this end, we introduce RankDist, an algorithm for efficiently computing probability of item position in a ranked recommendation. We show that rank-based (rather than score-based) methods that are computed using RankDist, which were not applied in recommender systems before, offer a guaranteed optimality by expectation and empirical superiority when tested on common benchmarks.

Interruptible pure exploration in multi-armed bandits (MABs) is a key component of Monte-Carlo tree search algorithms for sequential decision problems. We introduce Discriminative Bucketing (DB), a novel family of strategies for pure exploration in MABs, which allows for adapting recent advances in non-interruptible strategies to the interruptible setting, while guaranteeing exponential-rate performance improvement over time. Our experimental evaluation demonstrates that the corresponding instances of DB favorably compete both with the currently popular strategies UCB1 and Epsilon-Greedy, as well as with the conservative uniform sampling.

Fork decoupling is a recent approach to exploiting problem structure in state space search. The problem is assumed to take the form of a fork, where a single (large) center component provides preconditions for several (small) leaf components. The leaves are then conditionally independent in the sense that, given a fixed center path p, the compliant leaf moves - those leaf moves enabled by the preconditions supplied along p - can be scheduled independently for each leaf. Fork-decoupled state space search exploits this through conducting a regular search over center paths, augmented with maintenance of the compliant paths for each leaf individually. We herein show that the same ideas apply to much more general star-topology structures, where leaves may supply preconditions for the center, and actions may affect several leaves simultaneously as long as they also affect the center. Our empirical evaluation in planning, super-imposing star topologies by automatically grouping the state variables into suitable components, shows the merits of the approach.

While in classical planning the objective is to achieve one of the equally attractive goal states at as low total action cost as possible, the objective in deterministic oversubscription planning (OSP) is to achieve an as valuable as possible subset of goals within a fixed allowance of the total action cost. Although numerous applications in various fields share the latter objective, no substantial algorithmic advances have been made in deterministic OSP. Tracing the key sources of progress in classical planning, we identify a severe lack of effective domain-independent approximations for OSP. With our focus here on optimal planning, our goal is to bridge this gap. Two classes of approximation techniques have been found especially useful in the context of optimal classical planning: those based on state-space abstractions and those based on logical landmarks for goal reachability. The question we study here is whether some similar-in-spirit, yet possibly mathematically different, approximation techniques can be developed for OSP. In the context of abstractions, we define the notion of additive abstractions for OSP, study the complexity of deriving effective abstractions from a rich space of hypotheses, and reveal some substantial, empirically relevant islands of tractability. In the context of landmarks, we show how standard goal-reachability landmarks of certain classical planning tasks can be compiled into the OSP task of interest, resulting in an equivalent OSP task with a lower cost allowance, and thus with a smaller search space. Our empirical evaluation confirms the effectiveness of the proposed techniques, and opens a wide gate for further developments in oversubscription planning.

State-of-the-art Monte-Carlo tree search algorithms can be parametrized with any of the two information updating procedures: MC-backup and DP-backup. The dynamics of these two procedures is very different, and so far, their relative pros and cons have been poorly understood. Formally analyzing the dependency of MC- and DP-backups on various MDP parameters, we reveal numerous important issues that get hidden by the worst-case bounds on the algorithm performance, and reconfirm these findings by a systematic experimental test.

In the basic setup of oversubscription planning (OSP), the objective is to achieve an as valuable as possible subset of goals within a fixed allowance of the total action cost [32]. Continuing from the recent successes in exploiting logical goal-reachability landmarks in classical planning, we develop a framework for exploiting such landmarks in heuristic-search OSP. We show how standard landmarks of certain classical planning tasks can be compiled into the OSP task of interest, resulting in an equivalent OSP task with a lower budget, and thus with a smaller search space. We then show how such landmark-based task enrichment can be combined in a mutually stratifying way with the BFBB search used for OSP planning. Our empirical evaluation confirms the effectiveness of the proposed landmark-based budget reduction scheme.

Online planning algorithms are typically a tool of choice for dealing with sequential decision problems in combinatorial search spaces. Many such problems, however, also exhibit combinatorial actions, yet standard planning algorithms do not cope well with this type of “the curse of dimensionality”. Following a recently opened line of related work on combinatorial multi-armed bandit (CMAB) problems, we propose a novel CMAB planning scheme, as well as two specific instances of this scheme, dedicated to exploiting what is called linear side information. Using a representative strategy game as a benchmark, we show that the resulting algorithms very favorably compete with the state-of-the-art.

We consider online planning in Markov decision processes (MDPs). In online planning, the agent focuses on its current state only, deliberates about the set of possible policies from that state onwards and, when interrupted, uses the outcome of that exploratory deliberation to choose what action to perform next. Formally, the performance of algorithms for online planning is assessed in terms of simple regret, the agent's expected performance loss when the chosen action, rather than an optimal one, is followed. To date, state-of-the-art algorithms for online planning in general MDPs are either best effort, or guarantee only polynomial-rate reduction of simple regret over time. Here we introduce a new Monte-Carlo tree search algorithm, BRUE, that guarantees exponential-rate and smooth reduction of simple regret. At a high level, BRUE is based on a simple yet non-standard state-space sampling scheme, MCTS2e, in which different parts of each sample are dedicated to different exploratory objectives. We further extend BRUE with a variant of ``learning by forgetting.'' The resulting parametrized algorithm, BRUE(alpha), exhibits even more attractive formal guarantees than BRUE. Our empirical evaluation shows that both BRUE and its generalization, BRUE(alpha), are also very effective in practice and compare favorably to the state-of-the-art.

The increased demand for distributed computations on “big data” has led to solutions such as SCOPE, DryadLINQ, Pig, and Hive, which allow the user to specify queries in an SQL-like language, enriched with sets of user-defined operators. The lack of exact semantics for user-defined operators interferes with the query optimization process, thus putting the burden of suggesting, at least partial, query plans on the user. In an attempt to ease this burden, we propose a formal model that allows for data-parallel program synthesis (DPPS) in a semantically well-defined manner. We show that this model generalizes existing frameworks for data-parallel computation, while providing the flexibility of query plan generation that is currently absent from these frameworks. In particular, we show how existing, off-the-shelf, AI planning tools can be used for solving DPPS tasks.

Despite its success, the delete relaxation has significant pitfalls. Recent work has devised the red-black planning framework, where red variables take the relaxed semantics (accumulating their values), while black variables take the regular semantics. Provided the red variables are chosen so that red-black plan generation is tractable, one can generate such a plan for every search state, and take its length as the heuristic distance estimate. Previous results were not suitable for this purpose because they identified tractable fragments for red-black plan existence, as opposed to red-black plan generation. We identify a new fragment of red-black planning, that fixes this issue. We devise machinery to efficiently generate red-black plans, and to automatically select the red variables. Experiments show that the resulting heuristics can significantly improve over standard delete relaxation heuristics.

Monte-Carlo search is successfully used in simulation-based planning for various large-scale sequential decision problems, and the UCT algorithm seems to be the choice in most (if not all) such recent success stories. Based on some recent discoveries in theory and empirical analysis of Monte-Carlo search, here we argue that, if online sequential decision making is your problem, and Monte-Carlo tree search is your way to go, then UCT is unlikely to be the best fit for your needs.

For almost two decades, monotonic, or ``delete free,'' relaxation has been one of the key auxiliary tools in the practice of domain-independent deterministic planning. In the particular contexts of both satisficing and optimal planning, it underlies most state-of-the-art heuristic functions. While satisficing planning for monotonic tasks is polynomial-time, optimal planning for monotonic tasks is NP-equivalent. Here we establish both negative and positive results on the complexity of some wide fragments of optimal monotonic planning, with the fragments being defined around the causal graph topology. Our results shed some light on the link between the complexity of general optimal planning and the complexity of optimal planning for the respective monotonic relaxations.

Domain-independent planning is one of the foundational areas in the field of Artificial Intelligence. A description of a planning task consists of an initial world state, a goal, and a set of actions for modifying the world state. The objective is to find a sequence of actions, that is, a plan, that transforms the initial world state into a goal state. In optimal planning, we are interested in finding not just a plan, but one of the cheapest plans. A prominent approach to optimal planning these days is heuristic state-space search, guided by admissible heuristic functions. Numerous admissible heuristics have been developed, each with its own strengths and weaknesses, and it is well known that there is no single "best'' heuristic for optimal planning in general. Thus, which heuristic to choose for a given planning task is a difficult question. This difficulty can be avoided by combining several heuristics, but that requires computing numerous heuristic estimates at each state, and the tradeoff between the time spent doing so and the time saved by the combined advantages of the different heuristics might be high. We present a novel method that reduces the cost of combining admissible heuristics for optimal planning, while maintaining its benefits. Using an idealized search space model, we formulate a decision rule for choosing the best heuristic to compute at each state. We then present an active online learning approach for learning a classifier with that decision rule as the target concept, and employ the learned classifier to decide which heuristic to compute at each state. We evaluate this technique empirically, and show that it substantially outperforms the standard method for combining several heuristics via their pointwise maximum.

Various operational control systems (OCS) are naturally modeled as Markov Decision Processes. OCS often enjoy access to predictions of future events that have substantial impact on their operations. For example, reliable forecasts of extreme weather conditions are widely available, and such events can affect typical request patterns for customer response management systems, the flight and service time of airplanes, or the supply and demand patterns for electricity. The space of exogenous events impacting OCS can be very large, prohibiting their modeling within the MDP; moreover, for many of these exogenous events there is no useful predictive, probabilistic model. Realtime predictions, however, possibly with a short lead-time, are often available. In this work we motivate a model which combines offline MDP infinite horizon planning with realtime adjustments given specific predictions of future exogenous events, and suggest a framework in which such predictions are captured and trigger real-time planning problems. We propose a number of variants of existing MDP solution algorithms, adapted to this context, and evaluate them empirically.

Connecting between standard AI planning constructs and a classical cooperative model of transferable-utility coalition games, we introduce the notion of transferable-utility (TU) planning games. The key representational property of these games is that coalitions are valued implicitly based on their ability to carry out efficient joint plans. On the side of the expressiveness, we show that existing succinct representations of monotonic TU games can be efficiently compiled into TU planning games. On the side of computation, TU planning games allow us to provide some of the strongest to date tractability results for core-existence and core-membership queries in succinct TU coalition games.

It is well known that there cannot be a single "best" heuristic for optimal planning in general. One way of overcoming this is by combining admissible heuristics (e.g. by using their maximum), which requires computing numerous heuristic estimates at each state. However, there is a tradeoff between the time spent on computing these heuristic estimates for each state, and the time saved by reducing the number of expanded states. We present a novel method that reduces the cost of combining admissible heuristics for optimal search, while maintaining its benefits. Based on an idealized search space model, we formulate a decision rule for choosing the best heuristic to compute at each state. We then present an active online learning approach for that decision rule, and employ the learned model to decide which heuristic to compute at each state. We evaluate this technique empirically, and show that it substantially outperforms each of the individual heuristics that were used, as well as their regular maximum.

State-space search with explicit abstraction heuristics is at the state of the art of cost-optimal planning. These heuristics are inherently limited, nonetheless, because the size of the abstract space must be bounded by some, even if a very large, constant. Targeting this shortcoming, we introduce the notion of (additive) implicit abstractions, in which the planning task is abstracted by instances of tractable fragments of optimal planning. We then introduce a concrete setting of this framework, called fork-decomposition, that is based on two novel fragments of tractable cost-optimal planning. The induced admissible heuristics are then studied formally and empirically. This study testifies for the accuracy of the fork decomposition heuristics, yet our empirical evaluation also stresses the tradeoff between their accuracy and the runtime complexity of computing them. Indeed, some of the power of the explicit abstraction heuristics comes from precomputing the heuristic function offline and then determining h(s) for each evaluated state s by a very fast lookup in a ``database.'' By contrast, while fork-decomposition heuristics can be calculated in polynomial time, computing them is far from being fast. To address this problem, we show that the time-per-node complexity bottleneck of the fork-decomposition heuristics can be successfully overcome. We demonstrate that an equivalent of the explicit abstraction notion of a ``database'' exists for the fork-decomposition abstractions as well, despite their exponential-size abstract spaces. We then verify empirically that heuristic search with the ``databased" fork-decomposition heuristics favorably competes with the state of the art of cost-optimal planning.

Various tasks in decision making and decision support systems require selecting a preferred subset of a given set of items. Here we focus on problems where the individual items are described using a set of characterizing attributes, and a generic preference specification is required, that is, a specification that can work with an arbitrary set of items. For example, preferences over the content of an online newspaper should have this form: At each viewing, the newspaper contains a subset of the set of articles currently available. Our preference specification over this subset should be provided offline, but we should be able to use it to select a subset of any currently available set of articles, e.g., based on their tags. We present a general approach for lifting formalisms for specifying preferences over objects with multiple attributes into ones that specify preferences over subsets of such objects. We also show how we can compute an optimal subset given such a specification in a relatively efficient manner. We provide an empirical evaluation of the approach as well as some worst-case complexity results.

Planning as satisfiability, as implemented in, for instance, the SATPLAN tool, is a highly competitive method for finding parallel step-optimal plans. A bottleneck in this approach is to *prove the absence* of plans of a certain length. Specifically, if the optimal plan has N steps, then it is typically very costly to prove that there is no plan of length N-1. We pursue the idea of leading this proof within solution length preserving abstractions (over-approximations) of the original planning task. This is promising because the abstraction may have a much smaller state space; related methods are highly successful in model checking. In particular, we design a novel abstraction technique based on which one can, in several widely used planning benchmarks, construct abstractions that have exponentially smaller state spaces while preserving the length of an optimal plan. Surprisingly, the idea turns out to appear quite hopeless in the context of planning as satisfiability. Evaluating our idea empirically, we run experiments on almost all benchmarks of the international planning competitions up to IPC 2004, and find that even hand-made abstractions do not tend to improve the performance of SATPLAN. Exploring these findings from a theoretical point of view, we identify an interesting phenomenon that may cause this behavior. We compare various planning-graph based CNF encodings F of the original planning task with the CNF encodings F_abs of the abstracted planning task. We prove that, in many cases, the shortest resolution refutation for F_abs can never be shorter than that for F. This suggests a fundamental weakness of the approach, and motivates further investigation of the interplay between declarative transition-systems, over-approximating abstractions, and SAT encodings.

To improve the precision at the very top ranks of a document list presented in response to a query, researchers suggested to exploit information induced from clustering of documents highly ranked by some initial search. We propose a novel model for ranking such (query-specific) clusters by the presumed percentage of relevant documents that they contain. The model is based on (i) proposing a palette of "witness" cluster properties that purportedly correlate with this percentage, (ii) devising concrete quantitative measures for these properties, and (iii) ordering the clusters via aggregation of rankings induced by these individual measures. Empirical evaluation shows that our model is consistently more effective than previously suggested methods in detecting clusters containing a high relevant-document percentage. Furthermore, the precision-at-top-ranks performance of this model transcends that of standard document-based retrieval, and competes with that of a state-of-the-art document-based retrieval approach.

We study the complexity of cost-optimal classical planning over propositional state variables and unary-effect actions. We discover novel problem fragments for which such optimization is tractable, and identify certain conditions that differentiate between tractable and intractable problems. These results are based on exploiting both structural and syntactic characteristics of planning problems. Specifically, following Brafman and Domshlak (2003), we relate the complexity of planning and the topology of the causal graph. The main results correspond to tractability of cost-optimal planning for propositional problems with polytree causal graphs that either have O(1)-bounded in-degree, or are induced by actions having at most one prevail condition each. Almost all our tractability results are based on a constructive proof technique that connects between certain tools from planning and tractable constraint optimization, and we believe this technique is of interest on its own due to a clear evidence for its robustness.

Users tend to store huge amounts of files, of various formats, on their personal computers. As a result, finding a specific, desired file within the file system is a challenging task. This article addresses thedesktop searchproblem by considering various techniques for ranking results of a search query over the file system. First, basic ranking techniques, which are based on various file features (e.g., file name, access date, file size, etc.), are considered and their effectiveness is empirically analyzed. Next, two learning-based ranking schemes are presented, and are shown to be significantly more effective than the basic ranking methods. Finally, a novel ranking technique, based on query selectiveness, is considered for use during the cold-start period of the system. This method is also shown to be empirically effective, even though it does not involve any learning.

We propose a language-model-based approach for addressing the performance robustness problem -- with respect to free-parameters' values -- of pseudo-feedback-based query-expansion methods. Given a query, we create a set of language models representing different forms of its expansion by varying the parameters' values of some expansion method; then, we select a single model using criteria originally proposed for evaluating the performance of using the original query, or for deciding whether to employ expansion at all. Experimental results show that these criteria are highly effective in selecting relevance language models that are not only significantly more effective than poor performing ones, but that also yield performance that is almost indistinguishable from that of manually optimized relevance models.

This paper addresses the desktop search problem by considering varioustechniques for ranking results of a search query over thefile system. First, basic ranking techniques, which are based ona single file feature (e.g., file name, file content, access date, etc.)are considered. Next, two learning-based ranking schemes are presented, and are shown to be significantly more effective than the basic ranking methods. Finally, a novel ranking technique, based on query selectiveness is considered,for use during the cold-start period of the system. This method isalso shown to be empirically effective, even though it does notinvolve any learning.

We present a new algorithm for probabilistic planning with no observability. Our algorithm, called Probabilistic-FF, extends the heuristic forward-search machinery of Conformant-FF to problems with probabilistic uncertainty about both the initial state and action effects. Specifically, Probabilistic-FF combines Conformant-FF's techniques with a powerful machinery for weighted model counting in (weighted) CNFs, serving to elegantly define both the search space and the heuristic function. Our evaluation of Probabilistic-FF shows its fine scalability in a range of probabilistic domains, constituting a several orders of magnitude improvement over previous results in this area. We use a problematic case to point out the main open issue to be addressed by further research.

Schema matching is a basic operation of data integration, and several tools for automating it have been proposed and evaluated in the database community. Research in this area reveals that there is no single schema matcher that is guaranteed to succeed in finding a good mapping for all possible domains and, thus, an ensemble of schema matchers should be considered. In this paper, we introduce schema metamatching, a general framework for composing an arbitrary ensemble of schema matchers and generating a list of best ranked schema mappings. Informally, schema metamatching stands for computing a "consensus" ranking of alternative mappings between two schemata, given the "individual" graded rankings provided by several schema matchers. We introduce several algorithms for this problem, varying from adaptations of some standard techniques for general quantitative rank aggregation to novel techniques specific to the problem of schema matching, and to combinations of both. We provide a formal analysis of the applicability and relative performance of these algorithms and evaluate them empirically on a set of real-world schemata

In recent years, CP-nets have emerged as a useful tool for supporting preference elicitation, reasoning, and representation. CP-nets capture and support reasoning with qualitative conditional preference statements, statements that are relatively natural for users to express. In this paper, we extend the CP-nets formalism to handle another class of very natural qualitative statements one often uses in expressing preferences in daily life - statements of relative importance of attributes. The resulting formalism, TCP-nets, maintains the spirit of CP-nets, in that it remains focused on using only simple and natural preference statements, uses the ceteris paribus semantics, and utilizes a graphical representation of this information to reason about its consistency and to perform, possibly constrained, optimization using it. The extra expressiveness it provides allows us to better model tradeoffs users would like to make, more faithfully representing their preferences.

We present a novel approach to pseudo-feedback-based ad hoc retrieval that uses language models induced from both documents and clusters. First, we treat the pseudo-feedback documents produced in response to the original query as a set of pseudo-query that themselves can serve as input to the retrieval process. Observing that the documents returned in response to the pseudo-query can then act as pseudo-query for subsequent rounds, we arrive at a formulation of pseudo-query-based retrieval as an iterative process. Experiments show that several concrete instantiations of this idea, when applied in conjunction with techniques designed to heighten precision, yield performance results rivaling those of a number of previously-proposed algorithms, including the standard language-modeling approach. The use of cluster-based language models is a key contributing factor to our algorithms' success.

Multiagent opportunism refers to the ability of agents operating in a multiagent system (MAS) to recognize and respond to potential opportunities for mutual assistance in achieving individual goals. Two major potential obstacles in operationalizing multiagent opportunistic assistance in real-world systems are (i) low amounts of knowledge shared between the agents, and (ii) limited ability of the agents to re-plan dynamically. We have previously shown that even under these limiting conditions, systems of agents can benefit from multiagent opportunism. In this work we discuss how multi-agent systems can exploit shared knowledge for opportunistic predictive encoding using an approach based on an abstract plan representation called partial order plan graphs (POPGs). Further, we present several approaches for increasing system-level performance by improving the efficiency of the plans containing predictively encoded opportunities, as well as the results of an empirical analysis of their impact on the system performance.

Many artificial intelligence (AI) tasks, such as product configuration, decision support, and the construction of autonomous agents, involve a process of constrained optimization, that is, optimization of behavior or choices subject to given constraints. In this paper we present an approach for constrained optimization based on a set of hard constraints and a preference ordering represented using a CP‐network—a graphical model for representing qualitative preference information. This approach offers both pragmatic and computational advantages. First, it provides a convenient and intuitive tool for specifying the problem, and in particular, the decision maker's preferences. Second, it admits an algorithm for finding the most preferred feasible (Pareto‐optimal) outcomes that has the following anytime property: the set of preferred feasible outcomes are enumerated without backtracking. In particular, the first feasible solution generated by this algorithm is Pareto optimal.

Prior work has shown that context-specific independence (CSI) in Bayes networks can be exploited to speed up belief updating. We examine how networks with variables exhibiting mutual exclusion (e.g., “selector variables”), as well as CSI, can be efficiently updated. In particular, directed-path singly connected and polytree networks that have an additional common selector variable can be updated in linear time (given null and general conjunctive evidence, respectively), where quadratic time would be needed without the mutual exclusion requirement. The above results have direct applications, as such network topologies can be used in predicting the ramifications of user selection in some multimedia data browsing systems. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 703–725, 2004.

Information about user preferences plays a key role in automated decision making. In many domains it is desirable to assess such preferences in a qualitative rather than quantitative way. In this paper, we propose a qualitative graphical representation of preferences that reflects conditional dependence and independence of preference statements under a ceteris paribus (all else being equal) interpretation. Such a representation is often compact and arguably quite natural in many circumstances. We provide a formal semantics for this model, and describe how the structure of the network can be exploited in several inference tasks, such as determining whether one outcome dominates (is preferred to) another, ordering a set outcomes according to the preference relation, and constructing the best outcome subject to available evidence.

We present a new approach for adaptive presentation of structured information, based on preference-based constrained optimization techniques rooted in qualitative decision-theory. In this approach, document presentation is viewed as a configuration problem whose goal is to determine the optimal presentation of a document, while taking into account the preferences of the content provider, viewer interaction with the browser, and, possibly, some layout constraints. The preferences of the content provider are represented by a CP-net, a graphical, qualitative preference model developed in Boutilier et al. [1999]. The layout constraints are represented as geometric constraints, integrated within the optimization process. We discuss the theoretical basis of our approach, as well as implemented prototype systems for Web pages and for general media-rich document presentation.

Unary operator domains -- i.e., domains in which operators have a single effect -- arise naturally in many control problems. In its most general form, the problem of STRIPS planning in unary operator domains is known to be as hard as the general STRIPS planning problem -- both are PSPACE-complete. However, unary operator domains induce a natural structure, called the domain's causal graph. This graph relates between the preconditions and effect of each domain operator. Causal graphs were exploited by Williams and Nayak in order to analyze plan generation for one of the controllers in NASA's Deep-Space One spacecraft. There, they utilized the fact that when this graph is acyclic, a serialization ordering over any subgoal can be obtained quickly. In this paper we conduct a comprehensive study of the relationship between the structure of a domain's causal graph and the complexity of planning in this domain. On the positive side, we show that a non-trivial polynomial time plan generation algorithm exists for domains whose causal graph induces a polytree with a constant bound on its node indegree. On the negative side, we show that even plan existence is hard when the graph is a directed-path singly connected DAG. More generally, we show that the number of paths in the causal graph is closely related to the complexity of planning in the associated domain. Finally we relate our results to the question of complexity of planning with serializable subgoals.