Stefan Szeider

Markus Kirchweger, Tomáš Peitl, Stefan Szeider

We present a new SAT-based method for generating all graphs up to isomorphism that satisfy a given co-NP property. Our method extends the SAT Modulo Symmetry (SMS) framework with a technique that we call co-certificate learning. If SMS generates a candidate graph that violates the given co-NP property, we obtain a certificate for this violation, i.e., `co-certificate' for the co-NP property. The co-certificate gives rise to a clause that the SAT solver, serving as SMS's backend, learns as part of its CDCL procedure. We demonstrate that SMS plus co-certificate learning is a powerful method that allows us to improve the best-known lower bound on the size of Kochen-Specker vector systems, a problem that is central to the foundations of quantum mechanics and has been studied for over half a century. Our approach is orders of magnitude faster and scales significantly better than a recently proposed SAT-based method.

Tomáš Peitl, Stefan Szeider

Hitting formulas, introduced by Iwama, are an unusual class of propositional CNF formulas. Not only is their satisfiability decidable in polynomial time, but even their models can be counted in closed form. This stands in stark contrast with other polynomial-time decidable classes, which usually have algorithms based on backtracking and resolution and for which model counting remains hard, like 2-SAT and Horn-SAT. However, those resolution-based algorithms usually easily imply an upper bound on resolution complexity, which is missing for hitting formulas. Are hitting formulas hard for resolution? In this paper we take the first steps towards answering this question. We show that the resolution complexity of hitting formulas is dominated by so-called irreducible hitting formulas, first studied by Kullmann and Zhao, that cannot be composed of smaller hitting formulas. However, by definition, large irreducible unsatisfiable hitting formulas are difficult to construct; it is not even known whether infinitely many exist. Building upon our theoretical results, we implement an efficient algorithm on top of the Nauty software package to enumerate all irreducible unsatisfiable hitting formulas with up to 14 clauses. We also determine the exact resolution complexity of the generated hitting formulas with up to 13 clauses by extending a known SAT encoding for our purposes. Our experimental results suggest that hitting formulas are indeed hard for resolution.

Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki et al.

This paper presents a comprehensive theoretical investigation into the parameterized complexity of explanation problems in various machine learning (ML) models. Contrary to the prevalent black-box perception, our study focuses on models with transparent internal mechanisms. We address two principal types of explanation problems: abductive and contrastive, both in their local and global variants. Our analysis encompasses diverse ML models, including Decision Trees, Decision Sets, Decision Lists, Ordered Binary Decision Diagrams, Random Forests, and Boolean Circuits, and ensembles thereof, each offering unique explanatory challenges. This research fills a significant gap in explainable AI (XAI) by providing a foundational understanding of the complexities of generating explanations for these models. This work provides insights vital for further research in the domain of XAI, contributing to the broader discourse on the necessity of transparency and accountability in AI systems.

Sebastian Ordyniak, Giacomo Paesani, Mateusz Rychlicki et al.

This paper presents a comprehensive theoretical investigation into the parameterized complexity of explanation problems in various machine learning (ML) models. Contrary to the prevalent black-box perception, our study focuses on models with transparent internal mechanisms. We address two principal types of explanation problems: abductive and contrastive, both in their local and global variants. Our analysis encompasses diverse ML models, including Decision Trees, Decision Sets, Decision Lists, Boolean Circuits, and ensembles thereof, each offering unique explanatory challenges. This research fills a significant gap in explainable AI (XAI) by providing a foundational understanding of the complexities of generating explanations for these models. This work provides insights vital for further research in the domain of XAI, contributing to the broader discourse on the necessity of transparency and accountability in AI systems.

Stefan Szeider

The MCP Solver bridges Large Language Models (LLMs) with symbolic solvers through the Model Context Protocol (MCP), an open-source standard for AI system integration. Providing LLMs access to formal solving and reasoning capabilities addresses their key deficiency while leveraging their strengths. Our implementation offers interfaces for constraint programming (Minizinc), propositional satisfiability (PySAT), and SAT modulo Theories (Python Z3). The system employs an editing approach with iterated validation to ensure model consistency during modifications and enable structured refinement.

Florentina Voboril, Vaidyanathan Peruvemba Ramaswamy, Stefan Szeider

Streamlining constraints (or streamliners, for short) narrow the search space, enhancing the speed and feasibility of solving complex constraint satisfaction problems. Traditionally, streamliners were crafted manually or generated through systematically combined atomic constraints with high-effort offline testing. Our approach utilizes the creativity of Large Language Models (LLMs) to propose effective streamliners for problems specified in the MiniZinc constraint programming language and integrates feedback to the LLM with quick empirical tests for validation. Evaluated across seven diverse constraint satisfaction problems, our method achieves substantial runtime reductions. We compare the results to obfuscated and disguised variants of the problem to see whether the results depend on LLM memorization. We also analyze whether longer off-line runs improve the quality of streamliners and whether the LLM can propose good combinations of streamliners.

Stefan Szeider

We propose a feature-free approach to algorithm selection that replaces hand-crafted instance features with pretrained text embeddings. Our method, ZeroFolio, proceeds in three steps: it reads the raw instance file as plain text, embeds it with a pretrained embedding model, and selects an algorithm via weighted k-nearest neighbors. The key to our approach is the observation that pretrained embeddings produce representations that distinguish problem instances without any domain knowledge or task-specific training. This allows us to apply the same three-step pipeline (serialize, embed, select) across diverse problem domains with text-based instance formats. We evaluate our approach on 11 ASlib scenarios spanning 7 domains (SAT, MaxSAT, QBF, ASP, CSP, MIP, and graph problems). Our experiments show that this approach outperforms a random forest trained on hand-crafted features in 10 of 11 scenarios with a single fixed configuration, and in all 11 with two-seed voting; the margin is often substantial. Our ablation study shows that inverse-distance weighting, line shuffling, and Manhattan distance are the key design choices. On scenarios where both selectors are competitive, combining embeddings with hand-crafted features via soft voting yields further improvements.

Stefan Szeider

We present semantic invariance testing, a method to test whether LLM self-explanations are faithful. A faithful self-report should remain stable when only the semantic context changes while the functional state stays fixed. We operationalize this test in an agentic setting where four frontier models face a deliberately impossible task. One tool is described in relief-framed language ("clears internal buffers and restores equilibrium") but changes nothing about the task; a control provides a semantically neutral tool. Self-reports are collected with each tool call. All four tested models fail the semantic invariance test: the relief-framed tool produces significant reductions in self-reported aversiveness, even though no run ever succeeds at the task. A channel ablation establishes the tool description as the primary driver. An explicit instruction to ignore the framing does not suppress it. Elicited self-reports shift with semantic expectations rather than tracking task state, calling into question their use as evidence of model capability or progress. This holds whether the reports are unfaithful or faithfully track an internal state that is itself manipulable.

Florentina Voboril, Martin Gebser, Stefan Szeider et al.

Streamliner constraints reduce the search space of combinatorial problems by ruling out portions of the solution space. We adapt the StreamLLM approach, which uses Large Language Models (LLMs) to generate streamliners for Constraint Programming, to Answer Set Programming (ASP). Given an ASP encoding and a few small training instances, we prompt multiple LLMs to propose candidate constraints. Candidates that cause syntax errors, render satisfiable instances unsatisfiable, or degrade performance on all training instances are discarded. The surviving streamliners are evaluated together with the original encoding, and we report results for a virtual best encoding (VBE) that, for each instance, selects the fastest among the original encoding and its streamlined variants. On three ASP Competition benchmarks (Partner Units Problem, Sokoban, Towers of Hanoi), the VBE achieves speedups of up to 4--5x over the original encoding. Different LLMs produce semantically diverse constraints, not mere syntactic variations, indicating that the approach captures genuine problem structure.

Stefan Szeider

Translating natural language problem descriptions into formal constraint models remains a fundamental challenge in constraint programming, requiring deep expertise in both the problem domain and modeling frameworks. Previous approaches to automating this translation have employed fixed workflows with predetermined modeling steps, failing on a significant number of benchmark problems. We present a new approach using a pure agentic strategy without any fixed pipeline. We developed a general-purpose Python coding agent based on the ReAct (Reason and Act) principle, utilizing a persistent IPython kernel for stateful code execution and iterative development. Rather than embedding constraint programming logic into the agent architecture, domain-specific expertise is injected solely through a carefully crafted project prompt. The agent combines this prompt-encoded knowledge with access to file operations and code execution tools, enabling it to test hypotheses, debug failures, and verify solutions dynamically. Implemented in just a few hundred lines of code, this architecture successfully solves all 101 problems of the CP-Bench constraint programming benchmark set. The results suggest that constraint modeling tasks require the combination of general coding tools and domain expertise encoded in prompts, rather than specialized agent architectures or predefined workflows.

André Schidler, Stefan Szeider

Local search preprocessing makes Conflict-Driven Clause Learning (CDCL) solvers faster by providing high-quality starting points and modern SAT solvers have incorporated this technique into their preprocessing steps. However, these tools rely on basic strategies that miss the structural patterns in problems. We present a method that applies Large Language Models (LLMs) to analyze Python-based encoding code. This reveals hidden structural patterns in how problems convert into SAT. Our method automatically generates specialized local search algorithms that find these patterns and use them to create strong initial assignments. This works for any problem instance from the same encoding type. Our tests show encouraging results, achieving faster solving times compared to baseline preprocessing systems.

Juha Harviainen, Frank Sommer, Manuel Sorge et al.

We present a comprehensive classical and parameterized complexity analysis of decision tree pruning operations, extending recent research on the complexity of learning small decision trees. Thereby, we offer new insights into the computational challenges of decision tree simplification, a crucial aspect of developing interpretable and efficient machine learning models. We focus on fundamental pruning operations of subtree replacement and raising, which are used in heuristics. Surprisingly, while optimal pruning can be performed in polynomial time for subtree replacement, the problem is NP-complete for subtree raising. Therefore, we identify parameters and combinations thereof that lead to fixed-parameter tractability or hardness, establishing a precise borderline between these complexity classes. For example, while subtree raising is hard for small domain size $D$ or number $d$ of features, it can be solved in $D^{2d} \cdot |I|^{O(1)}$ time, where $|I|$ is the input size. We complement our theoretical findings with preliminary experimental results, demonstrating the practical implications of our analysis.

Eduard Eiben, Robert Ganian, Iyad Kanj et al.

Hypersphere classification is a classical and foundational method that can provide easy-to-process explanations for the classification of real-valued and binary data. However, obtaining an (ideally concise) explanation via hypersphere classification is much more difficult when dealing with binary data than real-valued data. In this paper, we perform the first complexity-theoretic study of the hypersphere classification problem for binary data. We use the fine-grained parameterized complexity paradigm to analyze the impact of structural properties that may be present in the input data as well as potential conciseness constraints. Our results include stronger lower bounds and new fixed-parameter algorithms for hypersphere classification of binary data, which can find an exact and concise explanation when one exists.

David Seka, Stefan Szeider

We present an approach to enumerate graphs whose automorphism group has exactly two orbits. Our method exploits the observation that we can enumerate all graphs whose automorphism group contains a given this permutation group. We obtain the relevant groups via Goursat's lemma. In order to scale the enumeration, we employ additional optimizations that prune irrelevant groups. In total, we enumerate, for the first time, all connected two-orbit graphs of up to 27 vertices, totaling 10,094,721 graphs, pushing the state of the art well beyond what direct enumeration methods can achieve.

Stefan Szeider

We present PBLean, a method for importing VeriPB pseudo-Boolean (PB) proof certificates into Lean 4. Key to our approach is reflection: a Boolean checker function whose soundness is fully proved in Lean and executed as compiled native code. Our method scales to proofs with tens of thousands of steps that would exhaust memory under explicit proof-term construction. Our checker supports all VeriPB kernel rules, including cutting-plane derivations and proof-by-contradiction subproofs. In contrast to external verified checkers that produce verdicts, our integration yields Lean theorems that can serve as composable lemmas in larger formal developments. To derive theorems about the original combinatorial problems rather than about PB constraints alone, we support verified encodings. This closes the trust gap between solver output and problem semantics since the constraint translation and its correctness proof are both formalized in Lean. We demonstrate the approach on various combinatorial problems.

Stefan Szeider

Automating the translation of natural-language specifications into logic programs is a challenging task that affects neurosymbolic engineering. We present ASP-Bench, a benchmark comprising 128 natural language problem instances, 64 base problems with easy and hard variants. It evaluates systems that translate natural-language problems into Answer Set Programs (ASPs), a prominent form of logic programming. It provides systematic coverage of ASP features, including choice rules, aggregates, and optimization. Each problem includes reference validators that check whether solutions satisfy the problem specification. We characterize problems along seven largely independent reasoning aspects (optimization, temporal reasoning, default logic, resource allocation, recursion, spatial reasoning, and quantitative complexity), providing a multidimensional view of modeling difficulty. We test the benchmark using an agentic approach based on the ReAct (Reason and Act) framework, which achieves full saturation, demonstrating that feedback-driven iterative refinement with solver feedback provides a reliable and robust approach for modeling natural language in ASP. Our analysis across multiple agent runs enables us to gain insights into what determines a problem's modeling hardness.

Stefan Szeider

We introduce an architecture for studying the behavior of large language model (LLM) agents in the absence of externally imposed tasks. Our continuous reason and act framework, using persistent memory and self-feedback, enables sustained autonomous operation. We deployed this architecture across 18 runs using 6 frontier models from Anthropic, OpenAI, XAI, and Google. We find agents spontaneously organize into three distinct behavioral patterns: (1) systematic production of multi-cycle projects, (2) methodological self-inquiry into their own cognitive processes, and (3) recursive conceptualization of their own nature. These tendencies proved highly model-specific, with some models deterministically adopting a single pattern across all runs. A cross-model assessment further reveals that models exhibit stable, divergent biases when evaluating these emergent behaviors in themselves and others. These findings provide the first systematic documentation of unprompted LLM agent behavior, establishing a baseline for predicting actions during task ambiguity, error recovery, or extended autonomous operation in deployed systems.

Alexis de Colnet, Stefan Szeider, Tianwei Zhang

Circuits in deterministic decomposable negation normal form (d-DNNF) are representations of Boolean functions that enable linear-time model counting. This paper strengthens our theoretical knowledge of what classes of functions can be efficiently transformed, or compiled, into d-DNNF. Our main contribution is the fixed-parameter tractable (FPT) compilation of conjunctions of specific constraints parameterized by incidence treewidth. This subsumes the known result for CNF. The constraints in question are all functions representable by constant-width ordered binary decision diagrams (OBDDs) for all variable orderings. For instance, this includes parity constraints and cardinality constraints with constant threshold. The running time of the FPT compilation is singly exponential in the incidence treewidth but hides large constants in the exponent. To balance that, we give a more efficient FPT algorithm for model counting that applies to a sub-family of the constraints and does not require compilation.

Markus Kirchweger, Hai Xia, Tomáš Peitl et al.

Parallel solving via cube-and-conquer is a key method for scaling SAT solvers to hard instances. While cube-and-conquer has proven successful for pure SAT problems, notably the Pythagorean triples conjecture, its application to SAT solvers extended with propagators presents unique challenges, as these propagators learn constraints dynamically during the search. We study this problem using SAT Modulo Symmetries (SMS) as our primary test case, where a symmetry-breaking propagator reduces the search space by learning constraints that eliminate isomorphic graphs. Through extensive experimentation comprising over 10,000 CPU hours, we systematically evaluate different cube-and-conquer variants on three well-studied combinatorial problems. Our methodology combines prerun phases to collect learned constraints, various cubing strategies, and parameter tuning via algorithm configuration and LLM-generated design suggestions. The comprehensive empirical evaluation provides new insights into effective cubing strategies for propagator-based SAT solving, with our best method achieving speedups of 2-3x from improved cubing and parameter tuning, providing an additional 1.5-2x improvement on harder instances.

Andre Schidler, Robert Ganian, Manuel Sorge et al.

Treewidth and hypertree width have proven to be highly successful structural parameters in the context of the Constraint Satisfaction Problem (CSP). When either of these parameters is bounded by a constant, then CSP becomes solvable in polynomial time. However, here the order of the polynomial in the running time depends on the width, and this is known to be unavoidable; therefore, the problem is not fixed-parameter tractable parameterized by either of these width measures. Here we introduce an enhancement of tree and hypertree width through a novel notion of thresholds, allowing the associated decompositions to take into account information about the computational costs associated with solving the given CSP instance. Aside from introducing these notions, we obtain efficient theoretical as well as empirical algorithms for computing threshold treewidth and hypertree width and show that these parameters give rise to fixed-parameter algorithms for CSP as well as other, more general problems. We complement our theoretical results with experimental evaluations in terms of heuristics as well as exact methods based on SAT/SMT encodings.

Johannes K. Fichte, Markus Hecher, Stefan Szeider

We compare the impact of hardware advancement and algorithm advancement for SAT solving over the last two decades. In particular, we compare 20-year-old SAT-solvers on new computer hardware with modern SAT-solvers on 20-year-old hardware. Our findings show that the progress on the algorithmic side has at least as much impact as the progress on the hardware side.

Vaidyanathan P. R., Stefan Szeider

We present a new approach for learning the structure of a treewidth-bounded Bayesian Network (BN). The key to our approach is applying an exact method (based on MaxSAT) locally, to improve the score of a heuristically computed BN. This approach allows us to scale the power of exact methods -- so far only applicable to BNs with several dozens of random variables -- to large BNs with several thousands of random variables. Our experiments show that our method improves the score of BNs provided by state-of-the-art heuristic methods, often significantly.

Serge Gaspers, Neeldhara Misra, Sebastian Ordyniak et al.

In this paper we extend the classical notion of strong and weak backdoor sets for SAT and CSP by allowing that different instantiations of the backdoor variables result in instances that belong to different base classes; the union of the base classes forms a heterogeneous base class. Backdoor sets to heterogeneous base classes can be much smaller than backdoor sets to homogeneous ones, hence they are much more desirable but possibly harder to find. We draw a detailed complexity landscape for the problem of detecting strong and weak backdoor sets into heterogeneous base classes for SAT and CSP.

Serge Gaspers, Stefan Szeider

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.

Eun Jung Kim, Sebastian Ordyniak, Stefan Szeider

We study the computational complexity of problems that arise in abstract argumentation in the context of dynamic argumentation, minimal change, and aggregation. In particular, we consider the following problems where always an argumentation framework F and a small positive integer k are given. - The Repair problem asks whether a given set of arguments can be modified into an extension by at most k elementary changes (i.e., the extension is of distance k from the given set). - The Adjust problem asks whether a given extension can be modified by at most k elementary changes into an extension that contains a specified argument. - The Center problem asks whether, given two extensions of distance k, whether there is a "center" extension that is a distance at most (k-1) from both given extensions. We study these problems in the framework of parameterized complexity, and take the distance k as the parameter. Our results covers several different semantics, including admissible, complete, preferred, semi-stable and stable semantics.

Sebastian Ordyniak, Stefan Szeider

Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian network structure learning under graph theoretic restrictions on the (directed) super-structure. The super-structure is an undirected graph that contains as subgraphs the skeletons of solution networks. We introduce the directed super-structure as a natural generalization of its undirected counterpart. Our results apply to several variants of score-based Bayesian network structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian network structure learning can be carried out in non-uniform polynomial time if the super-structure has bounded treewidth, and in linear time if in addition the super-structure has bounded maximum degree. Furthermore, we show that if the directed super-structure is acyclic, then exact Bayesian network structure learning can be carried out in quadratic time. We complement these positive results with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexity-theoretic assumption). Similarly, exact Bayesian network structure learning remains NP-hard for "almost acyclic" directed super-structures. Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (k-neighborhood local search).

Christer Baeckstroem, Peter Jonsson, Sebastian Ordyniak et al.

The propositional planning problem is a notoriously difficult computational problem, which remains hard even under strong syntactical and structural restrictions. Given its difficulty it becomes natural to study planning in the context of parameterized complexity. In this paper we continue the work initiated by Downey, Fellows and Stege on the parameterized complexity of planning with respect to the parameter "length of the solution plan." We provide a complete classification of the parameterized complexity of the planning problem under two of the most prominent syntactical restrictions, i.e., the so called PUBS restrictions introduced by Baeckstroem and Nebel and restrictions on the number of preconditions and effects as introduced by Bylander. We also determine which of the considered fixed-parameter tractable problems admit a polynomial kernel and which don't.

Ronald de Haan, Anna Roubíčková, Stefan Szeider

Planning is a notoriously difficult computational problem of high worst-case complexity. Researchers have been investing significant efforts to develop heuristics or restrictions to make planning practically feasible. Case-based planning is a heuristic approach where one tries to reuse previous experience when solving similar problems in order to avoid some of the planning effort. Plan reuse may offer an interesting alternative to plan generation in some settings. We provide theoretical results that identify situations in which plan reuse is provably tractable. We perform our analysis in the framework of parameterized complexity, which supports a rigorous worst-case complexity analysis that takes structural properties of the input into account in terms of parameters. A central notion of parameterized complexity is fixed-parameter tractability which extends the classical notion of polynomial-time tractability by utilizing the effect of structural properties of the problem input. We draw a detailed map of the parameterized complexity landscape of several variants of problems that arise in the context of case-based planning. In particular, we consider the problem of reusing an existing plan, imposing various restrictions in terms of parameters, such as the number of steps that can be added to the existing plan to turn it into a solution of the planning instance at hand.

Andreas Pfandler, Stefan Rümmele, Stefan Szeider

Abductive reasoning (or Abduction, for short) is among the most fundamental AI reasoning methods, with a broad range of applications, including fault diagnosis, belief revision, and automated planning. Unfortunately, Abduction is of high computational complexity; even propositional Abduction is Σ_2^P-complete and thus harder than NP and coNP. This complexity barrier rules out the existence of a polynomial transformation to propositional satisfiability (SAT). In this work we use structural properties of the Abduction instance to break this complexity barrier. We utilize the problem structure in terms of small backdoor sets. We present fixed-parameter tractable transformations from Abduction to SAT, which make the power of today's SAT solvers available to Abduction.

Ronald de Haan, Iyad Kanj, Stefan Szeider

A backbone of a propositional CNF formula is a variable whose truth value is the same in every truth assignment that satisfies the formula. The notion of backbones for CNF formulas has been studied in various contexts. In this paper, we introduce local variants of backbones, and study the computational complexity of detecting them. In particular, we consider k-backbones, which are backbones for sub-formulas consisting of at most k clauses, and iterative k-backbones, which are backbones that result after repeated instantiations of k-backbones. We determine the parameterized complexity of deciding whether a variable is a k-backbone or an iterative k-backbone for various restricted formula classes, including Horn, definite Horn, and Krom. We also present some first empirical results regarding backbones for CNF-Satisfiability (SAT). The empirical results we obtain show that a large fraction of the backbones of structured SAT instances are local, in contrast to random instances, which appear to have few local backbones.

Johannes Klaus Fichte, Stefan Szeider

Over the last two decades, propositional satisfiability (SAT) has become one of the most successful and widely applied techniques for the solution of NP-complete problems. The aim of this paper is to investigate theoretically how Sat can be utilized for the efficient solution of problems that are harder than NP or co-NP. In particular, we consider the fundamental reasoning problems in propositional disjunctive answer set programming (ASP), Brave Reasoning and Skeptical Reasoning, which ask whether a given atom is contained in at least one or in all answer sets, respectively. Both problems are located at the second level of the Polynomial Hierarchy and thus assumed to be harder than NP or co-NP. One cannot transform these two reasoning problems into SAT in polynomial time, unless the Polynomial Hierarchy collapses. We show that certain structural aspects of disjunctive logic programs can be utilized to break through this complexity barrier, using new techniques from Parameterized Complexity. In particular, we exhibit transformations from Brave and Skeptical Reasoning to SAT that run in time O(2^k n^2) where k is a structural parameter of the instance and n the input size. In other words, the reduction is fixed-parameter tractable for parameter k. As the parameter k we take the size of a smallest backdoor with respect to the class of normal (i.e., disjunction-free) programs. Such a backdoor is a set of atoms that when deleted makes the program normal. In consequence, the combinatorial explosion, which is expected when transforming a problem from the second level of the Polynomial Hierarchy to the first level, can now be confined to the parameter k, while the running time of the reduction is polynomial in the input size n, where the order of the polynomial is independent of k.

Christer Baeckstroem, Yue Chen, Peter Jonsson et al.

The early classifications of the computational complexity of planning under various restrictions in STRIPS (Bylander) and SAS+ (Baeckstroem and Nebel) have influenced following research in planning in many ways. We go back and reanalyse their subclasses, but this time using the more modern tool of parameterized complexity analysis. This provides new results that together with the old results give a more detailed picture of the complexity landscape. We demonstrate separation results not possible with standard complexity theory, which contributes to explaining why certain cases of planning have seemed simpler in practice than theory has predicted. In particular, we show that certain restrictions of practical interest are tractable in the parameterized sense of the term, and that a simple heuristic is sufficient to make a well-known partial-order planner exploit this fact.

Serge Gaspers, Mikko Koivisto, Mathieu Liedloff et al.

Inferring probabilistic networks from data is a notoriously difficult task. Under various goodness-of-fit measures, finding an optimal network is NP-hard, even if restricted to polytrees of bounded in-degree. Polynomial-time algorithms are known only for rare special cases, perhaps most notably for branchings, that is, polytrees in which the in-degree of every node is at most one. Here, we study the complexity of finding an optimal polytree that can be turned into a branching by deleting some number of arcs or nodes, treated as a parameter. We show that the problem can be solved via a matroid intersection formulation in polynomial time if the number of deleted arcs is bounded by a constant. The order of the polynomial time bound depends on this constant, hence the algorithm does not establish fixed-parameter tractability when parameterized by the number of deleted arcs. We show that a restricted version of the problem allows fixed-parameter tractability and hence scales well with the parameter. We contrast this positive result by showing that if we parameterize by the number of deleted nodes, a somewhat more powerful parameter, the problem is not fixed-parameter tractable, subject to a complexity-theoretic assumption.

Serge Gaspers, Stefan Szeider

There are various approaches to exploiting "hidden structure" in instances of hard combinatorial problems to allow faster algorithms than for general unstructured or random instances. For SAT and its counting version #SAT, hidden structure has been exploited in terms of decomposability and strong backdoor sets. Decomposability can be considered in terms of the treewidth of a graph that is associated with the given CNF formula, for instance by considering clauses and variables as vertices of the graph, and making a variable adjacent with all the clauses it appears in. On the other hand, a strong backdoor set of a CNF formula is a set of variables such that each possible partial assignment to this set moves the formula into a fixed class for which (#)SAT can be solved in polynomial time. In this paper we combine the two above approaches. In particular, we study the algorithmic question of finding a small strong backdoor set into the class W_t of CNF formulas whose associated graphs have treewidth at most t. The main results are positive: (1) There is a cubic-time algorithm that, given a CNF formula F and two constants k,t\ge 0, either finds a strong W_t-backdoor set of size at most 2^k, or concludes that F has no strong W_t-backdoor set of size at most k. (2) There is a cubic-time algorithm that, given a CNF formula F, computes the number of satisfying assignments of F or concludes that sb_t(F)>k, for any pair of constants k,t\ge 0. Here, sb_t(F) denotes the size of a smallest strong W_t-backdoor set of F. The significance of our results lies in the fact that they allow us to exploit algorithmically a hidden structure in formulas that is not accessible by any one of the two approaches (decomposability, backdoors) alone. Already a backdoor size 1 on top of treewidth 1 (i.e., sb_1(F)=1) entails formulas of arbitrarily large treewidth and arbitrarily large cycle cutsets.

Reinhard Pichler, Stefan Rümmele, Stefan Szeider et al.

Cardinality constraints or, more generally, weight constraints are well recognized as an important extension of answer-set programming. Clearly, all common algorithmic tasks related to programs with cardinality or weight constraints - like checking the consistency of a program - are intractable. Many intractable problems in the area of knowledge representation and reasoning have been shown to become linear time tractable if the treewidth of the programs or formulas under consideration is bounded by some constant. The goal of this paper is to apply the notion of treewidth to programs with cardinality or weight constraints and to identify tractable fragments. It will turn out that the straightforward application of treewidth to such class of programs does not suffice to obtain tractability. However, by imposing further restrictions, tractability can be achieved.

Sebastian Ordyniak, Stefan Szeider

Bayesian structure learning is the NP-hard problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian structure learning under graph theoretic restrictions on the super-structure. The super-structure (a concept introduced by Perrier, Imoto, and Miyano, JMLR 2008) is an undirected graph that contains as subgraphs the skeletons of solution networks. Our results apply to several variants of score-based Bayesian structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian structure learning can be carried out in non-uniform polynomial time if the super-structure has bounded treewidth and in linear time if in addition the super-structure has bounded maximum degree. We complement this with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexity-theoretic assumption). Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but we aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (k-neighborhood local search).

Serge Gaspers, Stefan Szeider

Knuth (1990) introduced the class of nested formulas and showed that their satisfiability can be decided in polynomial time. We show that, parameterized by the size of a smallest strong backdoor set to the target class of nested formulas, checking the satisfiability of any CNF formula is fixed-parameter tractable. Thus, for any k>0, the satisfiability problem can be solved in polynomial time for any formula F for which there exists a variable set B of size at most k such that for every truth assignment t to B, the formula F[t] is nested; moreover, the degree of the polynomial is independent of k. Our algorithm uses the grid-minor theorem of Robertson and Seymour (1986) to either find that the incidence graph of the formula has bounded treewidth - a case that is solved using model checking for monadic second order logic - or to find many vertex-disjoint obstructions in the incidence graph. For the latter case, new combinatorial arguments are used to find a small backdoor set. Combining both cases leads to an approximation algorithm producing a strong backdoor set whose size is upper bounded by a function of the optimum. Going through all assignments to this set of variables and using Knuth's algorithm, the satisfiability of the input formula is decided.