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Sebastian Ordyniak

Sebastian Ordyniak appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Data Structures and AlgorithmsArtificial IntelligenceComputational ComplexityDiscrete Mathematicsmath.COLogic in Computer ScienceMachine LearningFormal Languages and Automata Theory

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Published work

25 published item(s)

preprint2026arXiv

Clausal Deletion Backdoors for QBF: a Parameterized Complexity Approach

Determining the validity of a quantified Boolean formula (QBF) is a PSPACE-complete problem with rich expressive power. Despite interest in efficient solvers, there is, compared to problems in NP, a lack of positive theoretical results, and in the parameterized complexity setting one often has to restrict the quantifier prefix (e.g., bounding alternations) to obtain fixed parameter tractability (FPT). We propose a new parameter: the number of variables in clauses that has to be removed before reaching a tractable class (a clause covering (CC) backdoor). We are then interested in solving QBF in FPT time given a CC-backdoor of size $k$. We consider the three classical, tractable cases of QBF as base classes: Horn, 2-CNF, and linear equations. We establish W[1]-hardness for Horn but prove FPT for the others, and prove that in a precise, algebraic sense, we are only missing one important case for a full dichotomy. Our algorithms are non-trivial and depend on propagation, and Gaussian elimination, respectively, and are comparably unexplored for QBF.

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preprint2022arXiv

Almost Consistent Systems of Linear Equations

Checking whether a system of linear equations is consistent is a basic computational problem with ubiquitous applications. When dealing with inconsistent systems, one may seek an assignment that minimizes the number of unsatisfied equations. This problem is NP-hard and UGC-hard to approximate within any constant even for two-variable equations over the two-element field. We study this problem from the point of view of parameterized complexity, with the parameter being the number of unsatisfied equations. We consider equations defined over Euclidean domains - a family of commutative rings that generalize finite and infinite fields including the rationals, the ring of integers, and many other structures. We show that if every equation contains at most two variables, the problem is fixed-parameter tractable. This generalizes many eminent graph separation problems such as Bipartization, Multiway Cut and Multicut parameterized by the size of the cutset. To complement this, we show that the problem is W[1]-hard when three or more variables are allowed in an equation, as well as for many commutative rings that are not Euclidean domains. On the technical side, we introduce the notion of important balanced subgraphs, generalizing important separators of Marx [Theor. Comput. Sci. 2006] to the setting of biased graphs. Furthermore, we use recent results on parameterized MinCSP [Kim et al., SODA 2021] to efficiently solve a generalization of Multicut with disjunctive cut requests.

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preprint2022arXiv

An Algorithmic Framework for Locally Constrained Homomorphisms

A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.

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preprint2022arXiv

SAT Backdoors: Depth Beats Size

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. Bounded backdoor size implies bounded backdoor depth, but there are formulas of constant backdoor depth and arbitrarily large backdoor size. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. We base our FPT approximation algorithm on a sophisticated notion of obstructions, extending Mählmann et al.'s obstruction trees in various ways, including the addition of separator obstructions. We develop the algorithm through a new game-theoretic framework that simplifies the reasoning about backdoors. Finally, we show that bounded backdoor depth captures tractable classes of CNF formulas not captured by any known method.

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preprint2020arXiv

Solving Integer Linear Programs with a Small Number of Global Variables and Constraints

Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of "global" variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N -fold ILP instances. Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.

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preprint2016arXiv

Backdoors into Heterogeneous Classes of SAT and CSP

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.

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preprint2016arXiv

Clique-Width and Directed Width Measures for Answer-Set Programming

Disjunctive Answer Set Programming (ASP) is a powerful declarative programming paradigm whose main decision problems are located on the second level of the polynomial hierarchy. Identifying tractable fragments and developing efficient algorithms for such fragments are thus important objectives in order to complement the sophisticated ASP systems available to date. Hard problems can become tractable if some problem parameter is bounded by a fixed constant; such problems are then called fixed-parameter tractable (FPT). While several FPT results for ASP exist, parameters that relate to directed or signed graphs representing the program at hand have been neglected so far. In this paper, we first give some negative observations showing that directed width measures on the dependency graph of a program do not lead to FPT results. We then consider the graph parameter of signed clique-width and present a novel dynamic programming algorithm that is FPT w.r.t. this parameter. Clique-width is more general than the well-known treewidth, and, to the best of our knowledge, ours is the first FPT algorithm for bounded clique-width for reasoning problems beyond SAT.

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preprint2016arXiv

First order limits of sparse graphs: Plane trees and path-width

Nesetril and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On the positive side, every first order convergent sequence of trees or graphs with no long path (graphs with bounded tree-depth) has a limit modeling. We strengthen these results by showing that every first order convergent sequence of plane trees (trees with embeddings in the plane) and every first order convergent sequence of graphs with bounded path-width has a limit modeling.

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preprint2016arXiv

Strong Backdoors for Linear Temporal Logic

In the present paper we introduce the notion of strong backdoors into the field of temporal logic for the CNF-fragment of linear temporal logic introduced by Fisher. We study the parameterised complexity of the satisfiability problem parameterised by the size of the backdoor. We distinguish between backdoor detection and evaluation of backdoors into the fragments of horn and krom formulas. Here we classify the operator fragments of globally-operators for past or future, and the combination of both. Detection is shown to be in FPT whereas the complexity of evaluation behaves different. We show that for krom formulas the problem is paraNP-complete. For horn formulas the complexity is shown to be either fixed parameter tractable or paraNP-complete depending on the considered operator fragment.

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preprint2015arXiv

FO Model Checking on Posets of Bounded Width

Over the past two decades the main focus of research into first-order (FO) model checking algorithms have been sparse relational structures-culminating in the FPT-algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs [STOC'14], with dense structures starting to attract attention only recently. Bova, Ganian and Szeider [LICS'14] initiated the study of the complexity of FO model checking on partially ordered sets (posets). Bova, Ganian and Szeider showed that model checking existential FO logic is fixed-parameter tractable (FPT) on posets of bounded width, where the width of a poset is the size of the largest antichain in the poset. The existence of an FPT algorithm for general FO model checking on posets of bounded width, however, remained open. We resolve this question in the positive by giving an algorithm that takes as its input an $n$-element poset $\cal P$ of width $w$ and an FO logic formula $ϕ$, and determines whether $ϕ$ holds on $\cal P$ in time $f(ϕ,w)\cdot n^2$.

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preprint2015arXiv

Kernelization Using Structural Parameters on Sparse Graph Classes

Meta-theorems for polynomial (linear) kernels have been the subject of intensive research in parameterized complexity. Heretofore, meta-theorems for linear kernels exist on graphs of bounded genus, $H$-minor-free graphs, and $H$-topological-minor-free graphs. To the best of our knowledge, no meta-theorems for polynomial kernels are known for any larger sparse graph classes; e.g., for classes of bounded expansion or for nowhere dense ones. In this paper we prove such meta-theorems for the two latter cases. More specifically, we show that graph problems that have finite integer index (FII) have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. While our parameter may seem rather strong, we argue that a linear kernelization result on graphs of bounded expansion with a weaker parameter (than treedepth modulator) would fail to include some of the problems covered by our framework. Moreover, we only require the problems to have FII on graphs of constant treedepth. This allows us to prove linear kernels for problems such as Longest Path/Cycle, Exact $s,t$-Path, Treewidth, and Pathwidth, which do not have FII on general graphs (and the first two not even on bounded treewidth graphs).

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preprint2014arXiv

A Parameterized Study of Maximum Generalized Pattern Matching Problems

The generalized function matching (GFM) problem has been intensively studied starting with [Ehrenfeucht and Rozenberg, 1979]. Given a pattern p and a text t, the goal is to find a mapping from the letters of p to non-empty substrings of t, such that applying the mapping to p results in t. Very recently, the problem has been investigated within the framework of parameterized complexity [Fernau, Schmid, and Villanger, 2013]. In this paper we study the parameterized complexity of the optimization variant of GFM (called Max-GFM), which has been introduced in [Amir and Nor, 2007]. Here, one is allowed to replace some of the pattern letters with some special symbols "?", termed wildcards or don't cares, which can be mapped to an arbitrary substring of the text. The goal is to minimize the number of wildcards used. We give a complete classification of the parameterized complexity of Max-GFM and its variants under a wide range of parameterizations, such as, the number of occurrences of a letter in the text, the size of the text alphabet, the number of occurrences of a letter in the pattern, the size of the pattern alphabet, the maximum length of a string matched to any pattern letter, the number of wildcards and the maximum size of a string that a wildcard can be mapped to.

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preprint2014arXiv

Parameterized Complexity Results for Exact Bayesian Network Structure Learning

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).

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preprint2014arXiv

The Complexity of Repairing, Adjusting, and Aggregating of Extensions in Abstract Argumentation

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.

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preprint2014arXiv

Tree-depth and Vertex-minors

In a recent paper, Kwon and Oum claim that every graph of bounded rank-width is a pivot-minor of a graph of bounded tree-width (while the converse has been known true already before). We study the analogous questions for "depth" parameters of graphs, namely for the tree-depth and related new shrub-depth. We show that shrub-depth is monotone under taking vertex-minors, and that every graph class of bounded shrub-depth can be obtained via vertex-minors of graphs of bounded tree-depth. We also consider the same questions for bipartite graphs and pivot-minors.

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preprint2013arXiv

A Complete Parameterized Complexity Analysis of Bounded Planning

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.

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preprint2013arXiv

Parameterized Algorithms for Modular-Width

It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty. The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILPs and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path and Hamiltonian cycle), which are W-hard for both clique-width and its recently introduced restriction, shrub-depth. We thus argue that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.

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preprint2013arXiv

Parameterized Complexity and Kernel Bounds for Hard Planning Problems

The propositional planning problem is a notoriously difficult computational problem. Downey et al. (1999) initiated the parameterized analysis of planning (with plan length as the parameter) and Bäckström et al. (2012) picked up this line of research and provided an extensive parameterized analysis under various restrictions, leaving open only one stubborn case. We continue this work and provide a full classification. In particular, we show that the case when actions have no preconditions and at most $e$ postconditions is fixed-parameter tractable if $e\leq 2$ and W[1]-complete otherwise. We show fixed-parameter tractability by a reduction to a variant of the Steiner Tree problem; this problem has been shown fixed-parameter tractable by Guo et al. (2007). If a problem is fixed-parameter tractable, then it admits a polynomial-time self-reduction to instances whose input size is bounded by a function of the parameter, called the kernel. For some problems, this function is even polynomial which has desirable computational implications. Recent research in parameterized complexity has focused on classifying fixed-parameter tractable problems on whether they admit polynomial kernels or not. We revisit all the previously obtained restrictions of planning that are fixed-parameter tractable and show that none of them admits a polynomial kernel unless the polynomial hierarchy collapses to its third level.

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preprint2013arXiv

Upper and Lower Bounds for Weak Backdoor Set Detection

We obtain upper and lower bounds for running times of exponential time algorithms for the detection of weak backdoor sets of 3CNF formulas, considering various base classes. These results include (omitting polynomial factors), (i) a 4.54^k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Horn formulas; (ii) a 2.27^k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Krom formulas. These bounds improve an earlier known bound of 6^k. We also prove a 2^k lower bound for these problems, subject to the Strong Exponential Time Hypothesis.

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preprint2012arXiv

Algorithms and Complexity Results for Exact Bayesian Structure Learning

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).

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preprint2012arXiv

Don't Be Strict in Local Search!

Local Search is one of the fundamental approaches to combinatorial optimization and it is used throughout AI. Several local search algorithms are based on searching the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naïve brute-force search of the k-exchange neighborhood requires n to the power of O(k) time, which is not practical even for very small values of k. Fellows et al. (IJCAI 2009) studied whether this brute-force search is avoidable and gave positive and negative answers for several combinatorial problems. They used the notion of local search in a strict sense. That is, an improved solution needs to be found in the k-exchange neighborhood even if a global optimum can be found efficiently. In this paper we consider a natural relaxation of local search, called permissive local search (Marx and Schlotter, IWPEC 2009) and investigate whether it enhances the domain of tractable inputs. We exemplify this approach on a fundamental combinatorial problem, Vertex Cover. More precisely, we show that for a class of inputs, finding an optimum is hard, strict local search is hard, but permissive local search is tractable. We carry out this investigation in the framework of parameterized complexity.

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preprint2012arXiv

On Finding Optimal Polytrees

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.

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preprint2012arXiv

The Complexity of Planning Revisited - A Parameterized Analysis

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.

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preprint2011arXiv

Algorithms and Complexity Results for Persuasive Argumentation

The study of arguments as abstract entities and their interaction as introduced by Dung (Artificial Intelligence 177, 1995) has become one of the most active research branches within Artificial Intelligence and Reasoning. A main issue for abstract argumentation systems is the selection of acceptable sets of arguments. Value-based argumentation, as introduced by Bench-Capon (J. Logic Comput. 13, 2003), extends Dung's framework. It takes into account the relative strength of arguments with respect to some ranking representing an audience: an argument is subjectively accepted if it is accepted with respect to some audience, it is objectively accepted if it is accepted with respect to all audiences. Deciding whether an argument is subjectively or objectively accepted, respectively, are computationally intractable problems. In fact, the problems remain intractable under structural restrictions that render the main computational problems for non-value-based argumentation systems tractable. In this paper we identify nontrivial classes of value-based argumentation systems for which the acceptance problems are polynomial-time tractable. The classes are defined by means of structural restrictions in terms of the underlying graphical structure of the value-based system. Furthermore we show that the acceptance problems are intractable for two classes of value-based systems that where conjectured to be tractable by Dunne (Artificial Intelligence 171, 2007).

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preprint2011arXiv

Augmenting Tractable Fragments of Abstract Argumentation

We present a new and compelling approach to the efficient solution of important computational problems that arise in the context of abstract argumentation. Our approach makes known algorithms defined for restricted fragments generally applicable, at a computational cost that scales with the distance from the fragment. Thus, in a certain sense, we gradually augment tractable fragments. Surprisingly, it turns out that some tractable fragments admit such an augmentation and that others do not. More specifically, we show that the problems of credulous and skeptical acceptance are fixed-parameter tractable when parameterized by the distance from the fragment of acyclic argumentation frameworks. Other tractable fragments such as the fragments of symmetrical and bipartite frameworks seem to prohibit an augmentation: the acceptance problems are already intractable for frameworks at distance 1 from the fragments. For our study we use a broad setting and consider several different semantics. For the algorithmic results we utilize recent advances in fixed-parameter tractability.

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