Robert Ganian

AIJ Journal 2026 Journal Article

A structural complexity analysis of synchronous dynamical systems

• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• Viktoriia Korchemna

AAAI Conference 2026 Conference Paper

Matrix Editing Meets Fair Clustering: Parameterized Algorithms and Complexity

• Robert Ganian
• Hung P. Hoang
• Simon Wietheger

We study the computational problem of computing a fair means clustering of discrete vectors, which admits an equivalent formulation as editing a colored matrix into one with few distinct color-balanced rows by changing at most k values. While NP-hard in both the fairness-oblivious and the fair settings, the problem is well-known to admit a fixed-parameter algorithm in the former "vanilla" setting. As our first contribution, we exclude an analogous algorithm even for highly restricted fair means clustering instances. We then proceed to obtain a full complexity landscape of the problem, and establish tractability results which capture three means of circumventing our obtained lower bound: placing additional constraints on the problem instances, fixed-parameter approximation, or using an alternative parameterization targeting tree-like matrices.

IJCAI Conference 2025 Conference Paper

A Structural Complexity Analysis of Hierarchical Task Network Planning

• Cornelius Brand
• Robert Ganian
• Fionn Mc Inerney
• Simon Wietheger

We perform a refined complexity-theoretic analysis of three classical problems in the context of Hierarchical Task Network Planning: the verification of a provided plan, whether an executable plan exists, and whether a given state can be reached. Our focus lies on identifying structural properties which yield tractability. We obtain new polynomial algorithms for all three problems on a natural class of primitive networks, along with corresponding lower bounds. We also obtain an algorithmic meta-theorem for lifting polynomial-time solvability from primitive to general task networks, and prove that its preconditions are tight. Finally, we analyze the parameterized complexity of the three problems.

AAMAS Conference 2025 Conference Paper

Parameterized Algorithms for Multiagent Pathfinding on Trees

• Argyrios Deligkas
• Eduard Eiben
• Robert Ganian
• Iyad Kanj
• M. S. Ramanujan

The classical Multiagent Pathfinding problem has been extensively studied not only within the artificial intelligence research community, but also by scholars in the areas of theoretical computer science and computational geometry. The problem asks for a minimum-makespan schedule that routes 𝑘 agents (or robots) from their starting points to their destinations in a graph, while avoiding collisions, and is known to be NP-hard even on the fundamental class of trees. In this article we present two fixed parameter algorithms parameterized by 𝑘: the first yields a collision-free schedule on trees whose makespan deviates from the optimum by at most an additive polynomial function of 𝑘, and the second solves Multiagent Pathfinding optimally on the class of irreducible trees, i. e. , trees with no vertices of degree 2. Both results rely on novel tools and insights into the properties of optimal schedules.

AAAI Conference 2025 Conference Paper

Parameterized Complexity of Caching in Networks

• Robert Ganian
• Fionn Mc Inerney
• Dimitra Tsigkari

The fundamental caching problem in networks asks to find an allocation of contents to a network of caches with the aim of maximizing the cache hit rate. Despite the problem's importance to a variety of research areas - including not only content delivery, but also edge intelligence and inference - and the extensive body of work on empirical aspects of caching, very little is known about the exact boundaries of tractability for the problem beyond its general NP-hardness. We close this gap by performing a comprehensive complexity-theoretic analysis of the problem through the lens of the parameterized complexity paradigm, which is designed to provide more precise statements regarding algorithmic tractability than classical complexity. Our results include algorithmic lower and upper bounds which together establish the conditions under which the caching problem becomes tractable.

AAAI Conference 2025 Conference Paper

The Complexity of Extending Fair Allocations of Indivisible Goods

• Argyrios Deligkas
• Eduard Eiben
• Robert Ganian
• Tiger-Lily Goldsmith
• Stavros D. Ioannidis

We initiate the study of computing envy-free allocations of indivisible items in the extension setting, i.e., when some part of the allocation is fixed and the task is to allocate the remaining items. In view of the NP-hardness of the problem, we investigate whether - and under which conditions - one can obtain fixed-parameter algorithms for computing a solution in settings where most of the allocation is already fixed. Our results provide a broad complexity-theoretic classification of the problem which includes: (a) fixed-parameter algorithms tailored to settings with few distinct types of agents or items; (b) lower bounds which exclude the generalization of these positive results to more general settings. We conclude by showing that - unlike when computing allocations from scratch - the non-algorithmic question of whether more relaxed EF1 or EFX allocations exist can be completely resolved in the extension setting.

AIJ Journal 2025 Journal Article

The complexity of optimizing atomic congestion

• Cornelius Brand
• Robert Ganian
• Subrahmanyam Kalyanasundaram
• Fionn Mc Inerney

ICLR Conference 2025 Conference Paper

The Computational Complexity of Positive Non-Clashing Teaching in Graphs

• Robert Ganian
• Liana Khazaliya
• Fionn Mc Inerney
• Mathis Rocton

We study the classical and parameterized complexity of computing the positive non-clashing teaching dimension of a set of concepts, that is, the smallest number of examples per concept required to successfully teach an intelligent learner under the considered, previously established model. For any class of concepts, it is known that this problem can be effortlessly transferred to the setting of balls in a graph $G$. We establish (1) the NP-hardness of the problem even when restricted to instances with positive non-clashing teaching dimension $k=2$ and where all balls in the graph are present, (2) near-tight running time upper and lower bounds for the problem on general graphs, (3) fixed-parameter tractability when parameterized by the vertex integrity of $G$, and (4) a lower bound excluding fixed-parameter tractability when parameterized by the feedback vertex number and pathwidth of $G$, even when combined with $k$. Our results provide a nearly complete understanding of the complexity landscape of computing the positive non-clashing teaching dimension and answer open questions from the literature.

ICLR Conference 2025 Conference Paper

Training One-Dimensional Graph Neural Networks is NP-Hard

• Robert Ganian
• Mathis Rocton
• Simon Wietheger

We initiate the study of the computational complexity of training graph neural networks (GNNs). We consider the classical node classification setting; there, the intractability of training multidimensonal GNNs immediately follows from known lower bounds for training classical neural networks (and holds even for trivial GNNs). However, one-dimensional GNNs form a crucial case of interest: the computational complexity of training such networks depends on both the graphical structure of the network and the properties of the involved activation and aggregation functions. As our main result, we establish the NP-hardness of training ReLU-activated one-dimensional GNNs via a highly non-trivial reduction. We complement this result with algorithmic upper bounds for the training problem in the ReLU-activated and linearly-activated settings.

IJCAI Conference 2024 Conference Paper

Revisiting Causal Discovery from a Complexity-Theoretic Perspective

• Robert Ganian
• Viktoriia Korchemna
• Stefan Szeider

Causal discovery seeks to unveil causal relationships (represented as a so-called causal graph) from observational data. This paper investigates the complex relationship between the graph structure and the efficiency of constraint-based causal discovery algorithms. Our main contributions include (i) a near-tight characterization of which causal graphs admit a small d-separating set for each pair of vertices and thus can potentially be efficiently recovered by a constraint-based causal discovery algorithm, (ii) the explicit construction of a sequence of causal graphs on which the influential PC algorithm might need exponential time, although there is a small d-separating set between every pair of variables, and (iii) the formulation of a new causal discovery algorithm which achieves fixed-parameter running time by considering the maximum number of edge-disjoint paths between variables in the (undirected) super-structure as the parameter. A distinguishing feature of our investigation is that it is carried out within a more fine-grained model which more faithfully captures the infeasibility of performing accurate independence tests for large sets of conditioning variables.

AAAI Conference 2024 Conference Paper

The Complexity of Optimizing Atomic Congestion

• Cornelius Brand
• Robert Ganian
• Subrahmanyam Kalyanasundaram
• Fionn Mc Inerney

Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory, and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies - that is, a centrally planned routing that minimizes the average cost of agents - is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem.

AAAI Conference 2023 Conference Paper

A Parameterized Theory of PAC Learning

• Cornelius Brand
• Robert Ganian
• Kirill Simonov

Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP "dichotomy" in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT - the core concept at the center of the parameterized complexity paradigm - and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs.

AAAI Conference 2023 Conference Paper

A Structural Complexity Analysis of Synchronous Dynamical Systems

• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• Viktoriia Korchemna

Synchronous dynamical systems are well-established models that have been used to capture a range of phenomena in networks, including opinion diffusion, spread of disease and product adoption. We study the three most notable problems in synchronous dynamical systems: whether the system will transition to a target configuration from a starting configuration, whether the system will reach convergence from a starting configuration, and whether the system is guaranteed to converge from every possible starting configuration. While all three problems were known to be intractable in the classical sense, we initiate the study of their exact boundaries of tractability from the perspective of structural parameters of the network by making use of the more fine-grained parameterized complexity paradigm. As our first result, we consider treewidth - as the most prominent and ubiquitous structural parameter - and show that all three problems remain intractable even on instances of constant treewidth. We complement this negative finding with fixed-parameter algorithms for the former two problems parameterized by treedepth, a well-studied restriction of treewidth. While it is possible to rule out a similar algorithm for convergence guarantee under treedepth, we conclude with a fixed-parameter algorithm for this last problem when parameterized by treedepth and the maximum in-degree.

AIJ Journal 2023 Journal Article

Hedonic diversity games: A complexity picture with more than two colors

• Robert Ganian
• Thekla Hamm
• Dušan Knop
• Šimon Schierreich
• Ondřej Suchý

TARK Conference 2023 Conference Paper

Maximizing Social Welfare in Score-Based Social Distance Games

• Robert Ganian
• Thekla Hamm
• Dusan Knop
• Sanjukta Roy 0001
• Simon Schierreich
• Ondrej Suchý 0001

Social distance games have been extensively studied as a coalition formation model where the utilities of agents in each coalition were captured using a utility function u that took into account distances in a given social network. In this paper, we consider a non-normalized score-based definition of social distance games where the utility function u_v depends on a generic scoring vector v, which may be customized to match the specifics of each individual application scenario. As our main technical contribution, we establish the tractability of computing a welfare-maximizing partitioning of the agents into coalitions on tree-like networks, for every score-based function u_v. We provide more efficient algorithms when dealing with specific choices of u_v or simpler networks, and also extend all of these results to computing coalitions that are Nash stable or individually rational. We view these results as a further strong indication of the usefulness of the proposed score-based utility function: even on very simple networks, the problem of computing a welfare-maximizing partitioning into coalitions remains open for the originally considered canonical function u.

NeurIPS Conference 2023 Conference Paper

New Complexity-Theoretic Frontiers of Tractability for Neural Network Training

• Cornelius Brand
• Robert Ganian
• Mathis Rocton

In spite of the fundamental role of neural networks in contemporary machine learning research, our understanding of the computational complexity of optimally training neural networks remains limited even when dealing with the simplest kinds of activation functions. Indeed, while there has been a number of very recent results that establish ever-tighter lower bounds for the problem under linear and ReLU activation functions, little progress has been made towards the identification of novel polynomial-time tractable network architectures. In this article we obtain novel algorithmic upper bounds for training linear- and ReLU-activated neural networks to optimality which push the boundaries of tractability for these problems beyond the previous state of the art.

AIJ Journal 2023 Journal Article

Parameterized complexity of envy-free resource allocation in social networks

• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• Sebastian Ordyniak

ICML Conference 2023 Conference Paper

The Computational Complexity of Concise Hypersphere Classification

• Eduard Eiben
• Robert Ganian
• Iyad A. Kanj
• Sebastian Ordyniak
• Stefan Szeider

Hypersphere classification is a classical and foundational method that can provide easy-to-process explanations for the classification of real-valued as well as binary data. However, obtaining an (ideally concise) explanation via hypersphere classification is much more difficult when dealing with binary data as opposed to 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 not only stronger lower bounds but also a number of new fixed-parameter algorithms for hypersphere classification of binary data, which can find an exact and concise explanation when one exists.

AAAI Conference 2023 Conference Paper

The Parameterized Complexity of Network Microaggregation

• Václav Blažej
• Robert Ganian
• Dušan Knop
• Jan Pokorný
• Šimon Schierreich
• Kirill Simonov

Microaggregation is a classical statistical disclosure control technique which requires the input data to be partitioned into clusters while adhering to specified size constraints. We provide novel exact algorithms and lower bounds for the task of microaggregating a given network while considering both unrestricted and connected clusterings, and analyze these from the perspective of the parameterized complexity paradigm. Altogether, our results assemble a complete complexity-theoretic picture for the network microaggregation problem with respect to the most natural parameterizations of the problem, including input-specified parameters capturing the size and homogeneity of the clusters as well as the treewidth and vertex cover number of the network.

AIJ Journal 2022 Journal Article

An efficient algorithm for counting Markov equivalent DAGs

• Robert Ganian
• Thekla Hamm
• Topi Talvitie

AAAI Conference 2022 Conference Paper

Hedonic Diversity Games: A Complexity Picture with More than Two Colors

• Robert Ganian
• Thekla Hamm
• Dušan Knop
• Šimon Schierreich
• Ondřej Suchý

Hedonic diversity games are a variant of the classical Hedonic games designed to better model a variety of questions concerning diversity and fairness. Previous works mainly targeted the case with two diversity classes (represented as colors in the model) and provided some initial complexitytheoretic and existential results concerning Nash and individually stable outcomes. Here, we design new algorithms accompanied with lower bounds which provide a complete parameterized-complexity picture for computing Nash and individually stable outcomes with respect to the most natural parameterizations of the problem. Crucially, our results hold for general Hedonic diversity games where the number of colors is not necessarily restricted to two, and show that—apart from two trivial cases—a necessary condition for tractability in this setting is that the number of colors is bounded by the parameter. Moreover, for the special case of two colors we resolve an open question posed in previous work.

JAIR Journal 2022 Journal Article

Sum-of-Products with Default Values: Algorithms and Complexity Results

• Robert Ganian
• Eun Jung Kim
• Friedrich Slivovsky
• Stefan Szeider

Weighted Counting for Constraint Satisfaction with Default Values (#CSPD) is a powerful special case of the sum-of-products problem that admits succinct encodings of #CSP, #SAT, and inference in probabilistic graphical models. We investigate #CSPD under the fundamental parameter of incidence treewidth (i.e., the treewidth of the incidence graph of the constraint hypergraph). We show that if the incidence treewidth is bounded, #CSPD can be solved in polynomial time. More specifically, we show that the problem is fixed-parameter tractable for the combined parameter incidence treewidth, domain size, and support size (the maximum number of non-default tuples in a constraint). This generalizes known results on the fixed-parameter tractability of #CSPD under the combined parameter primal treewidth and domain size. We further prove that the problem is not fixed-parameter tractable if any of the three components is dropped from the parameterization.

IJCAI Conference 2022 Conference Paper

The Complexity of Envy-Free Graph Cutting

• Argyrios Deligkas
• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• Sebastian Ordyniak

We consider the problem of fairly dividing a set of heterogeneous divisible resources among agents with different preferences. We focus on the setting where the resources correspond to the edges of a connected graph, every agent must be assigned a connected piece of this graph, and the fairness notion considered is the classical envy freeness. The problem is NP-complete, and we analyze its complexity with respect to two natural complexity measures: the number of agents and the number of edges in the graph. While the problem remains NP-hard even for instances with 2 agents, we provide a dichotomy characterizing the complexity of the problem when the number of agents is constant based on structural properties of the graph. For the latter case, we design a polynomial-time algorithm when the graph has a constant number of edges.

ICML Conference 2022 Conference Paper

The Complexity of k-Means Clustering when Little is Known

• Robert Ganian
• Thekla Hamm
• Viktoriia Korchemna
• Karolina Okrasa
• Kirill Simonov

In the area of data analysis and arguably even in machine learning as a whole, few approaches have been as impactful as the classical k-means clustering. Here, we study the complexity of k-means clustering in settings where most of the data is not known or simply irrelevant. To obtain a more fine-grained understanding of the tractability of this clustering problem, we apply the parameterized complexity paradigm and obtain three new algorithms for k-means clustering of incomplete data: one for the clustering of bounded-domain (i. e. , integer) data, and two incomparable algorithms that target real-valued data. Our approach is based on exploiting structural properties of a graphical encoding of the missing entries, and we show that tractability can be achieved using significantly less restrictive parameterizations than in the complementary case of few missing entries.

JAIR Journal 2022 Journal Article

Threshold Treewidth and Hypertree Width

• Robert Ganian
• Andre Schidler
• Manuel Sorge
• Stefan Szeider

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.

SAT Conference 2022 Conference Paper

Weighted Model Counting with Twin-Width

• Robert Ganian
• Filip Pokrývka
• André Schidler
• Kirill Simonov
• Stefan Szeider

Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula F along with a bound k and asks for the weighted sum of all models with at most k positive literals. BWMC generalizes not only SAT but also (weighted) model counting. We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of F plus k. We show that this result is tight: it is neither possible to drop the bound k nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas.

AIJ Journal 2021 Journal Article

New width parameters for SAT and #SAT

• Robert Ganian
• Stefan Szeider

AIJ Journal 2021 Journal Article

The complexity landscape of decompositional parameters for ILP: Programs with few global variables and constraints

• Pavel Dvořák
• Eduard Eiben
• Robert Ganian
• Dušan Knop
• Sebastian Ordyniak

NeurIPS Conference 2021 Conference Paper

The Complexity of Bayesian Network Learning: Revisiting the Superstructure

• Robert Ganian
• Viktoriia Korchemna

We investigate the parameterized complexity of Bayesian Network Structure Learning (BNSL), a classical problem that has received significant attention in empirical but also purely theoretical studies. We follow up on previous works that have analyzed the complexity of BNSL w. r. t. the so-called superstructure of the input. While known results imply that BNSL is unlikely to be fixed-parameter tractable even when parameterized by the size of a vertex cover in the superstructure, here we show that a different kind of parameterization - notably by the size of a feedback edge set - yields fixed-parameter tractability. We proceed by showing that this result can be strengthened to a localized version of the feedback edge set, and provide corresponding lower bounds that complement previous results to provide a complexity classification of BNSL w. r. t. virtually all well-studied graph parameters. We then analyze how the complexity of BNSL depends on the representation of the input. In particular, while the bulk of past theoretical work on the topic assumed the use of the so-called non-zero representation, here we prove that if an additive representation can be used instead then BNSL becomes fixed-parameter tractable even under significantly milder restrictions to the superstructure, notably when parameterized by the treewidth alone. Last but not least, we show how our results can be extended to the closely related problem of Polytree Learning.

AAAI Conference 2021 Conference Paper

The Complexity of Object Association in Multiple Object Tracking

• Robert Ganian
• Thekla Hamm
• Sebastian Ordyniak

Object association, i. e. , the identification of which observations correspond to the same object, is a central task for the area of multiple object tracking. Two prominent models capturing this task have been introduced in the literature: the Lifted Multicut model and the more recent Lifted Paths model. Here, we carry out a detailed complexity-theoretic study of the problems arising from these two models that is aimed at complementing previous empirical work on object association. We obtain a comprehensive complexity map for both models that takes into account natural restrictions to instances such as possible bounds on the number of frames, number of tracked objects and branching degree, as well as less explicit structural restrictions such as having bounded treewidth. Our results include new fixed-parameter and XP algorithms for the problems as well as hardness proofs which altogether indicate that the Lifted Paths problem exhibits a more favorable complexity behavior than Lifted Multicut.

AAAI Conference 2021 Conference Paper

The Parameterized Complexity of Clustering Incomplete Data

• Eduard Eiben
• Robert Ganian
• Iyad Kanj
• Sebastian Ordyniak
• Stefan Szeider

We study fundamental clustering problems for incomplete data. Specifically, given a set of incomplete d-dimensional vectors (representing rows of a matrix), the goal is to complete the missing vector entries in a way that admits a partitioning of the vectors into at most k clusters with radius or diameter at most r. We give tight characterizations of the parameterized complexity of these problems with respect to the parameters k, r, and the minimum number of rows and columns needed to cover all the missing entries. We show that the considered problems are fixed-parameter tractable when parameterized by the three parameters combined, and that dropping any of the three parameters results in parameterized intractability. A byproduct of our results is that, for the complete data setting, all problems under consideration are fixed-parameter tractable parameterized by k + r.

IJCAI Conference 2021 Conference Paper

The Parameterized Complexity of Connected Fair Division

• Argyrios Deligkas
• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• Sebastian Ordyniak

We study the Connected Fair Division problem (CFD), which generalizes the fundamental problem of fairly allocating resources to agents by requiring that the items allocated to each agent form a connected subgraph in a provided item graph G. We expand on previous results by providing a comprehensive complexity-theoretic understanding of CFD based on several new algorithms and lower bounds while taking into account several well-established notions of fairness: proportionality, envy-freeness, EF1 and EFX. In particular, we show that to achieve tractability, one needs to restrict both the agents and the item graph in a meaningful way. We design (XP)-algorithms for the problem parameterized by (1) clique-width of G plus the number of agents and (2) treewidth of G plus the number of agent types, along with corresponding lower bounds. Finally, we show that to achieve fixed-parameter tractability, one needs to not only use a more restrictive parameterization of G, but also include the maximum item valuation as an additional parameter.

MFCS Conference 2020 Conference Paper

Extending Nearly Complete 1-Planar Drawings in Polynomial Time

• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• Fabian Klute
• Martin Nöllenburg

The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph G, a connected subgraph H of G and a drawing H of H, the extension problem asks whether H can be extended into a drawing of G while maintaining some desired property of the drawing (e. g. , planarity). In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the extension problem is polynomial-time solvable when the aim is to preserve planarity. Very recently we considered this problem for partial 1-planar drawings [ICALP 2020], which are drawings in the plane that allow each edge to have at most one crossing. The most important question identified and left open in that work is whether the problem can be solved in polynomial time when H can be obtained from G by deleting a bounded number of vertices and edges. In this work, we answer this question positively by providing a constructive polynomial-time decision algorithm.

KR Conference 2020 Conference Paper

Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

• Robert Ganian
• Tomáš Peitl
• Friedrich Slivovsky
• Stefan Szeider

We study dependency quantified Boolean formulas (DQBF), an extension of QBF in which dependencies of existential variables are listed explicitly rather than being implicit in the order of quantifiers. DQBF evaluation is a canonical NEXPTIME-complete problem, a complexity class containing many prominent problems that arise in Knowledge Representation and Reasoning. One approach for solving such hard problems is to identify and exploit structural properties captured by numerical parameters such that bounding these parameters gives rise to an efficient algorithm. This idea is captured by the notion of fixed-parameter tractability (FPT). We initiate the study of DQBF through the lens of fixed-parameter tractability and show that the evaluation problem becomes FPT under two natural parameterizations: the treewidth of the primal graph of the DQBF instance combined with a restriction on the interactions between the dependency sets, and also the treedepth of the primal graph augmented by edges representing dependency sets.

AAAI Conference 2020 Conference Paper

On the Parameterized Complexity of Clustering Incomplete Data into Subspaces of Small Rank

• Robert Ganian
• Iyad Kanj
• Sebastian Ordyniak
• Stefan Szeider

We consider a fundamental matrix completion problem where we are given an incomplete matrix and a set of constraints modeled as a CSP instance. The goal is to complete the matrix subject to the input constraints and in such a way that the complete matrix can be clustered into few subspaces with low rank. This problem generalizes several problems in data mining and machine learning, including the problem of completing a matrix into one with minimum rank. In addition to its ubiquitous applications in machine learning, the problem has strong connections to information theory, related to binary linear codes, and variants of it have been extensively studied from that perspective. We formalize the problem mentioned above and study its classical and parameterized complexity. We draw a detailed landscape of the complexity and parameterized complexity of the problem with respect to several natural parameters that are desirably small and with respect to several well-studied CSP fragments.

AAAI Conference 2020 Conference Paper

Parameterized Complexity of Envy-Free Resource Allocation in Social Networks

• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• Sebastian Ordyniak

We consider the classical problem of allocating resources among agents in an envy-free (and, where applicable, proportional) way. Recently, the basic model was enriched by introducing the concept of a social network which allows to capture situations where agents might not have full information about the allocation of all resources. We initiate the study of the parameterized complexity of these resource allocation problems by considering natural parameters which capture structural properties of the network and similarities between agents and items. In particular, we show that even very general fragments of the considered problems become tractable as long as the social network has bounded treewidth or bounded clique-width. We complement our results with matching lower bounds which show that our algorithms cannot be substantially improved.

IJCAI Conference 2020 Conference Paper

Stable Matchings with Diversity Constraints: Affirmative Action is beyond NP

• Jiehua Chen
• Robert Ganian
• Thekla Hamm

We investigate the following many-to-one stable matching problem with diversity constraints (SMTI-DIVERSE): Given a set of students and a set of colleges which have preferences over each other, where the students have overlapping types, and the colleges each have a total capacity as well as quotas for individual types (the diversity constraints), is there a matching satisfying all diversity constraints such that no unmatched student-college pair has an incentive to deviate? SMTI-DIVERSE is known to be NP-hard. However, as opposed to the NP-membership claims in the literature [Aziz et al. , AAMAS 2019; Huang, SODA 2010], we prove that it is beyond NP: it is complete for the complexity class Σ^P_2. In addition, we provide a comprehensive analysis of the problem’s complexity from the viewpoint of natural restrictions to inputs and obtain new algorithms for the problem.

IJCAI Conference 2020 Conference Paper

The Complexity Landscape of Resource-Constrained Scheduling

• Robert Ganian
• Thekla Hamm
• Guillaume Mescoff

The Resource-Constrained Project Scheduling Problem (RCPSP) and its extension via activity modes (MRCPSP) are well-established scheduling frameworks that have found numerous applications in a broad range of settings related to artificial intelligence. Unsurprisingly, the problem of finding a suitable schedule in these frameworks is known to be NP-complete; however, aside from a few results for special cases, we have lacked an in-depth and comprehensive understanding of the complexity of the problems from the viewpoint of natural restrictions of the considered instances. In the first part of our paper, we develop new algorithms and give hardness-proofs in order to obtain a detailed complexity map of (M)RCPSP that settles the complexity of all 1024 considered variants of the problem defined in terms of explicit restrictions of natural parameters of instances. In the second part, we turn to implicit structural restrictions defined in terms of the complexity of interactions between individual activities. In particular, we show that if the treewidth of a graph which captures such interactions is bounded by a constant, then we can solve MRCPSP in polynomial time.

IJCAI Conference 2020 Conference Paper

Threshold Treewidth and Hypertree Width

• Robert Ganian
• Andre Schidler
• Manuel Sorge
• Stefan Szeider

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.

AAAI Conference 2020 Conference Paper

Title

• Robert Ganian
• Thekla Hamm
• Topi Talvitie

We consider the problem of counting the number of DAGs which are Markov-equivalent, i. e. , which encode the same conditional independencies between random variables. The problem has been studied, among others, in the context of causal discovery, and it is known that it reduces to counting the number of so-called moral acyclic orientations of certain undirected graphs, notably chordal graphs. Our main empirical contribution is a new algorithm which outperforms previously known exact algorithms for the considered problem by a significant margin. On the theoretical side, we show that our algorithm is guaranteed to run in polynomial time on a broad class of chordal graphs, including interval graphs.

MFCS Conference 2019 Conference Paper

Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth

• Eduard Eiben
• Robert Ganian
• Thekla Hamm
• O-joung Kwon

We develop a framework for applying treewidth-based dynamic programming on graphs with "hybrid structure", i. e. , with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.

AAAI Conference 2019 Conference Paper

Solving Integer Quadratic Programming via Explicit and Structural Restrictions

• Eduard Eiben
• Robert Ganian
• Dusan Knop
• Sebastian Ordyniak

We study the parameterized complexity of Integer Quadratic Programming under two kinds of restrictions: explicit restrictions on the domain or coefficients, and structural restrictions on variable interactions. We argue that both kinds of restrictions are necessary to achieve tractability for Integer Quadratic Programming, and obtain four new algorithms for the problem that are tuned to possible explicit restrictions of instances that we may wish to solve. The presented algorithms are exact, deterministic, and complemented by appropriate lower bounds.

NeurIPS Conference 2019 Conference Paper

The Parameterized Complexity of Cascading Portfolio Scheduling

• Eduard Eiben
• Robert Ganian
• Iyad Kanj
• Stefan Szeider

Cascading portfolio scheduling is a static algorithm selection strategy which uses a sample of test instances to compute an optimal ordering (a cascading schedule) of a portfolio of available algorithms. The algorithms are then applied to each future instance according to this cascading schedule, until some algorithm in the schedule succeeds. Cascading algorithm scheduling has proven to be effective in several applications, including QBF solving and the generation of ImageNet classification models. It is known that the computation of an optimal cascading schedule in the offline phase is NP-hard. In this paper we study the parameterized complexity of this problem and establish its fixed-parameter tractability by utilizing structural properties of the success relation between algorithms and test instances. Our findings are significant as they reveal that in spite of the intractability of the problem in its general form, one can indeed exploit sparseness or density of the success relation to obtain non-trivial runtime guarantees for finding an optimal cascading schedule.

IJCAI Conference 2018 Conference Paper

A Structural Approach to Activity Selection

• Eduard Eiben
• Robert Ganian
• Sebastian Ordyniak

The general task of finding an assignment of agents to activities under certain stability and rationality constraints has led to the introduction of two prominent problems in the area of computational social choice: Group Activity Selection (GASP) and Stable Invitations (SIP). Here we introduce and study the Comprehensive Activity Selection Problem, which naturally generalizes both of these problems. In particular, we apply the parameterized complexity paradigm, which has already been successfully employed for SIP and GASP. While previous work has focused strongly on parameters such as solution size or number of activities, here we focus on parameters which capture the complexity of agent-to-agent interactions. Our results include a comprehensive complexity map for CAS under various restrictions on the number of activities in combination with restrictions on the complexity of agent interactions.

ICML Conference 2018 Conference Paper

Parameterized Algorithms for the Matrix Completion Problem

• Robert Ganian
• Iyad A. Kanj
• Sebastian Ordyniak
• Stefan Szeider

We consider two matrix completion problems, in which we are given a matrix with missing entries and the task is to complete the matrix in a way that (1) minimizes the rank, or (2) minimizes the number of distinct rows. We study the parameterized complexity of the two aforementioned problems with respect to several parameters of interest, including the minimum number of matrix rows, columns, and rows plus columns needed to cover all missing entries. We obtain new algorithmic results showing that, for the bounded domain case, both problems are fixed-parameter tractable with respect to all aforementioned parameters. We complement these results with a lower-bound result for the unbounded domain case that rules out fixed-parameter tractability w. r. t. some of the parameters under consideration.

AIJ Journal 2018 Journal Article

The complexity landscape of decompositional parameters for ILP

• Robert Ganian
• Sebastian Ordyniak

IJCAI Conference 2018 Conference Paper

Unary Integer Linear Programming with Structural Restrictions

• Eduard Eiben
• Robert Ganian
• Dušan Knop
• Sebastian Ordyniak

Recently a number of algorithmic results have appeared which show the tractability of Integer Linear Programming (ILP) instances under strong restrictions on variable domains and/or coefficients (AAAI 2016, AAAI 2017, IJCAI 2017). In this paper, we target ILPs where neither the variable domains nor the coefficients are restricted by a fixed constant or parameter; instead, we only require that our instances can be encoded in unary. We provide new algorithms and lower bounds for such ILPs by exploiting the structure of their variable interactions, represented as a graph. Our first set of results focuses on solving ILP instances through the use of a graph parameter called clique-width, which can be seen as an extension of treewidth which also captures well-structured dense graphs. In particular, we obtain a polynomial-time algorithm for instances of bounded clique-width whose domain and coefficients are polynomially bounded by the input size, and we complement this positive result by a number of algorithmic lower bounds. Afterwards, we turn our attention to ILPs with acyclic variable interactions. In this setting, we obtain a complexity map for the problem with respect to the graph representation used and restrictions on the encoding.

SAT Conference 2017 Conference Paper

Backdoor Treewidth for SAT

• Robert Ganian
• M. S. Ramanujan 0001
• Stefan Szeider

Abstract A strong backdoor in a CNF formula is a set of variables such that each possible instantiation of these variables moves the formula into a tractable class. The algorithmic problem of finding a strong backdoor has been the subject of intensive study, mostly within the parameterized complexity framework. Results to date focused primarily on backdoors of small size. In this paper we propose a new approach for algorithmically exploiting strong backdoors for SAT: instead of focusing on small backdoors, we focus on backdoors with certain structural properties. In particular, we consider backdoors that have a certain tree-like structure, formally captured by the notion of backdoor treewidth. First, we provide a fixed-parameter algorithm for SAT parameterized by the backdoor treewidth w. r. t. the fundamental tractable classes Horn, Anti-Horn, and 2CNF. Second, we consider the more general setting where the backdoor decomposes the instance into components belonging to different tractable classes, albeit focusing on backdoors of treewidth 1 (i. e. , acyclic backdoors). We give polynomial-time algorithms for SAT and #SAT for instances that admit such an acyclic backdoor.

AAAI Conference 2017 Conference Paper

Going Beyond Primal Treewidth for (M)ILP

• Robert Ganian
• Sebastian Ordyniak
• M. Ramanujan

Integer Linear Programming (ILP) and its mixed variant (MILP) are archetypical examples of NP-complete optimization problems which have a wide range of applications in various areas of artificial intelligence. However, we still lack a thorough understanding of which structural restrictions make these problems tractable. Here we focus on structure captured via so-called decompositional parameters, which have been highly successful in fields such as boolean satisfiability and constraint satisfaction but have not yet reached their full potential in the ILP setting. In particular, primal treewidth (an established decompositional parameter) can only be algorithmically exploited to solve ILP under restricted circumstances. Our main contribution is the introduction and algorithmic exploitation of two new decompositional parameters for ILP and MILP. The first, torso-width, is specifically tailored to the linear programming setting and is the first decompositional parameter which can also be used for MILP. The latter, incidence treewidth, is a concept which originates from boolean satisfiability but has not yet been used in the ILP setting; here we obtain a full complexity landscape mapping the precise conditions under which incidence treewidth can be used to obtain efficient algorithms. Both of these parameters overcome previous shortcomings of primal treewidth for ILP in unique ways, and consequently push the frontiers of tractability for these important problems.

SAT Conference 2017 Conference Paper

New Width Parameters for Model Counting

• Robert Ganian
• Stefan Szeider

Abstract We study the parameterized complexity of the propositional model counting problem #SAT for CNF formulas. As the parameter we consider the treewidth of the following two graphs associated with CNF formulas: the consensus graph and the conflict graph. Both graphs have as vertices the clauses of the formula; in the consensus graph two clauses are adjacent if they do not contain a complementary pair of literals, while in the conflict graph two clauses are adjacent if they do contain a complementary pair of literals. We show that #SAT is fixed-parameter tractable for the treewidth of the consensus graph but W[1]-hard for the treewidth of the conflict graph. We also compare the new parameters with known parameters under which #SAT is fixed-parameter tractable.

IJCAI Conference 2017 Conference Paper

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

• Pavel Dvořák
• Eduard Eiben
• Robert Ganian
• Dušan Knop
• Sebastian Ordyniak

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.

MFCS Conference 2017 Conference Paper

Towards a Polynomial Kernel for Directed Feedback Vertex Set

• Benjamin Bergougnoux
• Eduard Eiben
• Robert Ganian
• Sebastian Ordyniak
• M. S. Ramanujan 0001

In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

MFCS Conference 2016 Conference Paper

A Single-Exponential Fixed-Parameter Algorithm for Distance-Hereditary Vertex Deletion

• Eduard Eiben
• Robert Ganian
• O-joung Kwon

Vertex deletion problems ask whether it is possible to delete at most k vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our result with matching asymptotic lower bounds based on the exponential time hypothesis.

SODA Conference 2016 Conference Paper

Discovering Archipelagos of Tractability for Constraint Satisfaction and Counting

• Robert Ganian
• M. S. Ramanujan 0001
• Stefan Szeider

The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called “islands of tractability. ” A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language, whereas a constraint language is conservative if it contains all unary relations. Schaefer's famous Dichotomy Theorem (STOC 1978) identifies all islands of tractability in terms of tractable constraint languages over a Boolean domain of values. Since then many extensions and generalizations of this result have been obtained. Recently, Bulatov (TOCL 2011, JACM 2013) gave a full characterization of all islands of tractability for CSP and the counting version #CSP that are defined in terms of conservative constraint languages. This paper addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages isn't tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose we utilize the notion of a strong backdoor of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated moves the instance to an island of tractability, i. e. , to a tractable class of instances. We consider strong backdoors into scattered classes, consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Figuratively speaking, a scattered class constitutes an archipelago of tractability. The main difficulty lies in finding a strong backdoor of given size k; once it is found, we can try all possible instantiations of the backdoor variables and apply the polynomial time algorithms associated with the islands of tractability on the list component wise. Our main result is an algorithm that, given a CSP instance with n variables, finds in time f ( k ) n ℴ (1) a strong backdoor into a scattered class (associated with a list of finite conservative constraint languages) of size k or correctly decides that there isn't such a backdoor. This also gives the running time for solving (#)CSP, provided that (#)CSP is polynomial-time tractable for the considered constraint languages. Our result makes significant progress towards the main goal of the backdoor-based approach to CSPs – the identification of maximal base classes for which small backdoors can be detected efficiently.

MFCS Conference 2016 Conference Paper

On Existential MSO and its Relation to ETH

• Robert Ganian
• Ronald de Haan
• Iyad A. Kanj
• Stefan Szeider

Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k-CNF-Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this paper, we extend the framework of Impagliazzo et al. , and identify a larger set of problems that are equivalent to k-CNF-Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem. This research direction can be traced back to Fagin's celebrated theorem stating that NP coincides with the class of problems expressible in existential second order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter. We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat.

MFCS Conference 2016 Conference Paper

On the Complexity Landscape of Connected f-Factor Problems

• Robert Ganian
• N. S. Narayanaswamy
• Sebastian Ordyniak
• C. S. Rahul 0001
• M. S. Ramanujan 0001

Given an n-vertex graph G and a function f: V(G) -> {0, .. ., n-1}, an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. A classical result of Tutte (1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize Hamiltonian Cycle and hence is NP-complete. In fact, the Connected f-Factor problem remains NP-complete even when f(v) is at least n^epsilon for each vertex v and epsilon<1; on the other side of the spectrum, the problem was known to be polynomial-time solvable when f(v) is at least n/3 for every vertex v. In this paper, we extend this line of work and obtain new complexity results based on restricting the function f. In particular, we show that when f(v) is required to be at least n/(log n)^c, the problem can be solved in quasi-polynomial time in general and in randomized polynomial time if c <= 1. We also show that when c>1, the problem is NP-intermediate.

TCS Journal 2016 Journal Article

Quantified conjunctive queries on partially ordered sets

• Simone Bova
• Robert Ganian
• Stefan Szeider

AAAI Conference 2016 Conference Paper

The Complexity Landscape of Decompositional Parameters for ILP

• Robert Ganian
• Sebastian Ordyniak

Integer Linear Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems in artificial intelligence are solved in practice via a translation to ILP. Despite its huge range of applications, only few tractable fragments of ILP are known, probably the most prominent of which is based on the notion of total unimodularity. Using entirely different techniques, we identify new tractable fragments of ILP by studying structural parameterizations of the constraint matrix within the framework of parameterized complexity. In particular, we show that ILP is fixed-parameter tractable when parameterized by the treedepth of the constraint matrix and the maximum absolute value of any coefficient occurring in the ILP instance. Together with matching hardness results for the more general parameter treewidth, we draw a detailed complexity landscape of ILP w. r. t. decompositional parameters defined on the constraint matrix.

AAAI Conference 2016 Conference Paper

Using Decomposition-Parameters for QBF: Mind the Prefix!

• Eduart Eiben
• Robert Ganian
• Sebastian Ordyniak

Similar to the satisfiability (SAT) problem, which can be seen to be the archetypical problem for NP, the quantified Boolean formula problem (QBF) is the archetypical problem for PSPACE. Recently, Atserias and Oliva (2014) showed that, unlike for SAT, many of the well-known decompositional parameters (such as treewidth and pathwidth) do not allow efficient algorithms for QBF. The main reason for this seems to be the lack of awareness of these parameters towards the dependencies between variables of a QBF formula. In this paper we extend the ordinary pathwidth to the QBF-setting by introducing prefix pathwidth, which takes into account the dependencies between variables in a QBF, and show that it leads to an efficient algorithm for QBF. We hope that our approach will help to initiate the study of novel tailor-made decompositional parameters for QBF and thereby help to lift the success of these decompositional parameters from SAT to QBF.

SAT Conference 2015 Conference Paper

Community Structure Inspired Algorithms for SAT and #SAT

• Robert Ganian
• Stefan Szeider

Abstract We introduce h-modularity, a structural parameter of CNF formulas, and present algorithms that render the decision problem SAT and the model counting problem #SAT fixed-parameter tractable when parameterized by h-modularity. The new parameter is defined in terms of a partition of clauses of the given CNF formula into strongly interconnected communities which are sparsely interconnected with each other. Each community forms a hitting formula, whereas the interconnections between communities form a graph of small treewidth. Our algorithms first identify the community structure and then use them for an efficient solution of SAT and #SAT, respectively. We further show that h-modularity is incomparable with known parameters under which SAT or #SAT is fixed-parameter tractable.

Highlights Conference 2013 Conference Abstract

FO model checking of interval graphs

• Robert Ganian
• Petr Hliněný
• Daniel Kral
• Jan Obdržálek
• Jarett Schwartz
• Jakub Teska

We study the computational complexity of the model checking problem for the first order (FO) logic on interval graphs, i. e. , interscetion graphs of intervals on the real line. As the main result we show that for n-vertex interval graphs this problem can be solved in time O(n log n) if we take intervals with lengths from a fixed finite set. On the other hand, we show that this is no longer true once the interval lengths taken from any set that is dense in some open subset.

MFCS Conference 2013 Conference Paper

Meta-kernelization with Structural Parameters

• Robert Ganian
• Friedrich Slivovsky
• Stefan Szeider

Abstract Meta-kernelization theorems are general results that provide polynomial kernels for large classes of parameterized problems. The known meta-kernelization theorems, in particular the results of Bodlaender et al. (FOCS’09) and of Fomin et al. (FOCS’10), apply to optimization problems parameterized by solution size. We present meta-kernelization theorems that use structural parameters of the input and not the solution size. Let \(\mathcal{C}\) be a graph class. We define the \(\mathcal{C}\) - cover number of a graph to be the smallest number of modules the vertex set can be partitioned into such that each module induces a subgraph that belongs to the class \(\mathcal{C}\). We show that each graph problem that can be expressed in Monadic Second Order (MSO) logic has a polynomial kernel with a linear number of vertices when parameterized by the \(\mathcal{C}\) - cover number for any fixed class \(\mathcal{C}\) of bounded rank-width (or equivalently, of bounded clique-width, or bounded Boolean-width). Many graph problems such as c -Coloring, c -Domatic Number and c -Clique Cover are covered by this meta-kernelization result. Our second result applies to MSO expressible optimization problems, such as Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We show that these problems admit a polynomial annotated kernel with a linear number of vertices.

MFCS Conference 2012 Conference Paper

When Trees Grow Low: Shrubs and Fast MSO1

• Robert Ganian
• Petr Hlinený
• Jaroslav Nesetril
• Jan Obdrzálek
• Patrice Ossona de Mendez
• Reshma Ramadurai

Abstract Recent characterization [9] of those graphs for which coloured MSO 2 model checking is fast raised the interest in the graph invariant called tree-depth. Looking for a similar characterization for (coloured) MSO 1, we introduce the notion of shrub-depth of a graph class. To prove that MSO 1 model checking is fast for classes of bounded shrub-depth, we show that shrub-depth exactly characterizes the graph classes having interpretation in coloured trees of bounded height. We also introduce a common extension of cographs and of graphs with bounded shrub-depth — m-partite cographs (still of bounded clique-width), which are well quasi-ordered by the relation “is an induced subgraph of” and therefore allow polynomial time testing of hereditary properties.