arXiv subjects

Mateusz Rychlicki

Publications and source records attributed to Mateusz Rychlicki.

8 recordsLinked to original sources

arXiv2026

Structural Parameterizations for Eternal Vertex Cover

Eternal Vertex Cover (EVC) is a turn-based attacker-defender game on an undirected graph $G$. To begin with, the defender places $k$ guards on vertices of $G$. The attacker, on their turn, can choose an edge $e$ not already occupied at both endpoints to "attack". The edge $e$ is defended if a guard moves along the edge $e$. The defender, on their turn, can move any subset of guards. A guard can only move to a neighboring vertex. The minimum number of guards needed to indefinitely defend against any sequence of attacks is called the eternal vertex cover number, generalizing the classic vertex cover number. Determining this number is NP-hard in general, motivating the study of parameterized and approximation algorithms. The problem is known to be FPT when parameterized by the cover number, but structural parameters remain relatively unexplored in the literature. In this work, we explore structural parameterizations for EVC. We show that EVC is FPT parameterized by the cluster vertex deletion number, which generalizes the previously studied parameterization by vertex cover number. We next study the problem parameterized by vertex integrity, which is the smallest number of vertices we need to delete from $G$ so that the resulting graph is a disjoint union of constant-sized components. We first show that Eternal Vertex Cover is XP parameterized by vertex integrity. Then, we develop a polynomial-time approximation algorithm, which computes an additive $6k+1$ ($g(k)$) approximation, where $k$ is equal to the cluster vertex deletion number (vertex integrity). Finally, we show a FPT algorithm for when the deletion set produces "nice" connected components, which are components that are bounded in size and satisfy a technical condition.

cs.DM↗

arXiv2026

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.

cs.CC↗

arXiv2026

Backdoors for Quantified Boolean Formulas

The quantified Boolean formula problem (QBF) is a well-known PSpace-complete problem with rich expressive power, and is generally viewed as the SAT analogue for PSpace. Given that many problems today are solved in practice by reducing to SAT, and then using highly optimized SAT solvers, it is natural to ask whether problems in PSpace are amenable to this approach. While SAT solvers exploit hidden structural properties, such as backdoors to tractability, backdoor analysis for QBF is comparatively very limited. We present a comprehensive study of the (parameterized) complexity of QBF parameterized by backdoor size to the largest tractable syntactic classes: HORN, 2-SAT, and AFFINE. While SAT is in FPT under this parameterization, we prove that QBF remains PSpace-hard even on formulas with backdoors of constant size. Parameterizing additionally by the quantifier depth, we design FPT-algorithms for the classes 2-SAT and AFFINE, and show that 3-HORN is W[1]-hard. As our next contribution, we vastly extend the applicability of QBF backdoors not only for the syntactic classes defined above but also for tractable classes defined via structural restrictions, such as formulas with bounded incidence treewidth and quantifier depth. To this end, we introduce enhanced backdoors: these are separators S of size at most k in the primal graph such that S together with all variables contained in any purely universal component of the primal graph minus S is a backdoor. We design FPT-algorithms with respect to k for both evaluation and detection of enhanced backdoors to all tractable classes of QBF listed above and more.

cs.DS↗

arXiv2025

Explaining Decisions in ML Models: a Parameterized Complexity Analysis (Part I)

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

cs.AI↗

arXiv2024

Explaining Decisions in ML Models: a Parameterized Complexity Analysis

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

cs.AI↗

arXiv2024

Solving Quantified Boolean Formulas with Few Existential Variables

The quantified Boolean formula (QBF) problem is an important decision problem generally viewed as the archetype for PSPACE-completeness. Many problems of central interest in AI are in general not included in NP, e.g., planning, model checking, and non-monotonic reasoning, and for such problems QBF has successfully been used as a modelling tool. However, solvers for QBF are not as advanced as state of the art SAT solvers, which has prevented QBF from becoming a universal modelling language for PSPACE-complete problems. A theoretical explanation is that QBF (as well as many other PSPACE-complete problems) lacks natural parameters} guaranteeing fixed-parameter tractability (FPT). In this paper we tackle this problem and consider a simple but overlooked parameter: the number of existentially quantified variables. This natural parameter is virtually unexplored in the literature which one might find surprising given the general scarcity of FPT algorithms for QBF. Via this parameterization we then develop a novel FPT algorithm applicable to QBF instances in conjunctive normal form (CNF) of bounded clause length. We complement this by a W[1]-hardness result for QBF in CNF of unbounded clause length as well as sharper lower bounds for the bounded arity case under the (strong) exponential-time hypothesis.

cs.CC↗

arXiv2024

A Tight Subexponential-time Algorithm for Two-Page Book Embedding

A book embedding of a graph is a drawing that maps vertices onto a line and edges to simple pairwise non-crossing curves drawn into pages, which are half-planes bounded by that line. Two-page book embeddings, i.e., book embeddings into 2 pages, are of special importance as they are both NP-hard to compute and have specific applications. We obtain a 2^(O(\sqrt{n})) algorithm for computing a book embedding of an n-vertex graph on two pages -- a result which is asymptotically tight under the Exponential Time Hypothesis. As a key tool in our approach, we obtain a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph. We conclude by establishing the fixed-parameter tractability of computing minimum-page book embeddings when parameterized by the feedback edge number, settling an open question arising from previous work on the problem.

cs.DS↗

arXiv2022

Dynamic data structures for parameterized string problems

We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently, our goal is to design a data structure that efficiently maintains a solution, or reports a lack thereof, upon updates in the instance. We first consider the Closest String problem, for which we design randomized dynamic data structures with amortized update times $d^{\mathcal{O}(d)}$ and $|Σ|^{\mathcal{O}(d)}$, respectively, where $Σ$ is the alphabet and $d$ is the assumed bound on the maximum distance. These are obtained by combining known static approaches to Closest String with color-coding. Next, we note that from a result of Frandsen et al.~[J. ACM'97] one can easily infer a meta-theorem that provides dynamic data structures for parameterized string problems with worst-case update time of the form $\mathcal{O}(\log \log n)$, where $k$ is the parameter in question and $n$ is the length of the string. We showcase the utility of this meta-theorem by giving such data structures for problems Disjoint Factors and Edit Distance. We also give explicit data structures for these problems, with worst-case update times $\mathcal{O}(k2^{k}\log \log n)$ and $\mathcal{O}(k^2\log \log n)$, respectively. Finally, we discuss how a lower bound methodology introduced by Amarilli et al.~[ICALP'21] can be used to show that obtaining update time $\mathcal{O}(f(k))$ for Disjoint Factors and Edit Distance is unlikely already for a constant value of the parameter $k$.

cs.DS↗