arXiv subjects

Jan Dreier

Publications and source records attributed to Jan Dreier.

At least 19 recordsLinked to original sources

arXiv2026

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes

Monadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is \emph{monadically dependent} if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has \emph{almost linear neighborhood complexity}: for every graph $G$ in the class and every set $A\subseteq V(G)$, the family $\{N_G(v)\cap A : v\in V(G)\}$ has size $|A|^{1+o(1)}$. Second, every $n$-vertex graph in a monadically dependent class has radius-1 merge-width $n^{o(1)}$. Here, merge-width is the decomposition parameter of Dreier and Toru\'nczyk based on construction sequences; its radius-$r$ version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an $\mathcal{O}(n^5)$-time algorithm that, given an $n$-vertex graph $G$ such that $|\{N_G(v)\cap A : v\in V(G)\}|\le O(|A|^d)$ for every $A\subseteq V(G)$, computes a construction sequence witnessing radius-1 merge-width $\mathcal{O}(n^{1-1/d}\log n)$.

cs.DM↗

arXiv2026

A Rank-Preserving Locality Theorem

We prove a rank-preserving locality theorem for a syntactic variant of first-order logic, in the spirit of Gaifman's locality theorem and the rank-preserving locality theorem of Grohe, Kreutzer, and Siebertz. Our result allows for a weak form of scatter sentences, which can be evaluated more efficiently than usual scatter sentences considered in prior work. This is crucial in our application to graphs of bounded merge-width.

cs.LO↗

arXiv2026

Near-Linear Time Computation of Welzl Orders on Graphs with Linear Neighborhood Complexity

Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter functions correspond to graph classes with linear neighborhood complexity. For such systems, Welzl's theorem guarantees the existence of orders with only $\mathcal{O}(\log^2 n)$ crossings. A series of works has progressively improved the runtime for computing such orders, from Chazelle and Welzl's original $\mathcal{O}(|U|^3 |\mathcal{F}|)$ bound, through Har-Peled's $\mathcal{O}(|U|^2|\mathcal{F}|)$, to the recent sampling-based methods of Csik\'os and Mustafa. We present a randomized algorithm that computes Welzl orders for set systems with linear primal and dual shatter functions in time $\mathcal{O}(\|S\| \log \|S\|)$, where $\|S\| = |U| + \sum_{X \in \mathcal{F}} |X|$ is the size of the canonical input representation. As an application, we compute compact neighborhood covers in graph classes with (near-)linear neighborhood complexity in time $\mathcal{O}(n \log n)$ and improve the runtime of first-order model checking on monadically stable graph classes from $\mathcal{O}(n^{5+\varepsilon})$ to $\mathcal{O}(n^{3+\varepsilon})$.

cs.DS↗

arXiv2026

The Parameterized Complexity of Independent Set and More when Excluding a Half-Graph, Co-Matching, or Matching

A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured by the matching index, co-matching index, and half-graph index of a graph. Consequently, graph classes can be organized into the eight classes determined by which of the three indices are bounded. We completely classify the parameterized complexity of Independent Set, Clique, and Dominating Set across all eight of these classes. For this purpose, we first derive multiple tractability and hardness results from the existing literature, and then proceed to fill the identified gaps. Among our novel results, we show that Independent Set is fixed-parameter tractable on every graph class where the half-graph and co-matching indices are simultaneously bounded. Conversely, we construct a graph class with bounded half-graph index (but unbounded co-matching index), for which the problem is W[1]-hard. For the W[1]-hard cases of our classification, we review the state of approximation algorithms. Here, we contribute an approximation algorithm for Independent Set on classes of bounded half-graph index.

cs.CC↗

arXiv2026

Efficient reversal of transductions of sparse graph classes

(First-order) transductions are a basic notion capturing graph modifications that can be described in first-order logic. In this work, we propose an efficient algorithmic method to approximately reverse the application of a transduction, assuming the source graph is sparse. Precisely, for any graph class $\mathcal{C}$ that has structurally bounded expansion (i.e., can be transduced from a class of bounded expansion), we give an $O(n^4)$-time algorithm that given a graph $G\in \mathcal{C}$, computes a vertex-colored graph $H$ such that $G$ can be recovered from $H$ using a first-order interpretation and $H$ belongs to a graph class $\mathcal{D}$ of bounded expansion. This answers an open problem raised by Gajarsk\'y et al. In fact, for our procedure to work we only need to assume that $\mathcal{C}$ is monadically stable (i.e., does not transduce the class of all half-graphs) and has inherently linear neighborhood complexity (i.e., the neighborhood complexity is linear in all graph classes transducible from $\mathcal{C}$). This renders the conclusion that the graph classes satisfying these two properties coincide with classes of structurally bounded expansion.

cs.LO↗

arXiv2025

Merge-width and First-Order Model Checking

We introduce merge-width, a family of graph parameters that unifies several structural graph measures, including treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. Our parameters are based on new decompositions called construction sequences. These are sequences of ever coarser partitions of the vertex set, where each pair of parts has a specified default connection, and all vertex pairs of the graph that differ from the default are marked as resolved. The radius-$r$ merge-width is the maximum number of parts reached from a vertex by following a path of at most $r$ resolved edges. Graph classes of bounded merge-width -- for which the radius-$r$ merge-width parameter can be bounded by a constant, for each fixed $r=1,2,3,\ldots$ -- include all classes of bounded expansion or of bounded twin-width, thus unifying two central notions from the Sparsity and Twin-width frameworks. Furthermore, they are preserved under first-order transductions, which attests to their robustness. We conjecture that classes of bounded merge-width are equivalent to the previously introduced classes of bounded flip-width. As our main result, we show that the model checking problem for first-order logic is fixed-parameter tractable on graph classes of bounded merge-width, assuming the input includes a witnessing construction sequence. This unites and extends two previous model checking results: the result of Dvo\v{r}\'{a}k, Kr\'{a}l, and Thomas for classes of bounded expansion, and the result of Bonnet, Kim, Thomass\'e, and Watrigant for classes of bounded twin-width. Finally, we suggest future research directions that could impact the study of structural and algorithmic graph theory, in particular of monadically dependent graph classes, which we conjecture to coincide with classes of almost bounded merge-width.

math.CO↗

arXiv2024

Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph Classes

A conjecture in algorithmic model theory predicts that the model-checking problem for first-order logic is fixed-parameter tractable on a hereditary graph class if and only if the class is monadically dependent. Originating in model theory, this notion is defined in terms of logic, and encompasses nowhere dense classes, monadically stable classes, and classes of bounded twin-width. Working towards this conjecture, we provide the first two combinatorial characterizations of monadically dependent graph classes. This yields the following dichotomy. On the structure side, we characterize monadic dependence by a Ramsey-theoretic property called flip-breakability. This notion generalizes the notions of uniform quasi-wideness, flip-flatness, and bounded grid rank, which characterize nowhere denseness, monadic stability, and bounded twin-width, respectively, and played a key role in their respective model checking algorithms. Natural restrictions of flip-breakability additionally characterize bounded treewidth and cliquewidth and bounded treedepth and shrubdepth. On the non-structure side, we characterize monadic dependence by explicitly listing few families of forbidden induced subgraphs. This result is analogous to the characterization of nowhere denseness via forbidden subdivided cliques, and allows us to resolve one half of the motivating conjecture: First-order model checking is AW[$*$]-hard on every hereditary graph class that is monadically independent. The result moreover implies that hereditary graph classes which are small, have almost bounded twin-width, or have almost bounded flip-width, are monadically dependent. Lastly, we lift our result to also obtain a combinatorial dichotomy in the more general setting of monadically dependent classes of binary structures.

math.CO↗

arXiv2023

First-Order Model Checking on Monadically Stable Graph Classes

A graph class $\mathscr{C}$ is called monadically stable if one cannot interpret, in first-order logic, arbitrary large linear orders in colored graphs from $\mathscr{C}$. We prove that the model checking problem for first-order logic is fixed-parameter tractable on every monadically stable graph class. This extends the results of [Grohe, Kreutzer, and Siebertz; J. ACM '17] for nowhere dense classes and of [Dreier, M\"ahlmann, and Siebertz; STOC '23] for structurally nowhere dense classes to all monadically stable classes. As a complementary hardness result, we prove that for every hereditary graph class $\mathscr{C}$ that is edge-stable (excludes some half-graph as a semi-induced subgraph) but not monadically stable, first-order model checking is $\mathrm{AW}[*]$-hard on $\mathscr{C}$, and $\mathrm{W}[1]$-hard when restricted to existential sentences. This confirms, in the special case of edge-stable classes, an on-going conjecture that the notion of monadic NIP delimits the tractability of first-order model checking on hereditary classes of graphs. For our tractability result, we first prove that monadically stable graph classes have almost linear neighborhood complexity. Using this, we construct sparse neighborhood covers for monadically stable classes, which provides the missing ingredient for the algorithm of [Dreier, M\"ahlmann, and Siebertz; STOC '23]. The key component of this construction is the usage of orders with low crossing number [Welzl; SoCG '88], a tool from the area of range queries. For our hardness result, we prove a new characterization of monadically stable graph classes in terms of forbidden induced subgraphs. We then use this characterization to show that in hereditary classes that are edge-stable but not monadically stable, one can effectively interpret the class of all graphs using only existential formulas.

cs.LO↗

arXiv2023

Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond

It is known that first-order logic with some counting extensions can be efficiently evaluated on graph classes with bounded expansion, where depth-$r$ minors have constant density. More precisely, the formulas are $\exists x_1 ... x_k \#y \varphi(x_1,...,x_k, y)>N$, where $\varphi$ is an FO-formula. If $\varphi$ is quantifier-free, we can extend this result to nowhere dense graph classes with an almost linear FPT run time. Lifting this result further to slightly more general graph classes, namely almost nowhere dense classes, where the size of depth-$r$ clique minors is subpolynomial, is impossible unless FPT=W[1]. On the other hand, in almost nowhere dense classes we can approximate such counting formulas with a small additive error. Note those counting formulas are contained in FOC({<}) but not FOC1(P). In particular, it follows that partial covering problems, such as partial dominating set, have fixed parameter algorithms on nowhere dense graph classes with almost linear running time.

cs.LO↗

arXiv2023

Approximate Evaluation of Quantitative Second Order Queries

Courcelle's theorem and its adaptations to cliquewidth have shaped the field of exact parameterized algorithms and are widely considered the archetype of algorithmic meta-theorems. In the past decade, there has been growing interest in developing parameterized approximation algorithms for problems which are not captured by Courcelle's theorem and, in particular, are considered not fixed-parameter tractable under the associated widths. We develop a generalization of Courcelle's theorem that yields efficient approximation schemes for any problem that can be captured by an expanded logic we call Blocked CMSO, capable of making logical statements about the sizes of set variables via so-called weight comparisons. The logic controls weight comparisons via the quantifier-alternation depth of the involved variables, allowing full comparisons for zero-alternation variables and limited comparisons for one-alternation variables. We show that the developed framework threads the very needle of tractability: on one hand it can describe a broad range of approximable problems, while on the other hand we show that the restrictions of our logic cannot be relaxed under well-established complexity assumptions. The running time of our approximation scheme is polynomial in $1/\varepsilon$, allowing us to fully interpolate between faster approximate algorithms and slower exact algorithms. This provides a unified framework to explain the tractability landscape of graph problems parameterized by treewidth and cliquewidth, as well as classical non-graph problems such as Subset Sum and Knapsack.

cs.CC↗

arXiv2023

Pseudorandom Finite Models

We study pseudorandomness and pseudorandom generators from the perspective of logical definability. Building on results from ordinary derandomization and finite model theory, we show that it is possible to deterministically construct, in polynomial time, graphs and relational structures that are statistically indistinguishable from random structures by any sentence of first order or least fixed point logics. This raises the question of whether such constructions can be implemented via logical transductions from simpler structures with less entropy. In other words, can logical formulas be pseudorandom generators? We provide a complete classification of when this is possible for first order logic, fixed point logic, and fixed point logic with parity, and provide partial results and conjectures for first order logic with parity.

cs.LO↗

arXiv2023

First-Order Model Checking on Structurally Sparse Graph Classes

A class of graphs is structurally nowhere dense if it can be constructed from a nowhere dense class by a first-order transduction. Structurally nowhere dense classes vastly generalize nowhere dense classes and constitute important examples of monadically stable classes. We show that the first-order model checking problem is fixed-parameter tractable on every structurally nowhere dense class of graphs. Our result builds on a recently developed game-theoretic characterization of monadically stable graph classes. As a second key ingredient of independent interest, we provide a polynomial-time algorithm for approximating weak neighborhood covers (on general graphs). We combine the two tools into a recursive locality-based model checking algorithm. This algorithm is efficient on every monadically stable graph class admitting flip-closed sparse weak neighborhood covers, where flip-closure is a mild additional assumption. Thereby, establishing efficient first-order model checking on monadically stable classes is reduced to proving the existence of flip-closed sparse weak neighborhood covers on these classes - a purely combinatorial problem. We complete the picture by proving the existence of the desired covers for structurally nowhere dense classes: we show that every structurally nowhere dense class can be sparsified by contracting local sets of vertices, enabling us to lift the existence of covers from sparse classes.

cs.LO↗

arXiv2022

A logic-based algorithmic meta-theorem for mim-width

We introduce a logic called distance neighborhood logic with acyclicity and connectivity constraints ($\mathsf{A\&C~DN}$ for short) which extends existential $\mathsf{MSO_1}$ with predicates for querying neighborhoods of vertex sets and for verifying connectivity and acyclicity of vertex sets in various powers of a graph. Building upon [Bergougnoux and Kanté, ESA 2019; SIDMA 2021], we show that the model checking problem for every fixed $\mathsf{A\&C~DN}$ formula is solvable in $n^{O(w)}$ time when the input graph is given together with a branch decomposition of mim-width $w$. Nearly all problems that are known to be solvable in polynomial time given a branch decomposition of constant mim-width can be expressed in this framework. We add several natural problems to this list, including problems asking for diverse sets of solutions. Our model checking algorithm is efficient whenever the given branch decomposition of the input graph has small index in terms of the $d$-neighborhood equivalence [Bui-Xuan, Telle, and Vatshelle, TCS 2013]. We therefore unify and extend known algorithms for tree-width, clique-width and rank-width. Our algorithm has a single-exponential dependence on these three width measures and asymptotically matches run times of the fastest known algorithms for several problems. This results in algorithms with tight run times under the Exponential Time Hypothesis ($\mathsf{ETH}$) for tree-width, clique-width and rank-width; the above mentioned run time for mim-width is nearly tight under the $\mathsf{ETH}$ for several problems as well. Our results are also tight in terms of the expressive power of the logic: we show that already slight extensions of our logic make the model checking problem para-$\mathsf{NP}$-hard when parameterized by mim-width plus formula length.

cs.DS↗

arXiv2022

Combinatorial and Algorithmic Aspects of Monadic Stability

Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results. - In monadically stable classes the Ramsey numbers $R(s,t)$ are bounded from above by $\mathcal{O}(t^{s-1-δ})$ for some $δ>0$, improving the bound $R(s,t)\in \mathcal{O}(t^{s-1}/(\log t)^{s-1})$ known for general graphs and the bounds known for $k$-stable graphs when $s\leq k$. - For every monadically stable class $\mathcal{C}$ and every integer $r$, there exists $δ> 0$ such that every graph $G \in \mathcal{C}$ that contains an $r$-subdivision of the biclique $K_{t,t}$ as a subgraph also contains $K_{t^δ,t^δ}$ as a subgraph. This generalizes earlier results for nowhere dense graph classes. - We obtain a stronger regularity lemma for monadically stable classes of graphs. - Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes.

cs.DM↗

arXiv2022

Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes

Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth. Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class $\mathcal{C}$ of graphs is flip-flat if for every fixed radius $r$, every sufficiently large set of vertices of a graph $G \in \mathcal{C}$ contains a large subset of vertices with mutual distance larger than $r$, where the distance is measured in some graph $G'$ that can be obtained from $G$ by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms.

cs.LO↗

arXiv2022

Model Checking on Interpretations of Classes of Bounded Local Cliquewidth

We present a fixed-parameter tractable algorithm for first-order model checking on interpretations of graph classes with bounded local cliquewidth. Notably, this includes interpretations of planar graphs, and more generally, of classes of bounded genus. To obtain this result we develop a new tool which works in a very general setting of dependent classes and which we believe can be an important ingredient in achieving similar results in the future.

cs.DS↗

arXiv2022

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.

cs.DS↗

arXiv2022

Treelike decompositions for transductions of sparse graphs

We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs -- more precisely, from classes of bounded expansion and from nowhere dense classes. In both cases, the decomposition takes the form of a single colored rooted tree of bounded depth where, in addition, there can be links between nodes that are not related in the tree. The constraint is that the structure formed by the tree and the links has to be sparse. Using the decomposition theorem for transductions of nowhere dense classes, we show that they admit low-shrubdepth covers of size $O(n^\varepsilon)$, where $n$ is the vertex count and $\varepsilon>0$ is any fixed~real. This solves an open problem posed by Gajarský et al. (ACM TOCL '20) and also by Briański et al. (SIDMA '21).

cs.LO↗