Jan Dreier

STOC Conference 2025 Conference Paper

Merge-Width and First-Order Model Checking

• Jan Dreier
• Szymon Torunczyk

FOCS Conference 2024 Conference Paper

First-Order Model Checking on Monadically Stable Graph Classes

• Jan Dreier
• Ioannis Eleftheriadis
• Nikolas Mählmann
• Rose McCarty
• Michal Pilipczuk
• Szymon Torunczyk

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, Siebertz; J. ACM '17] for nowhere dense classes and of [Dreier, Mählmann, Siebertz; STOC '23] for structurally nowhere dense classes to all monadically stable classes. This result is complemented by a hardness result showing that monadic stability is precisely the dividing line between tractability and intractability of first-order model checking on hereditary classes that are edge-stable: exclude some half-graph as a semi-induced subgraph. Precisely, we prove that for every hereditary graph class $\mathscr{C}$ that is edge-stable but not monadically stable, first-order model checking is $\text{AW}[*]$ -hard on $\mathscr{C}$, and W[1]-hard when restricted to existential sentences. This confirms, in the special case of edge-stable classes, an open conjecture that the notion of monadic dependence 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, by combining tools from stability theory and from sparsity theory. We then use this result to construct sparse neighborhood covers for monadically stable graph classes, which provides the missing ingredient for the algorithm of [Dreier, Mählmann, 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 first 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 efficiently interpret the class of all graphs using only existential formulas; this implies W[1]-hardness of model checking already for existential formulas.

STOC Conference 2024 Conference Paper

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

• Jan Dreier
• Nikolas Mählmann
• Szymon Torunczyk

SODA Conference 2023 Conference Paper

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

• Benjamin Bergougnoux
• Jan Dreier
• Lars Jaffke

STOC Conference 2023 Conference Paper

First-Order Model Checking on Structurally Sparse Graph Classes

• Jan Dreier
• Nikolas Mählmann
• Sebastian Siebertz

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.

SODA Conference 2021 Conference Paper

Approximate Evaluation of First-Order Counting Queries

• Jan Dreier
• Peter Rossmanith

Kuske and Schweikardt introduced the very expressive first-order counting logic FOC( P ) to model database queries with counting operations. They showed that there is an efficient model-checking algorithm on graphs with bounded degree, while Grohe and Schweikardt showed that probably no such algorithm exists for trees of bounded depth. We analyze the fragment FO({>0}) of this logic. While we remove for example subtraction and comparison between two nonatomic counting terms, this logic remains quite expressive: We allow nested counting and comparison between counting terms and arbitrarily large numbers. Our main result is an approximation scheme of the model-checking problem for FO({>0}) that runs in linear fpt time on structures with bounded expansion. This scheme either gives the correct answer or says “I do not know. ” The latter answer may only be given if small perturbations in the number-symbols of the formula could make it both satisfied and unsatisfied. This is complemented by showing that exactly solving the model-checking problem for FO({>0}) is already hard on trees of bounded depth and just slightly increasing the expressiveness of FO({>0}) makes even approximation hard on trees.