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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., Comment: v2: added section "Conclusions and Future Work"

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.

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.

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., Comment: Extended version of LICS 2023 conference paper

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.

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 $\mathcal{O}(t^{s-1-\delta})$ for some $\delta>0$, 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 $\delta > 0$ such that every graph $G \in \mathcal{C}$ that contains an $r$-subdivision of the biclique $K_{t,t}$ as a subgraph also contains $K_{t^\delta,t^\delta}$ 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.

An indiscernible sequence $(\bar a_i)_{1\leq i\leq n}$ in a structure is an ordered sequence of tuples of elements which is very homogeneous in the sense that any two finite subsequences of the same length satisfy the same first-order formulas. We provide new characterizations of monadically stable and monadically NIP classes of structures in terms of indiscernible sequences by showing that they impose a strong structure on their neighborhoods. In particular, we show that every formula~$\phi(x,\bar y)$, where $x$ is a single free variable, has alternation rank at most $2$ over every sufficiently long indiscernible sequence in a monadically NIP class. We provide a second new characterization of monadically stable classes of graphs in terms of a new notion called flip-wideness. Flip-wideness 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. All our proofs are constructive and yield efficient algorithms.

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 Kant\'e, 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.

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., Comment: 28 pages, 5 figures

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 M\"{a}hlmann, 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 M\"{a}hlmann 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.

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 Gajarsk\'y et al. (ACM TOCL '20) and also by Bria\'nski et al. (SIDMA '21)., Comment: 39 pages, 2 figures

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts. In this paper, we show that one can encode graphs from a class with structurally bounded expansion via lacon-, shrub- and parity-decompositions from a class with bounded expansion. These decompositions are useful for lifting properties from sparse to structurally sparse graph classes.

In this paper, we prove that a graph $G$ with no $K_{s,s}$-subgraph and twin-width $d$ has $r$-admissibility and $r$-coloring numbers bounded from above by an exponential function of $r$ and that we can construct graphs achieving such a dependency in $r$.

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 non-atomic 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.

Complex networks are everywhere. They appear for example in the form of biological networks, social networks, or computer networks and have been studied extensively. Efficient algorithms to solve problems on complex networks play a central role in today's society. Algorithmic meta-theorems show that many problems can be solved efficiently. Since logic is a powerful tool to model problems, it has been used to obtain very general meta-theorems. In this work, we consider all problems definable in first-order logic and analyze which properties of complex networks allow them to be solved efficiently. The mathematical tool to describe complex networks are random graph models. We define a property of random graph models called $\alpha$-power-law-boundedness. Roughly speaking, a random graph is $\alpha$-power-law-bounded if it does not admit strong clustering and its degree sequence is bounded by a power-law distribution with exponent at least $\alpha$ (i.e. the fraction of vertices with degree $k$ is roughly $O(k^{-\alpha})$). We solve the first-order model-checking problem (parameterized by the length of the formula) in almost linear FPT time on random graph models satisfying this property with $\alpha \ge 3$. This means in particular that one can solve every problem expressible in first-order logic in almost linear expected time on these random graph models. This includes for example preferential attachment graphs, Chung-Lu graphs, configuration graphs, and sparse Erd\H{o}s-R\'{e}nyi graphs. Our results match known hardness results and generalize previous tractability results on this topic.

We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard. Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case EPC where the paths have to cover every edge in $G$ exactly once is already NP-hard for two paths on 4-regular graphs.

Preferential attachment graphs are random graphs designed to mimic properties of typical real world networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We use improved concentration bounds for vertex degrees to show that preferential attachment graphs contain asymptotically almost surely (a.a.s.) a one-subdivided clique of size at least $(\log n)^{1/4}$. Therefore, preferential attachment graphs are a.a.s somewhere-dense. This implies that algorithmic techniques developed for sparse graphs are not directly applicable to them. The concentration bounds state: Assuming that the exact degree $d$ of a fixed vertex (or set of vertices) at some early time $t$ of the random process is known, the probability distribution of $d$ is sharply concentrated as the random process evolves if and only if $d$ is large at time $t$.

We analyze some local properties of sparse Erdos-Renyi graphs, where $d(n)/n$ is the edge probability. In particular we study the behavior of very short paths. For $d(n)=n^{o(1)}$ we show that $G(n,d(n)/n)$ has asymptotically almost surely (a.a.s.~) bounded local treewidth and therefore is a.a.s.~nowhere dense. We also discover a new and simpler proof that $G(n,d/n)$ has a.a.s.~bounded expansion for constant~$d$. The local structure of sparse Erdos-Renyi Gaphs is very special: The $r$-neighborhood of a vertex is a tree with some additional edges, where the probability that there are $m$ additional edges decreases with~$m$. This implies efficient algorithms for subgraph isomorphism, in particular for finding subgraphs with small diameter. Finally we note that experiments suggest that preferential attachment graphs might have similar properties after deleting a small number of vertices.

Complex networks can be typically broken down into groups or modules. Discovering this "community structure" is an important step in studying the large-scale structure of networks. Many algorithms have been proposed for community detection and benchmarks have been created to evaluate their performance. Typically algorithms for community detection either partition the graph (non-overlapping communities) or find node covers (overlapping communities). In this paper, we propose a particularly simple semi-supervised learning algorithm for finding out communities. In essence, given the community information of a small number of "seed nodes", the method uses random walks from the seed nodes to uncover the community information of the whole network. The algorithm runs in time $O(k \cdot m \cdot \log n)$, where $m$ is the number of edges; $n$ the number of links; and $k$ the number of communities in the network. In sparse networks with $m = O(n)$ and a constant number of communities, this running time is almost linear in the size of the network. Another important feature of our algorithm is that it can be used for either non-overlapping or overlapping communities. We test our algorithm using the LFR benchmark created by Lancichinetti, Fortunato, and Radicchi specifically for the purpose of evaluating such algorithms. Our algorithm can compete with the best of algorithms for both non-overlapping and overlapping communities as found in the comprehensive study of Lancichinetti and Fortunato.

Communities are subsets of a network that are densely connected inside and share only few connections to the rest of the network. The aim of this research is the development and evaluation of an efficient algorithm for detection of overlapping, fuzzy communities. The algorithm gets as input some members of each community that we aim to discover. We call these members seed nodes. The algorithm then propagates this information by using random walks that start at non-seed nodes and end as they reach a seed node. The probability that a random walk starting at a non-seed node $v$ ends at a seed node $s$ is then equated with the probability that $v$ belongs to the communities of $s$. The algorithm runs in time $\tilde{O}(l \cdot m \cdot \log n)$, where $l$ is the number of communities to detect, $m$ is the number of edges, $n$ is the number of nodes. The $\tilde{O}$-notation hides a factor of at most $(\log \log n)^2$. The LFR benchmark proposed by Lancichinetti et al.\ is used to evaluate the performance of the algorithm. We found that, given a good set of seed nodes, it is able to reconstruct the communities of a network in a meaningful manner.