Related papers: Algorithms and data structures for first-order log…
First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem, parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question of…
The problem of designing connectivity oracles supporting vertex failures is one of the basic data structures problems for undirected graphs. It is already well understood: previous works [Duan--Pettie STOC'10; Long--Saranurak FOCS'22]…
Algorithmic meta-theorems explain the tractability of large classes of computational problems by linking logical expressibility with structural graph properties. While extensions of first-order logic such as FO+dp admit efficient model…
We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of $d\leq d_{\star}$ failed vertices in $\tilde{O}(d^3)$ time and thereafter…
We study first-order model checking, by which we refer to the problem of deciding whether or not a given first-order sentence is satisfied by a given finite structure. In particular, we aim to understand on which sets of sentences this…
In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order…
Tractability results for the model checking problem of logics yield powerful algorithmic meta theorems of the form: Every computational problem expressible in a logic $L$ can be solved efficiently on every class $\mathscr{C}$ of structures…
We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of…
Algorithmic meta-theorems provide an important tool for showing tractability of graph problems on graph classes defined by structural restrictions. While such results are well established for static graphs, corresponding frameworks for…
We revisit the vertex-failure connectivity oracle problem. This is one of the most basic graph data structure problems under vertex updates, yet its complexity is still not well-understood. We essentially settle the complexity of this…
In this paper we present an efficient reachability oracle under single-edge or single-vertex failures for planar directed graphs. Specifically, we show that a planar digraph $G$ can be preprocessed in $O(n\log^2{n}/\log\log{n})$ time,…
We revisit once more the problem of designing an oracle for answering connectivity queries in undirected graphs in the presence of vertex failures. Specifically, given an undirected graph $G$ with $n$ vertices and $m$ edges and an integer…
We show that the model-checking problem for successor-invariant first-order logic is fixed-parameter tractable on graphs with excluded topological subgraphs when parameterised by both the size of the input formula and the size of the…
Disjoint-paths logic, denoted $\mathsf{FO}$+$\mathsf{dp}$, extends first-order logic ($\mathsf{FO}$) with atomic predicates $\mathsf{dp}_r[(x_1,y_1),\ldots,(x_r,y_r)]$, expressing the existence of vertex-disjoint paths between $x_i$ and…
We study the problem of efficiently answering strong connectivity queries under two vertex failures. Given a directed graph $G$ with $n$ vertices, we provide a data structure with $O(nh)$ space and $O(h)$ query time, where $h$ is the height…
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…
For a graph $G$, the parameter treedepth measures the minimum depth among all forests $F$, called elimination forests, such that $G$ is a subgraph of the ancestor-descendant closure of $F$. We introduce a logic, called neighborhood operator…
Graph separation is a central tool in parameterized algorithm design, and important separators are among its most successful ingredients. They yield small, structured families of separators that can be enumerated efficiently, and underlie…
We give an improved connectivity oracle under vertex failures. After a set of $k$ vertices fails, our oracle performs an $O(k^{6})$-time update independent of the graph size $n$, and then answers pairwise connectivity queries in optimal…
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter…