Efficiently Finding and Counting Patterns with Distance Constraints in Sparse Graphs

Graph classes of bounded expansion were introduced by NešetÅ™il and de Mendez as a general model of structurally sparse graphs, which have received considerable attention from both combinatorial and algorithmic perspectives. A celebrated result of DvoÅ™ák et al. [JACM'13] showed that any first-order model checking problem on bounded-expansion graph classes is fixed-parameter tractable. A main drawback of the FPT algorithms resulted from this result is the high dependency of their time complexity on the parameter k: the algorithms run in time at least doubly exponential in k, even when the graph class is of polynomial expansion. It is natural to ask whether there exist FPT algorithms for these problem that run in singly exponential time, i.e., 2^kO(1) n^O(1) time. In this paper, we give a new algorithmic framework for a broad family of first-order model checking problems on sparse graphs, which results in algorithms with running time 2^kO(1) · n when the graph class is of exponential expansion (i.e., the expansion is bounded by a singly exponential function). This covers most well-studied instances of bounded-expansion graph classes, in particular, all polynomial-expansion graph classes. Our framework applies to all problems that can be formulated as finding k vertices in a host graph G with certain distance constraints. Furthermore, the framework can be generalized to give (1 ± ϵ)-approximation algorithms for the counting versions of these problems with running time 2^kO(1) · n (logn/ϵ)^O(1) on exponential-expansion graph classes. In terms of techniques, our framework differs entirely from the one of DvoÅ™ák et al. based on centered coloring. We develop various technical components based on the theory of sparse graphs and other tools such as representative sets/functions, tree decomposition, inclusion-exclusion, etc., which are of independent interest. Remarkably, some of our techniques can be applied to even more general graph classes, such as degenerate graph classes. Therefore, as a byproduct, we obtain a (1 ± ϵ)-approximation algorithm for approximately counting bounded-treewidth induced subgraphs in degenerate graphs with running time k^O(k) · (n/ϵ)^O(1). This resolves (in a much stronger form) an open problem of Bressan and Roth [FOCS'22], which asked whether such an algorithm exists for counting induced k-matching in degenerate graphs.