Polynomial Kernels and Wideness Properties of Nowhere Dense Graph Classes

Abstract

Nowhere dense classes of graphs [21,22] are very general classes of uniformly sparse graphs with several seemingly unrelated characterisations. From an algorithmic perspective, a characterisation of these classes in terms of uniform quasi-wideness, a concept originating in finite model theory, has proved to be particularly useful. Uniform quasi-wideness is used in many fpt-algorithms on nowhere dense classes. However, the existing constructions showing the equivalence of nowhere denseness and uniform quasi-wideness imply a non-elementary blow up in the parameter dependence of the fpt-algorithms, making them infeasible in practice.As a first main result of this paper, we use tools from logic, in particular from a sub-field of model theory known as stability theory, to establish polynomial bounds for the equivalence of nowhere denseness and uniform quasi-wideness.As an algorithmic application of our new methods, we obtain for every fixed value of r ∈ N a polynomial kernel for the distance-r dominating set problem on nowhere dense classes of graphs. This is particularly interesting, as it implies that for every subgraph-closed class C, the distance-r dominating set problem admits a kernel on C for every value of r if, and only if, it admits a polynomial kernel for every value of r (under the standard assumption of parameterized complexity theory that FPT ≠ W[2]).Finally, we demonstrate how to use the new methods to improve the parameter dependence of many fixedparameter algorithms. As an example we provide a single exponential parameterized algorithm for the Connected Dominating Set problem on nowhere dense graph

Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

“…Uniform quasi-wideness was introduced by Dawar in [19] and it was proved by Nešetřil and Ossona de Mendez in [60] that uniform quasi-wideness is equivalent to nowhere denseness. Very recently, it was shown that the function N in the above definition can be chosen to be polynomial in m [45,70]. A single exponential dependency was earlier established for classes of bounded expansion [44].…”

Parameterized circuit complexity of model-checking on sparse structures

“…For instance, consider the problem of computing the smallest distance-r dominating set in a given graph G, which is a subset of vertices D such that every vertex of G is at distance at most r from some vertex of D. This problem often serves as a benchmark for sparsity methods, and it was considered in the theory of sparse graphs from the points of view of parameterized algorithms [12], approximation [1,18], and kernelization [14,19,29]. In particular, Amiri et al [1] have recently used the results of Nešetřil and Ossona de Mendez [35] to give a distributed logarithmic-time constant-factor approximation algorithm for the distance-r dominating set problem on any class of bounded expansion.…”

Reconfiguration on Nowhere Dense Graph Classes

“…For us it will be important that the function N can be assumed to be polynomial in m (the degree of the polynomial may depend on r) and that the sets B and S can be efficiently computed. Polynomial bounds were first obtainend by Kreutzer et al [28], we refer to the improved bounds of Pilipczuk et al [39].…”

“…It was then shown that nowhere dense classes are the limit of tractability based on sparsity methods, more precisely, it was shown in [13] that if a class C is not nowhere dense and closed under taking subgraphs, then there is some r 1 such that Distance-r Dominating Set on C is W[2]-hard. It was later shown that the problem admits a polynomial kernel [28] and in fact an almost linear kernel [14] on nowhere dense classes.…”