On distance r-dominating and 2r-independent sets in sparse graphs

Dvořák [5] gave a bound on the minimum size of a distance r dominating set in the terms of the maximum size of a distance 2r independent set and generalized coloring numbers, thus obtaining a constant factor approximation algorithm for the parameters in any class of graphs with bounded expansion. We improve and clarify this dependence using an LP-based argument inspired by the work of Bansal and Umboh [3]. A set X of vertices of a graph G is dominating if each vertex of G either belongs to or has a neighbor in X, and it is independent if no two vertices of X are adjacent. The domination number γ(G) of G is the minimum size of a dominating set inG, and the independence number α(G) ofG is the maximum size of an independent set in G. Determining either of these parameters in a general graph is NP-complete [9]. Even approximating them is hard. No polynomialtime algorithm approximating the domination number of an n-vertex graph within a factor better than O(log n) exists [16], unless P = NP. Even worse, for every ε > 0, no polynomial-time algorithm approximating the independence number of an n-vertex graph within a factor better than O(n) exists [8], unless ZPP = NP. Both parameters become more tractable in sparse graphs—they have a PTAS in planar graphs [2] and other related graph classes, most generally in all graph classes with strongly sublinear separators [7]. To obtain constant-factor approximation, much weaker constraints suffice. Lenzen and Wattenhofer [12] proved that the domination number can be approximated within factor a + 3a + 1 on Charles University, Prague, Czech Republic. E-mail: [email protected]. Supported by project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foundation.