Online Dominating Set and Independent Set

Abstract: Finding minimum dominating set and maximum independent set for graphs in the classical online setup are notorious due to their disastrous $\Omega(n)$ lower bound of the competitive ratio that even holds for interval graphs, where $n$ is the number of vertices. In this paper, inspired by Newton number, first, we introduce the independent kissing number $\zeta$ of a graph. We prove that the well known online greedy algorithm for dominating set achieves optimal competitive ratio $\zeta$ for any graph. We show that the same greedy algorithm achieves optimal competitive ratio $\zeta$ for online maximum independent set of a class of graphs with independent kissing number $\zeta$. For minimum connected dominating set problem, we prove that online greedy algorithm achieves an asymptotic competitive ratio of $2(\zeta-1)$, whereas for a family of translated convex objects the lower bound is $\frac{2\zeta-1}{3}$. Finally, we study the value of $\zeta$ for some specific families of geometric objects: fixed and arbitrary oriented unit hyper-cubes in $I!!R^d$, congruent balls in $I!!R^3$, fixed oriented unit triangles, fixed and arbitrary oriented regular polygons in $I!!R^2$. For each of these families, we also present lower bounds of the minimum connected dominating set problem.