Online Dominating Set and Independent Set

TL;DR

This paper introduces the independent kissing number to analyze online dominating and independent set problems, establishing optimal competitive ratios for greedy algorithms across various graph classes and geometric objects.

Contribution

It defines the independent kissing number and proves its role in achieving optimal competitive ratios for online dominating and independent set algorithms.

Findings

Greedy algorithm achieves optimal ratio ζ for dominating set.

Greedy algorithm achieves optimal ratio ζ for maximum independent set.

Bounds are provided for specific geometric object families.

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.