Online Independent Set Beyond the Worst-Case: Secretaries, Prophets, and Periods

We investigate online algorithms for maximum (weight) independent set on graph classes with bounded inductive independence number like, e.g., interval and disk graphs with applications to, e.g., task scheduling and spectrum allocation. In the online setting, it is assumed that nodes of an unknown graph arrive one by one over time. An online algorithm has to decide whether an arriving node should be included into the independent set. Unfortunately, this natural and practically relevant online problem cannot be studied in a meaningful way within a classical competitive analysis as the competitive ratio on worst-case input sequences is lower bounded by Ω(n). This devastating lower bound holds even for randomized algorithms on unweighted interval graphs and, hence, for one of the most restricted graph class under consideration. As a worst-case analysis is pointless, we study online independent set in a stochastic analysis. Instead of focussing on a particular stochastic input model, we present a generic sampling approach that enables us to devise online algorithms achieving performance guarantees for a variety of input models. In particular, our analysis covers stochastic input models like the secretary model, in which an adversarial graph is presented in random order, and the prophet-inequality model, in which a randomly generated graph is presented in adversarial order. Our sampling approach bridges thus between stochastic input models of quite different nature. In addition, we show that our approach can be applied to a practically motivated admission control setting in which the algorithm uses the input from a preceding period as sample graph for the current period. Our sampling approach yields an online algorithm for maximum independent set on interval and disk graphs with competitive ratio O(1) with respect to all of the mentioned stochastic input models. More generally, for graph classes with inductive independence number ρ, the competitive ratio is O(ρ). The approach can be extended towards maximum-weight independent set by losing only a factor of O(log n) in the competitive ratio with n denoting the (expected) number of nodes. This upper bound is complemented by a lower bound of Ω(log n/ log logn) showing that our sampling approach achieves nearly the optimal competitive ratio in all of the considered models. Furthermore, we generalize our analysis to address several practically motivated extensions of the independent set problem, e.g., arrival and departure times of nodes or edge-weighted graphs capturing SINR-type interference conflicts in wireless networks. Dept. of Computer Science, RWTH Aachen University, Germany. {goebel,voecking}@cs.rwth-aachen.de. Supported by DFG Research Training Group AlgoSyn at RWTH Aachen University. Max-Planck-Institut für Informatik and Saarland University, Saarbrücken, Germany. [email protected]. Supported by DFG Cluster of Excellence M2CI at Saarland University and in part by DFG grant Ho 3831/3-1. Dept. of Computer Science, Cornell University University, Ithaca, NY, USA. [email protected]. Supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD) and by DFG through UMIC Research Center at RWTH Aachen University.