Parameterized Analysis of the Online Priority and Node-Weighted Steiner Tree Problems

In this paper we study the online variant of two well-known Steiner tree problems. In the online setting, the input consists of a sequence of terminals; upon arrival of a terminal, the online algorithm must irrevocably buy a subset of edges and vertices of the graph so as to guarantee the connectivity of the currently revealed part of the input. More precisely, we first study the online node-weighted Steiner tree problem, in which both edges and vertices are weighted, and the objective is to minimize the total cost of edges and vertices in the solution. We then address the online priority Steiner tree problem, in which each edge and each request are associated with a priority value, which corresponds to their bandwidth support and requirement, respectively. Both problems have applications in the domain of multicast network communications and have been studied from the point of view of approximation algorithms. Motivated by the observation that competitive analysis gives very pessimistic and unsatisfactory results when the only relevant parameter is the number of terminals, we introduce an approach based on parameterized analysis of online algorithms. In particular, we base the analysis on additional parameters that help reveal the true complexity of the underlying problem, and allow a much finer classification of online algorithms based on their performance. More specifically, for the online node-weighted Steiner tree problem, we show a tight bound of Θ(max{min{α,k},log k}) on the competitive ratio, where α is the ratio of the maximum node weight to the minimum node weight and k is the number of terminals. For the online priority Steiner tree problem, we show corresponding tight bounds of Θ(blogkb)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Theta }(b\log \frac {k}{b})$\end{document}, when k > b and Θ(k), when k ≤ b, where b is the number of priority levels and k is the number of terminals. Our main results apply to both deterministic and randomized algorithms, as well as to generalized versions of the problems (i.e., to Steiner forest variants).