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This paper addresses the competitiveness of online algorithms for two Steiner Tree problems. In the first problem, the underlying graph is directed and has bounded asymmetry, namely the maximum weight of antiparallel links in the graph does not exceed a parameter α. Previous work on this problem has left a gap on the competitive ratio which is as large as logarithmic in k. We present a refined analysis, both in terms of the upper and the lower bounds, that closes the gap and shows that a greedy algorithm is optimal for all values of the parameter α. The second part of the paper addresses the Euclidean Steiner tree problem on the plane. Alon and Azar [SoCG 1992, Disc. Comp. Geom. 1993] gave an elegant lower bound on the competitive ratio of any deterministic algorithm equal to Ώ(logk/ loglogk); however, the best known upper bound is the trivial bound O(logk). We give the first analysis that makes progress towards closing this long-standing gap. In particular, we present an online algorithm with competitive ratio O(logk/ loglogk), provided that the optimal offline Steiner tree belongs in a class of trees with relatively simple structure. This class comprises not only the adversarial instances of Alon and Azar, but also all rectilinear Steiner trees which can be decomposed in a polylogarithmic number of rectilinear full Steiner trees. [ABSTRACT FROM AUTHOR]

The edge asymmetry of a directed, edge-weighted graph is defined as the maximum ratio of the weight of antiparallel edges in the graph, and can be used as a measure of the heterogeneity of links in a data communication network. In this paper we provide a near-tight upper bound on the competitive ratio of the Online Steiner Tree problem in graphs of bounded edge asymmetry α. This problem has applications in efficient multicasting over networks with non-symmetric links. We show an improved upper bound of ]> on the competitive ratio of a simple greedy algorithm, for any request sequence of k terminals. The result almost matches the lower bound of ]> (where ε is an arbitrarily small constant) due to Faloutsos et al. [8] and Angelopoulos [2]. [ABSTRACT FROM AUTHOR]

It is known that in practice, request sequences for the list update problem exhibit a certain degree of locality of reference. Motivated by this observation we apply the locality of reference model for the paging problem due to Albers et al. [STOC 2002/JCSS 2005] in conjunction with bijective analysis [SODA 2007] to list update. Using this framework, we prove that Move-to-Front (MTF) is the unique optimal algorithm for list update. This addresses the open question of defining an appropriate model for capturing locality of reference in the context of list update [Hester and Hirschberg ACM Comp. Surv. 1985]. Our results hold both for the standard cost function of Sleator and Tarjan [CACM 1985] and the improved cost function proposed independently by Martínez and Roura [TCS 2000] and Munro [ESA 2000]. This result resolves an open problem of Martínez and Roura, namely proposing a measure which can successfully separate MTF from all other list-update algorithms. [ABSTRACT FROM AUTHOR]

In this paper we study a variant of the Node-Weighted Steiner Tree problem in which the weights (costs) of vertices are restricted, in the sense that the ratio of the maximum node weight to the minimum node weight is bounded by a quantity α. This problem has applications in multicast routing where the cost of participating routers must be taken into consideration and the network is relatively homogenous in terms of the cost of the routers. We consider both online and offline versions of the problem. For the offline version we show an upper bound of O( min {logα, logk}) on the approximation ratio of deterministic algorithms (where k is the number of terminals). We also prove that the bound is tight unless P=NP. For the online version we show a tight bound of Θ( max { min {α, k}, logk }), which applies to both deterministic and randomized algorithms. We also show how to apply (and extend to node-weighted graphs) recent work of Alon et al. so as to obtain a randomized online algorithm with competitive ratio O(logm logk), where m is the number of the edges in the graph, independently of the value of α. All our bounds also hold for the Generalized Node-Weighted Steiner Problem, in which only connectivity between pairs of vertices must be guaranteed. [ABSTRACT FROM AUTHOR]

We consider a variation of the classical problem of finding prices which guarantee equilibrium in linear markets consisting of divisible goods and agents with money. Specifically, we consider on-line algorithms for this problem in which goods are considered on-line, and each good is assigned an irrevocable price. Since exact equilibria will not be found in such a setting, we appeal to the concept of approximate equilibrium defined in previous studies of the problem, to characterize the quality of our solutions. We consider both deterministic and randomized algorithms for finding approximate equilibria. We prove a tight bound on the performance of deterministic algorithms, and show that under certain natural conditions, randomized algorithms lead to market prices which are closer to equilibrium. [ABSTRACT FROM AUTHOR]

Borodin, Nielsen and Rackoff [5] proposed a framework for abstracting the main properties of greedy-like algorithms with emphasis on scheduling problems, and Davis and Impagliazzo [6] extended it so as to make it applicable to graph optimization problems. In this paper we propose a related model which places certain reasonable restrictions on the power of the greedy-like algorithm. Our goal is to define a model in which it is possible to filter out certain overly powerful algorithms, while still capturing a very rich class of greedy-like algorithms. We argue that this approach better motivates the lower-bound proofs and possibly yields better bounds. To illustrate the techniques involved we apply the model to the well-known problems of (complete) facility location and dominating set. Keywords: Priority algorithms, inapproximability results, facility location, dominating set. [ABSTRACT FROM AUTHOR]