This paper studies an extension of the k-median problem under uncertain demand. We are given an n-vertex metric space V d and m client sets. The goal is to open a set of k facilities F such that the worst-case connection cost over all the client sets is minimized, i.e., minThis is a "min-max" or "robust" version of the k-median problem. Note that in contrast to the recent papers on robust and stochastic problems, we have only one stage of decision-making where we select a set of k facilities to open. Once a set of open facilities is fixed, each client in the uncertain client-set connects to the closest open facility. We present a simple, combinatorial O log n + log m -approximation algorithm for the robust k-median problem that is based on reweighting/Lagrangean-relaxation ideas. In fact, we give a general framework for (minimization) k-facility location problems where there is a bound on the number of open facilities. We show that if the location problem satisfies a certain "projection" property, then both the robust and stochastic versions of the location problem admit approximation algorithms with logarithmic ratios. We use our framework to give the first approximation algorithms for robust and stochastic versions of several location problems such as k-tree, capacitated k-median, and fault-tolerant k-median.

Abstract. Consider the following Steiner Tree leasing problem. Given a graph G = (V, E) with root r, and a sequence of terminal sets Dt ⊆ V for each day t ∈ [T ]. A feasible solution to the problem is a set of edges Et for each t connecting Dt to r. Instead of obtaining edges for a single day at a time, or for infinitely long (both of which give Steiner tree problems), we lease edges for say, { a day, a week, a month, a year }. Naturally, leasing an edge for a longer period costs less per unit of time. What is a good leasing strategy? In this paper, we give a general approach to solving a wide class of such problems by showing a close connection between deterministic leasing problems and problems in multistage stochastic optimization. All our results are in the offline setting.

The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete r-partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour, Akbari and Ebrahimi [2].

We consider the online bottleneck matching problem, where k serververtices lie in a metric space and k request-vertices that arrive over time each must immediately be permanently assigned to a server-vertex. The goal is to minimize the maximum distance between any request and its server. Because no algorithm can have a competitive ratio better than O(k) for this problem, we use resource augmentation analysis to examine the performance of three algorithms: the naive GREEDY algorithm, PERMUTATION, and BALANCE. We show that while the competitive ratio of GREEDY improves from exponential (when each server-vertex has one server) to linear (when each server-vertex has two servers), the competitive ratio of PERMU-TATION remains linear when an extra server is introduced at each server-vertex. The competitive ratio of BALANCE is also linear with an extra server at each server-vertex, even though it has been shown that an extra server makes it constant-competitive for the min-weight matching problem.

Web-based Doodle polls, where respondents indicate their availability for a collection of times provided by the poll initiator, are an increasingly common way of selecting a time for an event or meeting. Yet group dynamics can markedly influence an individual's response, and thus the overall solution quality. Via theoretical worst-case analysis, we analyze certain common behaviors of Doodle poll respondents, including when participants are either more generous with or more protective of their time, showing that deviating from one's "true availability" can have a substantial impact on the overall quality of the selected time. We show perhaps counter-intuitively that being more generous with your time can lead to inferior time slots being selected, and being more protective of your time can lead to superior time slots being selected. We also bound the improvement and degradation of outcome quality under both types of behaviors.