Optimal Multiple Stopping Rule for WarmStarting Sequential Selection. In this paper we present the Warmstarting Dynamic Thresholding algorithm, developed using dynamic programming, for a variant of the standard online selection problem. The problem allows job positions to be either free or already occupied at the beginning of the process. Throughout the selection process, the decision maker interviews one after the other the new candidates and reveals a quality score for each of them. Based on that information, she can (re)assign each job at most once by taking immediate and irrevocable decisions. We relax the hard requirement of the class of dynamic programming algorithms to perfectly know the distribution from which the scores of candidates are drawn, by presenting extensions for the partial and noinformation cases, in which the decision maker can learn the underlying score distribution sequentially while interviewing candidates.

Online Multistage Subset Maximization Problems. Numerous combinatorial optimization problems (knapsack, maximumweight matching, etc.) can be expressed as emphsubset maximization problems: One is given a ground set N1,dots,n, a collection mathcalFsubseteq 2N of subsets thereof such that emptysetinmathcalF, and an objective (profit) function p:mathcalFrightarrowmathbbR_. The task is to choose a set SinmathcalF that maximizes p(S). We consider the emphmultistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p_t (and possibly the set of feasible solutions mathcalF_t) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the tradeoff between good pertime solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the emphonline setting, that is, p_t (along with possibly mathcalF_t) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constantcompetitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one.