Resource allocation and subset selection: new approaches at the interface between discrete and continuous optimization

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Torrico Palacios, Alfredo Ignacio
Singh, Mohit
Pokutta, Sebastian
Toriello, Alejandro
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Resource allocation and subset selection are two relevant classes of problems in the core of combinatorial optimization. Over the past decade, there has been an increased interest in these areas due to their significant impact in real-world applications. Online advertising, sharing-economy systems, and kidney exchange are a few examples of the applicability of resource allocation models. On the other hand, data summarization, influence modeling in social networks, and sensor location have extensively motivated the study of subset selection. In this thesis, we propose new insights to two classical problems of these areas: the online bipartite matching problem and the constrained submodular function maximization problem. In the first part of this thesis, we consider novel polyhedral approaches for the i.i.d. online variant of the classical bipartite matching problem defined as follows: one side of the bipartition is fixed and known in advance, while nodes from the other side appear one at a time as i.i.d. realizations of a uniform distribution and must immediately be matched or discarded. First, we obtain various static relaxations of the polyhedral set of achievable probabilities, introduce valid inequalities, and discuss when they are facet-defining. We also show how several of these relaxations correspond to ranking policies and their time-dependent generalizations. Later, we focus on dynamic polyhedral relaxations. We show how they theoretically dominate the static relaxations from the previous part and perform a polyhedral study to theoretically examine the strength of the new relaxations. We also discuss how to derive heuristic policies from the dual prices of the relaxations, in a similar fashion to dynamic resource prices used in network revenue management. In both approaches, static and dynamic, we demonstrate the empirical quality of these relaxations and policies via computational experiments. In the second part of this thesis, we focus on new approaches to the classical constrained submodular maximization problem. In the first section, we consider the robust variant of this problem, in which we optimize several submodular functions simultaneously subject to a wide class of combinatorial constraints. We consider both the offline setting where the data defining the problem is known in advance, as well as, the online setting where the input data is revealed over time. For both models we provide efficient bi-criteria algorithms with provable guarantees. In the second section, we explore the concept of sharpness in submodular function maximization. Empirical studies have shown that the performance of the greedy algorithm is substantially better in practice, even though its worst-case guarantee. Our goal is to define sharpness for submodular functions as a candidate explanation for this phenomenon. We show that the greedy algorithm provably performs better as the sharpness of the submodular function increases. Lastly, in both sections we present an exhaustive computational study to support our theoretical results.
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