Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the writeup on Fixed-Threshold algorithm

Online Knapsack Problem under Expected Capacity Constraint

Online knapsack problem is considered, where items arrive in a sequential fashion that have two attributes; value and weight. Each arriving item has to be accepted or rejected on its arrival irrevocably. The objective is to maximize the sum of the value of the accepted items such that the sum of their weights is below a budget/capacity. Conventionally a hard budget/capacity constraint is considered, for which variety of results are available. In modern applications, e.g., in wireless networks, data centres, cloud computing, etc., enforcing the capacity constraint in expectation is sufficient. With this motivation, we consider the knapsack problem with an expected capacity constraint. For the special case of knapsack problem, called the secretary problem, where the weight of each item is unity, we propose an algorithm whose probability of selecting any one of the optimal items is equal to $1-1/e$ and provide a matching lower bound. For the general knapsack problem, we propose an algorithm whose competitive ratio is shown to be $1/4e$ that is significantly better than the best known competitive ratio of $1/10e$ for the knapsack problem with the hard capacity constraint.Comment: To appear in IEEE INFOCOM 2018, April 2018, Honolulu H