Submodular secretary problem and extensions

Abstract

Online auction is an essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic secretary problem, is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the multiple-choice secretary problem were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the submodular secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log 2 r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of subadditive secretary problem, there is noõ( √ n)-competitive algorithm and thus submodular functions are the most general functions to consider for constant competitiveness in our setting. We complement this result by giving a matching O( √ n)-competitive algorithm for the subadditive case. At the end, we consider some special cases of our general setting as well. *

Streaming Algorithms for Submodular Function Maximization

“…In the secretary model, the elements arrive according to a random permutation of the ground set and an element added to S cannot be discarded later. In the secretary model, constant factor algorithms are known for the cardinality constraint and some special cases of a single matroid constraint [GRST10,BHZ13]. These algorithms assume the stream is randomly ordered and their performance degrades badly against adversarial streams; the best competitive ratio for a single general matroid is O(log k) (where k is the rank of the matroid).…”

Geometry of Online Packing Linear Programs

“…The random permutation model has grown in popularity [4,11,16] since it avoids strong lower bounds of the pessimistic adversarial-order model [8] while still capturing the lack of total information about the input. Different online problems have already been studied in this model, including bin-packing [19], matchings [16,18], the AdWords Problem [11] and different generalizations of the Secretary Problem [2,4,5,17,23]. Closest to our work are packing problems with a single knapsack constraint.…”

Combinatorial Prophet Inequalities

“…As noted earlier, this algorithm circumvents the impossibility result of Bateni et al [10] for efficient algorithms 2 .…”

“…For the secretary problem, there has also been significant work on optimizing more general, combinatorial objective functions. A line of great works [10,26,9,29] on secretary problem with submodular valuations culminated with a general reduction by Feldman and Zenklusen [29] from any submodular function to additive (linear) valuations with only O(1) loss. Going beyond submodular is an important problem [25], but for subadditive objective functions there is a daunting Ω( √ n) lower bound on the competitive ratio for restricted value queries [10].…”