Online prophet-inequality matching with applications to ad allocation

Abstract

We study the problem of online prophet-inequality matching in bipartite graphs. There is a static set of bidders and an online stream of items. We represent the interest of bidders in items by a weighted bipartite graph. Each bidder has a capacity, i.e., an upper bound on the number of items that can be allocated to her. The weight of a matching is the total weight of edges matched to the bidders. Upon the arrival of an item, the online algorithm should either allocate it to a bidder or discard it. The objective is to maximize the weight of the resulting matching. We consider this model in a stochastic setting where we know the distribution of the incoming items in advance. Furthermore, we allow the items to be drawn from different distributions, i.e., we may assume that the t th item is drawn from distribution Dt. In contrast to i.i.d. model, this allows us to model the change in the distribution of items throughout the time. We call this setting the Prophet-Inequality Matching because of the possibility of having a different distribution for each time. We generalize the classic prophet inequality by presenting an algorithm with the approximation ratio of 1 − 1 √ k+3 where k is the minimum capacity. In case of k = 2, the algorithm gives a tight ratio of 1 2 which is a different proof of the prophet inequality. We also consider a model in which the bidders do not have a capacity, instead each bidder has a budget. The weight of a matching is the minimum of the budget of each vertex and the total weight of edges matched to it, when summed over all bidders. We show that if the bid to the budget ratio of every bidder is at most 1 k then a natural randomized online algorithm has an approximation ratio of 1 − k k e k k! ≈ 1 − 1 √ 2πk compared to the optimal offline (in which the ratio goes to 1 as k becomes large).We also present the applications of our model in Adword Allocation, Display Ad Allocation, and AdCell Model.

The Online Stochastic Generalized Assignment Problem

“…Therefore our algorithm saves a logarithmic factor in the loss of revenue. Indeed these drawbacks were also the motivation for improving the loss factor in the special cases of online GAP in previous papers [2,3].…”

“…Recently, Alaei et al [3,2] 6 use a combination of expected linear programming approach and dynamic programming to achieve a 1 − -competitive algorithm for adword and display ad. They use a relatively simple dynamic programming in combination with the LP solution to check whether they should assign an item to a bidder or discard it.…”

“…However, analysis of the dynamic programming even with uniform sizes is involved. Furthermore, it is not easy to generalize the approach of [2] to the model of Goemans et al [7] where the given sizes show only the expected size of an item.…”

Combinatorial Prophet Inequalities

“…The matroid prophet inequality was first explicitly formulated by Kleinberg and Weinberg [38], who also gave a tight 2-competitive algorithm. In a different direction, Alaei, Hajiaghayi, and Liaghat [3] considered a variant they call prophet-inequality matching, which is useful for online ad allocation. More generally, for intersection of a constant number of matroid, knapsack, and matching constraints, Feldman, Svensson, and Zenklusen [28] gave an O(1)-competitive algorithm; this is a corollary of their online contention reslution schemes (OCRS), which we also use heavily (see Section 3.1.2).…”

Online Degree-Bounded Steiner Network Design

“…In the worst case model, the celebrated result of Karp et al ([26], STOC'90) gives a (1 − 1/e)-competitive algorithm. Different variants of this problem have been extensively studied in the past decade, e.g., for the random arrival model see [16,25,32,36], for the full information model see [33,38], and for the prophetinequality model see [5,3,4]. We also refer the reader to the comprehensive survey by Mehta [35].…”