Prophet Secretary for Combinatorial Auctions and Matroids

Abstract

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). and Feldman et al. [17] 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 [45] and Esfandiari et al. [15] 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.

Beating Greedy for Stochastic Bipartite Matching

“…We conclude this section with Theorem 7 which shows that our algorithm Approx-QC is (1 − 1/e)-approximate. The proof follows exactly the same lines (except for the definition of base price c b ) as in Ehsani et al [EHKS18] to show that the expected weight of the edge adjacent to a fixed vertex b ∈ B in the output of Approx-QC is at least (1 − 1/e) · c b . Then by the linearity of expectation, the expected utility of Approx-QC is at least (1 − 1/e) · b∈B c b = (1 − 1/e) · e∈E x * e · w(e) OPT (recall that c b = e∈δ(b) x * e · w(e)).…”

“…Effectively, for each vertex in v ∈ B, its neighbor u arrives independently with probability x * uv and weight w uv , and it has to pick one neighbor so that its expected utility is close to uv x * uv · w uv . This setting is similar to the prophet secretary problem, and we extend the ideas of Ehsani et al [EHKS18] to achieve this.…”

“…In their recent work, Ehsani et al [EHKS18] gave a (1 − 1/e)-competitive algorithm for the prophet secretary problem. Inspired by this result, we design our algorithm so that for each b ∈ B, it recovers at least (1 − 1/e)-fraction of the expected weight of the edge incident to b in an optimal algorithm's output.…”

“…In Section 2.2, we analyze the structure of the LP polytope and show how to construct the distributions D QC a for each a ∈ A. Finally in Section 2.3, we present our proposed algorithm and adapt the ideas of Ehsani et al [EHKS18] to analyze its approximation guarantee.…”

“…The (1 − 1/e)-competitive algorithm algorithm given by Ehsani et al [EHKS18] for the prophet secretary problem first sets a base price for the item. If some buyer comes at time t ∈ [0, 1], and if the item is not already sold, then the algorithm sells the item to this buyer if the offered price is at least (1 − e t−1 ) times the base price.…”