Marek Adamczyk

Improved analysis of the greedy algorithm for stochastic matching

The stochastic matching problem with applications in online dating and kidney exchange was introduced by Chen et al. (2009) [1] together with a simple greedy strategy. They proved it is a 4-approximation, but conjectured that the greedy algorithm is in fact a 2-approximation. In this paper we confirm this hypothesis.

Sequential Posted-Price Mechanisms with Correlated Valuations

We study the revenue performance of sequential posted-price mechanisms and some natural extensions for a setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted-price mechanisms are conceptually simple mechanisms that work by proposing a “take-it-or-leave-it” offer to each buyer. We apply sequential posted-price mechanisms to single-parameter multiunit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted-price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted-price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the case of a continuous valuation distribution when various standard assumptions hold simultaneously (i.e., everywhere-supported, continuous, symmetric, and normalized (conditional) distributions that satisfy regularity, the MHR condition, and affiliation). In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted-price mechanism is proportional to the ratio of the highest and lowest possible valuation. We prove that a simple generalization of these mechanisms achieves a better revenue performance; namely, if the sequential posted-price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Ω (1/max { 1,d}) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence (d=0) to arbitrary dependence (d = n-1).

Improved Approximation Algorithms for Stochastic Matching

In this paper we consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. We are given an undirected graph in which every edge is assigned a probability of existence and a positive profit, and each node is assigned a positive integer called timeout. We know whether an edge exists or not only after probing it. On this random graph we are executing a process, which one-by-one probes the edges and gradually constructs a matching. The process is constrained in two ways: once an edge is taken it cannot be removed from the matching, and the timeout of node v upper-bounds the number of edges incident to v that can be probed. The goal is to maximize the expected profit of the constructed matching.

Submodular Stochastic Probing on Matroids

In a stochastic probing problem we are given a universe E, where each element e in E is active independently with probability p in [0,1], and only a probe of e can tell us whether it is active or not. On this universe we execute a process that one by one probes elements - if a probed element is active, then we have to include it in the solution, which we gradually construct. Throughout the process we need to obey inner constraints on the set of elements taken into the solution, and outer constraints on the set of all probed elements. This abstract model was presented in [Gupta and Nagaraja, IPCO 2013], and provides a unified view of a number of problems. Thus far all the results in this general framework pertain only to the case in which we are maximizing a linear objective function of the successfully probed elements. In this paper we generalize the stochastic probing problem by considering a monotone submodular objective function. We give a (1-1/e)/(k_in+k_out+1)-approximation algorithm for the case in which we are given k_in greater than 0 matroids as inner constraints and k_out greater than 1 matroids as outer constraints. There are two main ingredients behind this result. First is a previously unpublished stronger bound on the continuous greedy algorithm due to Vondrak. Second is a rounding procedure that also allows us to obtain an improved 1/(k_in+k_out)-approximation for linear objective functions.