Approximation Algorithms For The Maximum Carpool Matching Problem

The MAXIMUM CARPOOL MATCHING problem is a star packing problem in directed graphs. Formally, given a directed graph G = (V, A), a capacity function c : V -> N, and a weight function w : A -> R, a feasible carpool matching is a triple (P, D, M), where P (passengers) and D (drivers) form a partition of V, and M is a subset of A boolean AND (P x D), under the constraints that for every vertex d is an element of D, deg(in)(M) (d) <= c(d), and for every vertex p is an element of P, dego(ut)(M) (p) <= 1. In the MAXIMUM CARPOOL MATCHING problem we seek for a matching (P, D, M) that maximizes the total weight of M.The problem arises when designing an online carpool service, such as Zimride [1], that tries to connect between passengers and drivers based on (arbitrary) similarity function. The problem is known to be NP-hard, even for uniform weights and without capacity constraints.We present a 3-approximation algorithm for the problem and 2-approximation algorithm for the unweighted variant of the problem.