Approximation Algorithms for the Maximum Carpool Matching Problem

Abstract

The Maximum Carpool Matching problem is a star packing problem in directed graphs. Formally, given a directed graph $G = (V, A)$, a capacity function $c: V \rightarrow {\mathbb {N}}$, and a weight function $w : A \rightarrow {\mathbb {R}}$, 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 $A \cap (P \times D)$, under the constraints that for every vertex $d \in D$, $deg^M_{in}(d) \le c(d)$, and for every vertex $p \in P$, $deg^M_{out}(p) \le 1$. 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.