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 $c:V\to N$, and a weight function $w:A\to R$, a feasible \emph{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 $d\in D$, $din(d)\le c(d)$, and for every vertex $p\in P$, $dout(p)\le 1$. 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~\cite{zimride}, 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.