Carpooling in Social Networks

Abstract

We consider the online carpool fairness problem of Fagin–Williams (IBM J Res Dev 27(2):133–139, 1983), where an online algorithm is presented with a sequence of pairs drawn from a group of n potential drivers. The online algorithm must select one driver from each pair, with the objective of partitioning the driving burden as fairly as possible for all drivers. The unfairness of an online algorithm is a measure of the worst-case deviation between the number of times a person has driven and the number of times they would have driven if life was completely fair.

We consider the version of the problem in which drivers only carpool with their neighbors in a given social network graph; this is a generalization of the original problem, which corresponds to the social network of the complete graph. We show that, for graphs of degree d, the unfairness of deterministic algorithms against adversarial sequences is exactly d∕2. For randomized algorithms, we show that static algorithms, a natural class of online algorithms, have unfairness $\tilde{\Theta }(\sqrt{d})$. For random sequences on stars and in bounded-genus graphs, we give a deterministic algorithm with logarithmic unfairness. Interestingly, restricting the random sequences to sparse social network graphs increases the unfairness of the natural greedy algorithm. In particular, for the line social network, this algorithm has expected unfairness $\Omega (\log ^{1/3}n)$, whereas for the clique social network its expected unfairness is O(loglogn); see Ajtai–Aspnes–Naor–Rabani–Schulman–Waarts (J Algorithm 29(2):306–357, 1998).