Controlled Dynamic Fair Division.

In the single-resource dynamic fair division framework there is a homogeneous resource that is shared between agents dynamically arriving and departing over time. When n agents are present, there is only one truly ``fair'' allocation: each agent receives 1/n of the resource. Implementing this static solution in the dynamic world is notoriously impractical; there are too many disruptions to existing allocations: for a new agent to get her fair share, all other agents must give up a small piece. A natural remedy is simply to restrict the number of allowed disruptions when a new agent arrives. [16] considered this setting, and introduced a natural benchmark - the fairness ratio - the ratio of the minimal share to the ideal share (1/k when there are k agents in the system). They described an algorithm that obtains the optimal fairness ratio when d ≥ 1 disruptions are allowed per arriving agent. However, in systems with high arrival rates even one disruption per arrival can be too costly. We consider the scenario when fewer than one disruption per arrival is allowed. We show that we can maintain high levels of fairness even with significantly fewer than one disruption per arrival. In particular, we present an instance-optimal algorithm (the input to the algorithm is a vector of allowed disruptions) and show that the fairness ratio of this algorithm decays logarithmically with c, where c is the longest number of consecutive time steps in which we are not allowed any disruptions. We then consider dynamic fair division with multiple, heterogeneous resources. In this model, agents demand the resources in fixed proportions, known in economics as Leontief preferences. We show that the general problem is NP-hard, even if the resource demands are binary and known in advance. We study the case where the fairness criterion is Dominant Resource Fairness (DRF), and the demand vectors are binary. We design a generic algorithm for this setting using a reduction to the single-resource case. To prove an impossibility result, we take an integer program for the problem and analyze an algorithm for constructing dual solutions to a ``residual'' linear program; this approach may be of independent interest.