Fair solutions for some multiagent optimization problems

We consider optimization problems in a multiagent setting where a solution is evaluated with a vector. Each coordinate of this vector represents an agent's utility for the solution. Due to the possible conflicts, it is unlikely that one feasible solution is optimal for all agents. Then, a natural aim is to find solutions that maximize the satisfaction of the least satisfied agent, where the satisfaction of an agent is defined as his relative utility, i.e., the ratio between his utility for the given solution and his maximum possible utility. This criterion captures a classical notion of fairness since it focuses on the agent with lowest relative utility. We study worst-case bounds on this ratio: for which ratio a feasible solution is guaranteed to exist, i.e., to what extend can we find a solution that satisfies all agents? How can we build these solutions in polynomial time? For several optimization problems, we give polynomial-time deterministic algorithms which (almost always) achieve the best possible ratio.