# Computing Convex Coverage Sets for Multi−Objective Coordination Graphs

*Diederik Roijers‚ Shimon Whiteson and Frans Oliehoek*

### Abstract

Many real-world decision problems require making trade-offs between multiple objectives. However, in some cases, the relative importance of the objectives is not known when the problem is solved, precluding the use of single-objective methods. Instead, multi-objective methods, which compute the set of all potentially useful solutions, are required. This paper proposes new multi-objective algorithms for cooperative multi-agent settings. Following previous approaches, we exploit loose couplings, as expressed in graphical models, to coordinate efficiently. Existing methods, however, calculate only the *Pareto coverage set* (PCS), which we argue is inappropriate for stochastic strategies and unnecessarily large when the objectives are weighted in a linear fashion. In these cases, the typically much smaller *convex coverage set* (CCS) should be computed instead. A key insight of this paper is that, while computing the CCS is more expensive in unstructured problems, in many loosely coupled settings it is in fact cheaper to compute because the local solutions are more compact. We propose *convex multi-objective variable elimination*, which exploits this insight. We analyze its correctness and complexity and demonstrate empirically that it scales much better in the number of agents and objectives than alternatives that compute the PCS.