Governing convergence of Max-sum on DCOPs through damping and splitting

Abstract Max-sum is a version of Belief Propagation, used for solving DCOPs. In tree-structured problems, Max-sum converges to the optimal solution in linear time. Unfortunately, when the constraint graph representing the problem includes multiple cycles (as in many standard DCOP benchmarks), Max-sum does not converge and explores low quality solutions. Recent attempts to address this limitation proposed versions of Max-sum that guarantee convergence, while ignoring some of the problem's constraints. Damping is a method that is often used for increasing the chances that Belief Propagation will converge. That being said, it has not been suggested for inclusion in the algorithms that propose Max-sum for solving DCOPs. In this paper we advance the research on incomplete-inference DCOP algorithms by: 1) investigating the effect of damping on Max-sum. We prove that, while damping slows down the propagation of information among agents, on tree-structured graphs, Max-sum with damping is guaranteed to converge to the optimal solution in weakly polynomial time; and 2) proposing a novel method for adjusting the level of asymmetry in the factor graph, in order to achieve a balance between exploitation and exploration, when using Max-sum for solving DCOPs. By converting a standard factor graph to an equivalent split constraint factor graph (SCFG), in which each function-node is split into two function-nodes, we can control the level of asymmetry for each constraint. Our empirical results demonstrate a drastic improvement in the performance of Max-sum when using damping (referred to herein as Damped Max-sum, DMS). However, in contrast to the common assumption that Max-sum performs best when converging, we demonstrate that non converging versions perform efficient exploration, and produce high quality results, when implemented within an anytime framework. On most standard benchmarks, the best results were achieved using versions with a high damping factor, which outperformed existing incomplete DCOP algorithms. In addition, our results imply that by applying DMS to SCFGs with a minor level of asymmetry, we can find high quality solutions within a small number of iterations, even without using an anytime framework. We prove that for a factor graph with a single constraint, if this constraint is split symmetrically, Max-sum applied to the resulting cycle is guaranteed to converge to the optimal solution. We further demonstrate that for an asymmetric split, convergence is not guaranteed.