In a recent result, Weitz (D. Weitz, 2006) established equivalence between the marginal distribution of a node, say G, in any binary pair-wise Markov random field (MRF), say G, with the marginal distribution of the root node in the self-avoid walk tree of the G starting at v. Analogous result for max-marginal distribution holds for the reason that addition and multiplication commute in the same way as addition and maximum. This remarkable connection suggests a message-passing algorithm for computing exact marginal and max-marginal in any binary MRF G. In this paper, we exploit this property along with appropriate graph partitioning scheme to design approximate message passing algorithms for computing max-marginal of nodes or maximum a-posteriori assignment (MAP) in a binary MRF G. Our algorithm can provide provably arbitrarily small error for a large class of graphs including planar graphs. Our algorithms are linear in number of nodes G with constant dependent on the approximation error. For precise evaluation of computation cost of algorithm, we obtain a novel tight characterization of the size of self-avoiding walk tree for any connected graph as a function of number of edges and nodes.