Augmentative Message Passing for Traveling Salesman Problem and Graph Partitioning
This work addresses combinatorial optimization challenges for researchers and practitioners, but it is incremental as it applies an existing method to new problems.
The authors tackled combinatorial optimization problems by adapting the cutting plane method to message passing, achieving competitive solutions for Traveling Salesman Problem (TSP) and graph partitioning. For TSP, they found near-optimal solutions with empirical time scaling as N^3.
The cutting plane method is an augmentative constrained optimization procedure that is often used with continuous-domain optimization techniques such as linear and convex programs. We investigate the viability of a similar idea within message passing -- which produces integral solutions -- in the context of two combinatorial problems: 1) For Traveling Salesman Problem (TSP), we propose a factor-graph based on Held-Karp formulation, with an exponential number of constraint factors, each of which has an exponential but sparse tabular form. 2) For graph-partitioning (a.k.a., community mining) using modularity optimization, we introduce a binary variable model with a large number of constraints that enforce formation of cliques. In both cases we are able to derive surprisingly simple message updates that lead to competitive solutions on benchmark instances. In particular for TSP we are able to find near-optimal solutions in the time that empirically grows with N^3, demonstrating that augmentation is practical and efficient.