A Combinatoric Interpretation of Dual Variables for Weighted Matching Problems

We consider four weighted matching-type problems: the bipartite graph and general graph versions of matching and f -factors. The linear program duals for these problems are shown to be weights of certain subgraphs. Specifically the so-called y duals are the weights of certain maximum matchings or f -factors; z duals (used for general graphs) are the weights of certain 2-factors or 2f -factors. The y duals are canonical in a well-defined sense; z duals are canonical for matching and more generally for b-matchings (a special case of f -factors) but for f -factors their support can vary. As weights of combinatorial objects the duals are integral for given integral edge weights, and so they give new proofs that the linear programs for these problems are TDI.