Abstract
We consider the solution of matching problems with a convex cost function via a network flow algorithm. We review the general mapping between matching problems and flow problems on skew symmetric networks and revisit several results on optimality of network flows. We use these results to derive a balanced capacity scaling algorithm for matching problems with a linear cost function. The latter is later generalized to a balanced capacity scaling algorithm also for a convex cost function. We prove the correctness and discuss the complexity of our solution.
Highlights
We consider the solution of matching problems with a convex cost function via a network flow algorithm
We review the general mapping between matching problems and flow problems on skew symmetric networks and revisit several results on optimality of network flows
Skew symmetric networks have become an important tool for the efficient solution of matching problems [1]
Summary
Skew symmetric networks have become an important tool for the efficient solution of matching problems [1]. Going over from a matching problem to a problem of flow optimization often allows for simplification and speed-up of solution algorithms. Typical examples include minconvexproblems as studied in [2,3] For such minconvex problems a convex function in the number of matchings has to be minimized for each participant. In this paper we consider the problem of minimizing a separable convex objective function over a skew-symmetric network with a balanced flow. This problem can be mapped on the aforementioned matching problems and allows for an efficient solution of the latter.
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