Abstract
The maximum weighted matching problem in bipartite graphs is one of the classic combinatorial optimization problems, and arises in many different applications. Ordered binary decision diagram (OBDD) or algebraic decision diagram (ADD) or variants thereof provides canonical forms to represent and manipulate Boolean functions and pseudo-Boolean functions efficiently. ADD and OBDD-based symbolic algorithms give improved results for large-scale combinatorial optimization problems by searching nodes and edges implicitly. We present novel symbolic ADD formulation and algorithm for maximum weighted matching in bipartite graphs. The symbolic algorithm implements the Hungarian algorithm in the context of ADD and OBDD formulation and manipulations. It begins by setting feasible labelings of nodes and then iterates through a sequence of phases. Each phase is divided into two stages. The first stage is building equality bipartite graphs, and the second one is finding maximum cardinality matching in equality bipartite graph. The second stage iterates through the following steps: greedily searching initial matching, building layered network, backward traversing node-disjoint augmenting paths, updating cardinality matching and building residual network. The symbolic algorithm does not require explicit enumeration of the nodes and edges, and therefore can handle many complex executions in each step. Simulation experiments indicate that symbolic algorithm is competitive with traditional algorithms.
Highlights
The matching problems find their applications in many settings where we often wish to find the proper way to pair objects or people together to achieve some desired goal
We present novel symbolic algebraic decision diagram (ADD) formulation and algorithm for maximum weighted matching in bipartite graphs
The matching problems are classified into maximum cardinality matching in bipartite graphs, maximum cardinality matching in general graphs, maximum weighted matching in bipartite graphs, and maximum weighted matching in general graphs [1,2]
Summary
The matching problems find their applications in many settings where we often wish to find the proper way to pair objects or people together to achieve some desired goal. The classical Hungarian method was invented by Kuhn [3], which solves maximum weighted matching problems in strongly polynomial time of O n m n log n It was revised by Munkres, and has been known since as the Hungarian algorithm or the. Finding maximum weighted matching in bipartite graphs is one of typical combinatorial optimization problems, where the size of graphs is a significant and often prohibitive difficulty. This phenomenon is known as combinatorial state explosion, resulting in that large graphs cannot be stored and operated on even the largest contemporary computers.
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