Semi-matchings for bipartite graphs and load balancing

Abstract

We consider the problem of fairly matching the left-hand vertices of a bipartite graph to the right-hand vertices. We refer to this problem as the optimal semimatching problem; it is a relaxation of the known bipartite matching problem. We present a way to evaluate the quality of a given semi-matching and show that, under this measure, an optimal semi-matching balances the load on the right hand vertices with respect to any L p -norm. In particular, when modeling a job assignment system, an optimal semi-matching achieves the minimal makespan and the minimal flow time for the system. The problem of finding optimal semi-matchings is a special case of certain scheduling problems for which known solutions exist. However, these known solutions are based on general network optimization algorithms, and are not the most efficient way to solve the optimal semi-matching problem. To compute optimal semi-matchings efficiently, we present and analyze two new algorithms. The first algorithm generalizes the Hungarian method for computing maximum bipartite matchings, while the second, more efficient algorithm is based on a new notion of cost-reducing paths. Our experimental results demonstrate that the second algorithm is vastly superior to using known network optimization algorithms to solve the optimal semi-matching problem. Furthermore, this same algorithm can also be used to find maximum bipartite matchings and is shown to be roughly as efficient as the best known algorithms for this goal.

Workload Balancing in Parallel Machines based on Semi-matching Theory

“…For more detailed information about semi-matching theory, the reader is referred to the papers [5,6]. …”

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“…Semi-matching problem. In the semi-matching problem the nodes of a graph are divided into clients and servers [14]. Each client has to be assigned to an adjacent server.…”

Efficient scheduling for minimum latency aggregation in wireless sensor networks

“…The BSPT algorithm builds a shortest path tree by traversing the graph in a Breadth First Search (BFS) manner. As it builds the tree, the BSPT algorithm builds a bipartite graph for all the nodes of two consecutive levels in the tree, and it uses the bipartite semi-matching algorithm from Harvey et al [11] to distribute as evenly as possible the lower level nodes between the upper level nodes. This is done for every pair of consecutive levels in the tree.…”