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Matrix representation and gradient flows for NP-hard problems

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Over the past decade, a number of connections between continuous flows and numerical algorithms were established. Recently, Brockett and Wong reported a connection between gradient flows on the special orthogonal groupLO(n) and local search solutions for the assignment problem. In this paper, we describe a uniform formulation for certain NP-hard combinatorial optimization problems in matrix form and examine their connection with gradient flows onLO(n). For these problems, there is a correspondence between the so-called 2-opt solutions and asymptotically stable critical points of an associated gradient flow.

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Communicated by W. B. Gong

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Wong, W.S. Matrix representation and gradient flows for NP-hard problems. J Optim Theory Appl 87, 197–220 (1995). https://doi.org/10.1007/BF02192047

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Key Words

  • Gradient flows
  • assignment problem
  • traveling salesman problem
  • graph partitioning problem
  • local search