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Algorithms for the constrained maximum‐weight connected graph problem

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  • Heungsoon Felix Lee
  • Daniel R. Dooly

Abstract

Given a positive integer R and a weight for each vertex in a graph, the maximum‐weight connected graph (MCG) problem is to find a connected subgraph with R vertices that maximizes the sum of the weights. The MCG problem is strongly NP‐complete, and we study a special case of it: the constrained MCG (CMCG) problem, which is the MCG problem with a constraint of having a predetermined vertex included in the solution. We first show that the Steiner tree problem is a special case of the CMCG problem. Then we present three optimization algorithms for the CMCG problem. The first two algorithms deal with special graphs (tree and layered graphs) and employ different dynamic programming techniques, solving the CMCG problem in polynomial times. The third one deals with a general graph and uses a variant of the Balas additive method with an imbedded connectivity test and a pruning method. We also present a heuristic algorithm for the CMCG problem with a general graph and its bound analysis. We combine the two algorithms, heuristic and optimization, and present a practical solution method to the CMCG problem. Computational results are reported and future research issues are discussed. © 1996 John Wiley & Sons, Inc.

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  • Heungsoon Felix Lee & Daniel R. Dooly, 1996. "Algorithms for the constrained maximum‐weight connected graph problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(7), pages 985-1008, October.
    • Handle: RePEc:wly:navres:v:43:y:1996:i:7:p:985-1008
    DOI: 10.1002/(SICI)1520-6750(199610)43:73.0.CO;2-9
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    References listed on IDEAS

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    1. D. S. Johnson & K. A. Niemi, 1983. "On Knapsacks, Partitions, and a New Dynamic Programming Technique for Trees," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 1-14, February.
    2. Egon Balas, 1965. "An Additive Algorithm for Solving Linear Programs with Zero-One Variables," Operations Research, INFORMS, vol. 13(4), pages 517-546, August.
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    1. Heungsoon Felix Lee & Daniel R. Dooly, 1998. "Decomposition algorithms for the maximum‐weight connected graph problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(8), pages 817-837, December.

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