An LP-based heuristic algorithm for the node capacitated in-tree packing problem

  • Authors:

  • Yuma Tanaka;Shinji Imahori;Mihiro Sasaki;Mutsunori Yagiura

  • Affiliations:

  • Department of Computer Science and Mathematical Informatics, Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan;Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan;Department of Information Systems and Mathematical Sciences, Faculty of Information Sciences and Engineering, Nanzan University, 27 Seirei, Seto, Aichi 489-0863, Japan;Department of Computer Science and Mathematical Informatics, Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

  • Venue:
  • Computers and Operations Research
  • Year:
  • 2012

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Abstract

In this paper, we deal with the node capacitated in-tree packing problem. The input consists of a directed graph, a root node, a node capacity function and edge consumption functions for heads and tails. The problem is to find a subset of rooted spanning in-trees and their packing numbers, where the packing number of an in-tree is the number of times it is packed, so as to maximize the sum of packing numbers under the constraint that the total consumption of the packed in-trees at each node does not exceed the capacity of the node. This problem is known to be NP-hard. We propose a two-phase heuristic algorithm for this problem. In the first phase, it generates candidate spanning in-trees to be packed. The node capacitated in-tree packing problem can be formulated as an IP (integer programming) problem, and the proposed algorithm employs the column generation method for the LP (linear programming) relaxation problem of the IP to generate promising candidate in-trees. In the second phase, the algorithm computes the packing number of each in-tree. Our algorithm solves this second-phase problem by first modifying feasible solutions of the LP relaxation problem and then improving them with a greedy algorithm. We analyze upper and lower bounds on the solution quality of such LP-based algorithms for this problem. We conducted computational experiments on graphs used in related papers and on randomly generated graphs. The results indicate that our algorithm has a better performance than other existing methods.