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

• Published in: Computers & operations research Vol. 39; no. 3; pp. 637 - 646
• Main Authors: Tanaka, Yuma, Imahori, Shinji, Sasaki, Mihiro, Yagiura, Mutsunori
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.03.2012; Elsevier; Pergamon Press Inc
• Subjects: Algorithms; Applied sciences; Column generation; Column packings; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Exact sciences and technology; Flows in networks. Combinatorial problems; Graph theory; Graphs; Heuristic; Heuristic methods; Information retrieval. Graph; Integer programming; IP (Internet Protocol); Linear programming; LP relaxation; Mathematical analysis; Mathematical models; Mathematical programming; Operational research and scientific management; Operational research. Management science; Packing problem; Relaxation heuristics; Sensor network; Software; Studies; Theoretical computing; Trees; Wireless ad hoc network; Column generation; LP relaxation; Sensor network; Relaxation heuristics; Wireless ad hoc network; Hypergraph; Tree(graph); Bounded solution; Relaxation method; Ad hoc network; Consumption; Lower bound; Head; Graph node; Bin packing problem; Linear programming; Layout problem; Integer programming; Computational geometry; Upper bound; Treeing; Spanning tree; Heuristic method; NP hard problem; Greedy algorithm; Capacity constraint; Digraph; Sensor array; Directed graph
• ISSN: 0305-0548; 1873-765X; 0305-0548
• DOI: 10.1016/j.cor.2011.05.019

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. ► Introducing node capacities to a spanning arborescence packing problem on digraphs. ► Generating in-trees by the column generation method. ► Solving the minimum weight rooted arborescence problem as the pricing problem. ► Packing in-trees by a greedy algorithm with efficient data structures. ► The approximation guarantee of LP-based algorithms.