A filter-and-fan algorithm for the capacitated minimum spanning tree problem César Rego ⇑ , Frank Mathew School of Business Administration, University of Mississippi, University, MS 38677, USA article info Article history: Received 24 February 2010 Received in revised form 4 October 2010 Accepted 5 October 2010 Available online 14 October 2010 Keywords: Capacitated minimum spanning tree Compound neighborhoods Strategic oscillation Variable-depth neighborhood search Filter-and-fan abstract The capacitated minimum spanning tree (CMST) is a notoriously difficult problem in combinatorial opti- mization. Extensive investigation has been devoted to developing efficient algorithms to find optimal or near-optimal solutions. This paper proposes a new CMST heuristic algorithm that effectively combines the classical node-based and tree-based neighborhoods embodied in a filter-and-fan (F&F) approach, a local search procedure that generates compound moves in a tree search fashion. The overall algorithm is guided by a multi-level oscillation strategy used to trigger each type of neighborhood while allowing the search to cross feasibility boundaries. Computational results carried out on a standard set of 135 benchmark problems show that a simple F&F design competes effectively with prior CMST metaheuris- tics, rivaling the best methods, which are significantly more complex. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction The capacitated minimum spanning tree (CMST) problem is fundamental to the design of communication networks and has been widely studied for its importance in practical applications. A classical application consists of finding a minimum cost design of a capacitated centralized processing network where a central node of limited capacity must be linked via a tree topology to a number of remote terminals with a specified demand (see e.g. Kawatra & Bricker, 2000; Woolston & Albin, 1988). In the context of teleprocessing system design (Chandy & Russell, 1972; Esau & Williams, 1966; Kershenbaum & Boorstyn, 1983) the remote ter- minals are data terminals and the central node is the data process- ing (or control) center. The terminals communicate with the center along communication links of limited capacity. The capacity of a link is the maximum traffic the link can carry while maintaining acceptable response times. The tree topology stems from the fact that traffic from a single terminal cannot be simultaneously trans- mitted along several links without additional synchronizing cir- cuitry. The problem is a special case of the so-called Telpak problem, which is a fundamental design problem in fiber-optic lo- cal access networks (Gavish, 1991). The CMST also finds applica- tion in a variety of other settings in distribution, transportation, and logistics (see Amberg, Domschke, & Voß, 2000; Gavish, 1982, 1991). In addition, the problem provides a relaxation of the classi- cal capacitated vehicle routing problem, which is central to many other complex problems, including the design of communications networks with topological ring structures (Klincewicz, Luss, & Yan, 1998). The CMST problem has been widely studied over the last four decades with proposals of a variety of solution approaches. In this paper we are interested in heuristic algorithms, focusing on local search methods and guiding strategies superimposed by metaheu- ristic approaches. Local search defines a general class of heuristic methods that explore the solution space by iteratively generating new solutions derived from a neighborhood structure, and whose effectiveness is typically amplified by embedding them in meta- heuristic strategies. Earlier proposals for metaheuristic strategies applied to the CMST problem are due to Amberg, Domschke, and Voß (1996) who introduced a number of variants of simulated annealing and tabu search algorithms. In this work the authors consider two basic types of neighborhoods: a shift neighborhood that transfers a node from one sub-tree to another, and a swap neighborhood that interchanges nodes between sub-trees. Sharaiha, Gendreau, Laporte, and Osman (1997) proposed another tabu search approach based on a sub-tree neighborhood structure, which splits the current spanning tree into two sub-trees and reconnects them by adding an arc different from the one that had been deleted in the original tree. These neighborhoods that modify at most two sub-trees are generally called two-exchange neighborhoods. Patt- erson, Pirkul, and Rolland (1999) proposed an adaptive reasoning technique drawing on principles of adaptive memory programming of the type used in tabu search and joining them with constructive neighborhood search processes. Their approach iteratively exe- cutes the classical constructive heuristic of Esau and Williams (1966) under the guidance of tabu restrictions, which are probabi- listically modified at each iteration of the method. 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.10.003 ⇑ Corresponding author. E-mail addresses: crego@bus.olemiss.edu (C. Rego), fmathew@bus.olemiss.edu (F. Mathew). Computers & Industrial Engineering 60 (2011) 187–194 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie