Abstract
We are given a digraph G = (N,A), where each arc is colored with one among k given colors. We look for a spanning arborescence T of G rooted (wlog) at node 1 and having minimum changeover cost. We call this the Minimum Changeover Cost Arborescence problem. To the authors’ knowledge, it is a new problem. The concept of changeover costs is similar to the one, already considered in the literature, of reload costs, but the latter depend also on the amount of commodity flowing in the arcs and through the nodes, whereas this is not the case for the changeover costs. Here, given any node j ≠ 1, if a is the color of the single arc entering node j in arborescence T, and b is the color of an arc (if any) leaving node j, then these two arcs contribute to the total changeover cost of T by the quantity d ab , an entry of a k-dimensional square matrix D. We first prove that our problem is NPO-complete and very hard to approximate. Then we present Integer Programming formulations together with a combinatorial lower bound, a greedy heuristic and an exact solution approach. Finally, we report extensive computational results and exhibit a set of challenging instances.
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References
Wirth, H., Steffan, J.: Reload cost problems: minimum diameter spaning tree. Discrete Applied Mathematics 113, 73–85 (2001)
Galbiati, G.: The complexity of a minimum reload cost diameter problem. Discrete Applied Mathematics 156, 3494–3497 (2008)
Gamvros, I., Gouveia, L., Raghavan, S.: Reload cost trees and network design. In: Proc. International Network Optimization Conference, paper n.117 (2007)
Amaldi, E., Galbiati, G., Maffioli, F.: On minimum reload cost. Networks (to appear)
Gourvès, L., Lyra, A., Martinhon, C., Monnot, J.: The minimum reload s-t path, trail and walk problems. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., Tůma, P., Valencia, F. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 621–632. Springer, Heidelberg (2009)
Kann, V.: Polynomially bounded minimization problems that are hard to approximate. Nordic J. Comp. 1, 317–331 (1994)
Jonsson, P.: Near-optimal nonapproximability results for some NPO PB-complete problems. Inform. Process. Lett. 68, 249–253 (1997)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)
Edmonds, J.: Optimum branchings. J. Res. Nat. Bur. Stand. (B) 71, 233–240 (1967)
Schrijver, A.: Combinatorial optimization: polyhedra and efficiency. Springer, Heidelberg (2003)
Lawler, E.: The quadratic assignment problem. Manag. Sci. 9(4), 586–599 (1963)
Caprara, A.: Constrained 0–1 quadratic programming: Basic approaches and extensions. European Journal of Operational Research 187, 1494–1503 (2008)
Gabow, H., Galil, Z., Spencer, T., Tarjan, R.: Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6(2), 109–122 (1986)
Buchheim, C., Liers, F., Oswald, M.: Speeding up ip-based algorithms for constrained quadratic 0–1 optimization. Math. Progr. (B) 124(1-2), 513–535 (2010)
Hao, J., Orlin, J.: A faster algorithm for finding the minimum cut in a graph. In: Proc. of the 3rd ACM-SIAM Symposium on Discrete Algorithms, pp. 165–174 (1992)
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Galbiati, G., Gualandi, S., Maffioli, F. (2011). On Minimum Changeover Cost Arborescences. In: Pardalos, P.M., Rebennack, S. (eds) Experimental Algorithms. SEA 2011. Lecture Notes in Computer Science, vol 6630. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20662-7_10
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DOI: https://doi.org/10.1007/978-3-642-20662-7_10
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