In the k-labeled Spanning Forest Problem (kLSF), given a graph G with a label (color) assigned to each edge, and an integer positive value kmax we look for the minimum number of connected components that can be obtained by using at most kmax different labels. The problem is strictly related to the Minimum Labelling Spanning Tree Problem (MLST), since a spanning tree of the graph (i.e. a single connected component) would obviously be an optimal solution for the kLSF, if it can be obtained without violating the bound on kmax. In this work we present heuristic and exact approaches to solve this new problem
This paper introduces an additive branch-and-bound algorithm for two variants of the pickup and delivery traveling salesman problem in which loading and unloading operations have to be performed either in a Last-In-First-Out (LIFO) or in a First-In-First-Out (FIFO) order. Two relaxations are used within the additive approach: the assignment problem and the shortest spanning r-arborescence problem. The quality of the lower bounds is further improved by a set of elimination rules applied at each node of the search tree to remove from the problem arcs that cannot belong to feasible solutions because of precedence relationships. The performance of the algorithm and the effectiveness of the elimination rules are assessed on instances from the literature
We model and solve the Rainbow Cycle Cover Problem (RCCP). Given a connected and undirected graph G = ( V , E , L ) and a coloring function ℓ that assigns a color to each edge of G from the finite color set L , a cycle whose edges have all different colors is called a rainbow cycle. The RCCP consists of finding the minimum number of disjoint rainbow cycles covering G . The RCCP on general graphs is known to be NP-complete. We model the RCCP as an integer linear program, we derive valid inequalities and we solve it by branch-and-cut. Computational results are reported on randomly generated instances
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