Geodesic
Balanced problems on graphs with categorization of edges
Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 5-21

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Suppose a graph G = (V,E) with edge weights w(e) and edges partitioned into disjoint categories S₁,...,Sₚ is given. We consider optimization problems on G defined by a family of feasible sets (G) and the following objective function:
Keywords: algorithms on graphs, categorization of edges, NP-completeness 
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Berežný, Štefan; Lacko, Vladimír. Balanced problems on graphs with categorization of edges. Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 5-21. http://geodesic.mathdoc.fr/item/DMGT_2003_23_1_a0/