We present an average case model of classical problems in combinatorial optimization where there are color constraints. In all cases we seek some (spanning) sub-structure of a complete graph of minimum cost. The edges are randomly colored either red or blue. We bias against the red edges by placing a bound on the number of them that are allowed in our structure. This bound will be lower w.h.p. than what would occur without discrimination. We examine the effect of this bias on the minimum cost of a desired structure. We consider minimum cost spanning trees, shortest paths, minimum cost perfect matchings and the asymmetric traveling salesperson problem.
The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and N\"agele gave...
This paper fully characterize the set of requests of an optimal solution to the MARPG problem and shows how to compute in constant time its cardinality, and establishes upper bounds on the number of anomalies.
J. Bermond, Michel Cosnard, D. Coudert et al.· Networks· 1 citation
We consider a combinatorial optimization problem arising when a set of pick-up and delivery orders must be satisfied within an Automated Storage/Retrieval System. The computational complexity of the problem is still open, but it is conjectured to be
N
P
-hard. We point out some of its relevant properties a...
Nicola Bianchessi, Dario Ostuni, G. Righini· Open Journal of Mathematical...· 0 citations
We formalize the Steiner Traveling Salesman Problem (Steiner-TSP) on Graphs of Convex Sets (GCS), which seeks a minimum-cost closed trajectory through required convex sets while allowing optional transit vertices and revisits. To explore the resulting infinite solution space, we propose a unified branch-and-bound searc...
This paper proves that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph, and shows that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number.
Narek Bojikian, Alexander Firbas, Robert Ganian et al.· 0 citations
A
cubic tree
is a tree with leaves in which every internal vertex has degree exactly 3. Any such tree can be encoded by a
Path‐Length Matrix
(PLM), that is, an integer matrix whose th entry gives the number of edges in the unique path between leaves and in . The convex hull of all PLMs associated with cubic tre...
D. Catanzaro, G. Joret, Brieuc Pierre et al.· Networks· 0 citations
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