Weighted matching with pair restrictions
The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in many other directions, extending our results further is impossible (unless P = NP). One consequence of the new polynomial time algorithm is that it demonstrates, for the first time, that a prize-collecting assignment problem with “pair restriction” is solved in polynomial time. The prize collecting assignment problem is a relaxation of the prize-collecting traveling salesman problem which requires, for any prescribed pair of nodes, either both nodes of the pair are matched or none of them are. It is shown that the prize collecting assignment problem is equivalent to the prize collecting cycle cover problem which is hence solvable in polynomial time as well.
KeywordsAssignment Traveling salesman problem Prize collecting Weighted graph matching Matching matroid Gammoid Weighted matroid parity
The topic of prize collecting 2-factor arose from a discussion the first author had with Alejandro Toriello concerning relaxations for the prize collecting traveling salesman problem.
- 1.Archetti, C., Speranza, M.G., Vigo, D.: Vehicle routing problems with profits. In: Vehicle Routing: Problems, Methods, and Applications, SIAM, pp. 273–298 (2014)Google Scholar
- 3.Balas, E.: The prize collecting traveling salesman problem and its applications. In: The Traveling Salesman Problem and Its Variations, Springer, pp. 663–695 (2007)Google Scholar
- 6.Cheung, H.Y., Lau, L.C., Leung, K.M.: Algebraic algorithms for linear matroid parity problems. ACM Trans. Algorithms (TALG), 10(3), article number 10 (2014)Google Scholar
- 11.Goemans, M.X.: Combining approximation algorithms for the prize-collecting TSP. arXiv:0910.0553v1 (2009)
- 12.Hoffman, K.L., Padberg, M., Rinaldi, G.: Traveling salesman problem. In: Encyclopedia of Operations Research and Management Science, Springer, pp. 1573–1578 (2013)Google Scholar
- 14.Lovász, L.: The matroid matching problem. In: Algebraic Methods in Graph Theory, pp. 495–517, Szeged (1978)Google Scholar