Abstract
An even factor in a digraph, introduced by Cunningham and Geelen (Vertex-disjoint dipaths and even dicircuits. manuscript, 2001), is a collection of vertex-disjoint dipaths and even dicycles, which generalizes a path-matching of Cunningham and Geelen (Combinatorica 17, 315–337, 1997). In a restricted class of digraphs, called odd-cycle-symmetric, Pap (Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 3509, pp. 66–80, Springer, Heidelberg, 2005) presented a combinatorial algorithm to find a maximum even factor. For odd-cycle-symmetric weighted digraphs, which are odd-cycle-symmetric digraphs accompanied by a weight vector satisfying a certain property, Király and Makai (Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 3064, pp. 416–430, Springer, Heidelberg, 2004) provided a linear program that describes the maximum weight even factor problem, and proved the dual integrality. In this paper, we present a primal-dual algorithm to find a maximum weight even factor for an odd-cycle-symmetric weighted digraph. This algorithm is based on the weighted matching algorithm of Edmonds and the maximum even factor algorithm of Pap. The running time of the algorithm is O(n 3 m), where n and m are the numbers of the vertices and arcs, respectively, which is better than that of the existing algorithms for the special cases. The algorithm also gives a constructive proof for the dual integrality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cunningham W.H. (2002). Matchings, matroids and extensions. Math. Program. Ser. B 91: 515–542
Cunningham W.H. and Geelen J.F. (1997). The optimal path-matching problem. Combinatorica 17: 315–337
Cunningham, W.H., Geelen, J.F.: Vertex-disjoint dipaths and even dicircuits. manuscript (2001)
Cunningham W.H. and Marsh III A.B. (1978). A primal algorithm for optimum matching. Math. Program. Study 8: 50–72
Edmonds J. (1965). Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. Sect. B 69: 125–130
Edmonds J. (1965). Paths, trees and flowers. Can. J. Math. 17: 449–467
Frank A. and Szegő L. (2002). Note on the path-matching formula. J. Graph Theory 41: 110–119
Harvey, N.J.A.: Algebraic structures and algorithms for matching and matroid problems. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 531–542 (2006)
Király T. and Makai M. (2004). On polyhedra related to even factors. In: Bienstock, D. and Nemhauser, G.L. (eds) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science 3064., pp 416–430. Springer, Heidelberg
Nemhauser G.L. and Wolsey L.A. (1988). Integer and Combinatorial Optimization. Wiley, New York
Pap G. (2005). A combinatorial algorithm to find a maximum even factor. In: Jünger, M. and Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 3509., pp 66–80. Springer, Heidelberg
Pap G. (2007). Combinatorial algorithms for matchings, even factors and square-free 2-factors. Math. Program. Ser. B 110: 57–69
Pap G. and Szegő L. (2004). On the maximum even factor in weakly symmetric graphs. J. Comb. Theory Ser. B 91: 201–213
Schrijver A. (1983). Min-max results in combinatorial optimization. In: Bachem, A., Grötschel, M. and Korte, B. (eds) Mathematical Programming—The State of the Art, pp 439–500. Springer, Heidelberg
Schrijver A. (1983). Short proofs on the matching polyhedron. J. Comb. Theory Ser. B 34: 104–108
Spille B. and Szegő L. (2004). A Gallai-Edmonds-type structure theorem for path-matchings. J. Graph Theory 46: 93–102
Spille B. and Weismantel R. (2002). A generalization of Edmonds’ matching and matroid intersection algorithms. In: Cook, W.J. and Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, 2337., pp 9–20. Springer, Heidelberg