Mathematical Programming

, Volume 132, Issue 1–2, pp 261–276

# A weighted independent even factor algorithm

Full Length Paper Series A

## Abstract

An even factor in a digraph is a vertex-disjoint collection of directed cycles of even length and directed paths. An even factor is called independent if it satisfies a certain matroid constraint. The problem of finding an independent even factor of maximum size is a common generalization of the nonbipartite matching and matroid intersection problems. In this paper, we present a primal-dual algorithm for the weighted independent even factor problem in odd-cycle-symmetric weighted digraphs. Cunningham and Geelen have shown that this problem is solvable via valuated matroid intersection. Their method yields a combinatorial algorithm running in O(n 3 γn 6 m) time, where n and m are the number of vertices and edges, respectively, and γ is the time for an independence test. In contrast, combining the weighted even factor and independent even factor algorithms, our algorithm works more directly and runs in O(n 4 γ + n 5) time. The algorithm is fully combinatorial, and thus provides a new dual integrality theorem which commonly extends the total dual integrality theorems for matching and matroid intersection.

### Keywords

Independent even factor Combinatorial algorithm Dual integrality Nonbipartite matching Matroid intersection

### Mathematics Subject Classification (2000)

90C27 05C70 52B40

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### References

1. 1.
Bouchet A., Cunningham W.H.: Delta-matroids, jump systems, and bisubmodular polyhedra. SIAM J. Discrete Math. 8, 17–32 (1995)
2. 2.
Cunningham W.H.: Matching, matroids, and extensions. Math. Program. 91, 515–542 (2002)
3. 3.
Cunningham W.H., Geelen J.F.: The optimal path-matching problem. Combinatorica 17, 315–337 (1997)
4. 4.
Cunningham W.H., Marsh A.B. III: A primal algorithm for optimum matching. Math. Program. Study 8, 50–72 (1978)
5. 5.
Edmonds J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. Sect. B 69, 125–130 (1965)
6. 6.
Edmonds J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)
7. 7.
Edmonds J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York (1970)Google Scholar
8. 8.
Edmonds J.: Matroid intersection. Ann. Discrete Math. 4, 39–49 (1979)
9. 9.
Frank A., Szegő L.: Note on the path-matching formula. J. Graph Theory 41, 110–119 (2002)
10. 10.
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)Google Scholar
11. 11.
Iwata S., Takazawa K.: The independent even factor problem. SIAM J. Discrete Math. 22, 1411–1427 (2008)
12. 12.
Király T., Makai M.: On polyhedra related to even factors. In: Bienstock, D., Nemhauser, G.L. (eds) Integer Programming and Combinatorial Optimization: Proceedings of the 10th International IPCO Conference, Lecture Notes on Computer Science 3064, pp. 416–430. Springer, Heidelberg (2004)Google Scholar
13. 13.
Kobayashi Y., Takazawa K.: Even factors, jump systems, and discrete convexity. J. Comb. Theory Ser. B 99, 139–161 (2009)
14. 14.
Lawler E.L.: Matroid intersection algorithms. Math. Program. 9, 31–56 (1975)
15. 15.
Murota K.: Valuated matroid intersection I: optimality criteria. SIAM J. Discrete Math. 9, 545–561 (1996)
16. 16.
Murota K.: Valuated matroid intersection II: algorithms. SIAM J. Discrete Math. 9, 562–576 (1996)
17. 17.
Murota K.: M-convex functions on jump systems: a general framework for minsquare graph factor problem. SIAM J. Discrete Math. 20, 213–226 (2006)
18. 18.
Oxley J.G.: Matroid Theory. Oxford University Press, Oxford (1992)
19. 19.
Pap, G.: A Constructive Approach to Matching and Its Generalizations. Ph.D. thesis, Eötvös Loránd University (2006)Google Scholar
20. 20.
Pap G.: Combinatorial algorithms for matchings, even factors and square-free 2-factors. Math. Program. 110, 57–69 (2007)
21. 21.
Pap G., Szegő L.: On the maximum even factor in weakly symmetric graphs. J. Comb. Theory Ser. B 91, 201–213 (2004)
22. 22.
Spille B., Szegő L.: A Gallai-Edmonds-type structure theorem for path-matchings. J. Graph Theory 46, 93–102 (2004)
23. 23.
Spille B., Weismantel R.: A generalization of Edmonds’ matching and matroid intersection algorithms. In: Cook, W.J., Schulz, A.S. (eds) Integer Programming and Combinatorial Optimization: Proceedings of the 9th International IPCO Conference, Lecture Notes in Computer Science 2337, pp. 9–20. Springer, Heidelberg (2002)Google Scholar
24. 24.
Takazawa K.: A weighted even factor algorithm. Math. Program. 115, 223–237 (2008)