International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 128-139

# Solving Matching Problems Efficiently in Bipartite Graphs

• Selma Djelloul
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)

## Abstract

We investigate the problem maxDMM of computing a largest set of pairwise disjoint maximum matchings in undirected graphs. In this paper, n, m denote, respectively, the number of vertices and the number of edges. We solve maxDMM for bipartite graphs, by providing an $$O(n^{1.5}\sqrt{m/\log n} + mn\log n)$$-time algorithm. We design better algorithms for complete bipartite graphs, and bisplit graphs. (Bisplit graphs are bipartite graphs with the nested neighborhood property.) Specifically, we prove that the problem maxDMM is solvable in complete bipartite graphs in time O(m). A sequence $$S=(s_1,\cdots ,s_t)$$ of positive integers is said to be color-feasible for a graph G, if there exists a proper edge-coloring of G with colors $$1,\cdots ,t$$, such that precisely $$s_i$$ edges have color i, for every $$i=1,\cdots ,t$$. Actually, for complete bipartite graphs, we prove that, for any sequence S of integers which is color-feasible for a complete bipartite graph G, an edge-coloring of G corresponding to S can be obtained in time O(m). For bisplit graphs, (1) we solve maxDMM in time $$O(mn\log n)$$, and (2) we design an $$O(n^2\log n)$$-time algorithm to count all maximum matchings. This latter time is the same time in which runs the best known algorithm computing the number of maximum matchings in bisplit graphs [17], but our algorithm is much simpler than the one given in [17]. The key idea underlying both results is that bisplit graphs have an O(n)-time enumeration of their minimal vertex covers.

## Notes

### Acknowledgements

I thank Odile Favaron for her helpful idea to handle the proof of Theorem 3 by arithmetic. I thank Pierre Fraigniaud for his careful reading and advices to improve the paper writing.

## References

1. 1.
Alt, H., Blum, N., Mehlhorn, K., Paul, M.: Computing a maximum cardinality matching in a bipartite graph in time $${O}(n^{1.5}\sqrt{m/\log n})$$. Inf. Process. Lett. 37(4), 237–240 (1991)
2. 2.
Asratian, A.S.: Some results on an edge-coloring problem of Folkman and Fulkerson. Discret. Math. 223, 13–25 (2000)
3. 3.
Chartrand, G., Zhang, P.: A First Course in Graph Theory. Dover Publications, New York (2012)Google Scholar
4. 4.
Cole, R., Ost, K., Schirra, S.: Edge-coloring bipartite multigraphs in $${O(E \log D)}$$ time. Combinatorica 21(1), 5–12 (2001)
5. 5.
Couturier, J.F., Golovach, P.A., Kratsch, D., Liedloff, M., Pyatkin, A.: Colorings with few colors: counting, enumeration and combinatorial bounds. Theory Comput. Syst. 52, 645–667 (2013). Springer-Verlag New York. Secaucus, NJ, USA
6. 6.
Ding, G.: Covering the edges with consecutive sets. J. Graph Theory 15(5), 559–562 (1991)
7. 7.
Even, S., Kariv, O.: An $${O}(n^{2.5})$$ algorithm for maximum matching in general graphs. In: IEEE 16th annual Symposium on Foundations of Computer Science (FOCS), pp. 100–112 (1975)Google Scholar
8. 8.
Folkman, J., Fulkerson, D.R.: Edge-colorings in bipartite graphs. In: Bose, R., Dowling, T. (eds.) Combinatorial Mathematics and its Applications, pp. 561–577. University of North Carolina Press, Chapel Hill (1969)Google Scholar
9. 9.
Frost, H., Jacobson, M., Kabell, J., Morris, F.R.: Bipartite analogues of split graphs and related topics. Ars Combinatoria 29, 283–288 (1990)
10. 10.
Golumbic, M.R., Goss, C.F.: Perfect elimination and chordal bipartite graphs. J. Graph Theory 2(2), 155–163 (1978)
11. 11.
Hammer, P.L., Peled, U.N., Sun, X.: Difference graphs. Discret. Appl. Math. 28, 35–44 (1990)
12. 12.
Heggernes, P., Kratsch, D.: Linear-time certifying recognition algorithms and forbidden induced subgraphs. Nordic J. Comput. 14, 87–108 (2007)
13. 13.
Heinrich, K., Hell, P., Kirkpatrick, D.G., Liu, G.: A simple existence criterion for ($$g<f$$)-factors. Discrete Mathematics 85, 313–317 (1990)
14. 14.
Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10(4), 718–720 (1981)
15. 15.
Kawarabayashi, K., Kobayashi, Y., Reed, B.: The disjoint paths problem in quadratic time. J. Comb. Theory Ser. B 102, 424–435 (2012)
16. 16.
Kowalik, L.: Improved edge-coloring with three colors. Theoret. Comput. Sci. 410, 3733–3742 (2009)
17. 17.
Okamoto, Y., Uehara, R., Uno, T.: Counting the number of matchings in chordal and chordal bipartite graph classes. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911, pp. 296–307. Springer, Heidelberg (2010)
18. 18.
Petersen, J.: Die theorie der regulären graphen. Acta Mathematica 15, 193–220 (1891)
19. 19.
Robertson, N., Sanders, D., Seymour, P., Thomas, R.: Efficiently four-coloring planar graphs. In: Proceedings of the 28th annual ACM Symposium on Theory of Computing, (STOC), pp. 571–575 (1996)Google Scholar
20. 20.
Robertson, N., Seymour, P., Thomas, R.: Tutte’s edge-coloring conjecture. J. Comb. Theory Ser. B 70, 166–183 (1997)
21. 21.
Schrijver, A.: Combinatorial Optimization, vol. 1. Springer-Verlag, Berlin (2003)
22. 22.
Slater, P.: A constructive characterization of trees with at least $$k$$ disjoint maximum matchings. J. Comb. Theory Ser. B 25, 326–338 (1978)
23. 23.
Thomas, R.: Recent excluded minor theorem for graphs. In: Surveys in Combinatorics, vol. 267, pp. 201–222 (1999). The electronic journal of combinatorics 8 (2001)Google Scholar
24. 24.
Yannakakis, M.: Node deletion problems on bipartite graphs. SIAM J. Comput. 10(2), 310–327 (1981)