Acta Informatica

, Volume 15, Issue 4, pp 329–346

# Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems

• W. LipskiJr.
• F. P. Preparata
Article

## Summary

A bipartite graph G=(A, B, E) is convex on the vertex set A if A can be ordered so that for each element b in the vertex set B the elements of A connected to b form an interval of A; G is doubly convex if it is convex on both A and B. Letting ¦A¦=m and ¦B¦=n, in this paper we describe maximum matching algorithms which run in time O(m + nA(n)) on convex graphs (where A(n) is a very slowly growing function related to a functional inverse of Ackermann's function), and in time O(m+n) on doubly convex graphs. We also show that, given a maximum matching in a convex bipartite graph G, a corresponding maximum set of independent vertices can be found in time O(m+n). Finally, we briefly discuss some generalizations of convex bipartite graphs and some extensions of the previously discussed techniques to instances in scheduling theory.

## Keywords

Information System Operating System Data Structure Communication Network Information Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The design and analysis of computer algorithms. Reading, MA: Addison-Wesley, 1974Google Scholar
2. 2.
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. System Sci. 13, 335–379 (1976)Google Scholar
3. 3.
Emde Boas, P. van: Preserving order in a forest in less than logarithmic time. Proc. 16th Annual Symp. on Foundations of Comp. Sci., Univ. of California, Berkeley, Oct. 1975, pp. 75–84Google Scholar
4. 4.
Emde Boas, P. van: Preserving order in a forest in less than logarithmic time and linear space. Information Processing Lett. 6, 80–82 (1977)Google Scholar
5. 5.
Gale, D.: Optimal assignments in an ordered set: an application of matroid theory. J. Combinatorial Theory 4, 176–180 (1968)Google Scholar
6. 6.
Gavril, F.: Testing for equality between maximum matching and minimum node covering. Information Processing Lett. 6, 199–202 (1977)Google Scholar
7. 7.
Glover, F.: Maximum matching in convex bipartite graph. Naval Res. Logist. Quart. 14, 313–316 (1967)Google Scholar
8. 8.
Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2, 225–231 (1973)Google Scholar
9. 9.
Lawler, E.L.: Combinatorial Optimization: Networks and matroids. New York, NY: Holt, Rinehart and Winston, 1976Google Scholar
10. 10.
Lipski, W.: Information storage and retrieval — mathematical foundations II (Combinatorial problems). Theor. Comput. Sci. 3, 183–211 (1976)Google Scholar
11. 11.
Lipski, W., Lodi, E., Luccio, F., Mugnai, C., Pagli, L.: On two dimensional data organization II. Fundamenta Informaticae 2, 227–243 (1977)Google Scholar
12. 12.
Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. Assoc. Comput. Mach. 22, 215–224 (1975)Google Scholar

## Authors and Affiliations

• W. LipskiJr.
• 1
• F. P. Preparata
• 1
1. 1.Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

## Personalised recommendations

### Citearticle 