# Distributed near-optimal matching

• Xiaotie Deng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 920)

## Abstract

In this paper, we consider the following distributed bipartite matching problems: Let G=(L, R; E) be a bipartite graph in which boys are part L (left nodes), and girls are part R (right nodes.) There is an edge (li, rj) ∈ E iff boy li is interested in girl rj. Every boy li will propose to a girl rj among all those he is interested in, i.e., his adjacent right nodes in the bipartite graph G. If several boys propose to the same girl, only one of them will be accepted by the girl. We assume that none of the boys communicate with others. This matching problem is typical of distributed computing under incomplete information: Each boy only knows his own preference but none of the others. We first study the one-round matching problem: each boy proposes to at most one girl, so that the total number of girls receiving a proposal is maximized. If the maximum matching is M, then no deterministic algorithm can produce a matching of size not bounded by a constant, but a randomized algorithm achieves √M—and this is shown optimal by an adversary argument. If we allow many rounds in matching, with the senders learning from their failures, then the ratio is unbounded when the number of rounds is smaller than Δ, and becomes bounded (two) at the Δ-th round. In contrast, an extension of the one-round randomized algorithm establishes that there is no such discontinuity in the randomized case. This randomized result is also matched by an upper bound of asymptotically the same order.

## References

1. 1.
N. Alon and J. H. Spencer: Probabilistic Method. Viley-Interscience Publication. John Wiley & Sons, Inc. Toronto. (1992)Google Scholar
2. 2.
A. Blum and P. Chalasani: An Online Algorithm for Improving Performance in Navigation. FOCS93. (1993) 2–11Google Scholar
3. 3.
B. Bollobas, Extremal Graph Theory. Academic Press Inc. New York. (1978)Google Scholar
4. 4.
5. 5.
R. L. Graham, B.L. Rothschild, and J.H. Spencer: Ramsey Theory. John Wiley & Sons. New York. (1980)Google Scholar
6. 6.
S. Irani, and Y. Rabani: On the Value of Information in Coordination Games. FOCS93. (1993) 12–21Google Scholar
7. 7.
E. Koutsoupias, and C. H. Papadimitriou: On the k-server conjecture. Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing. (1994) 507–511Google Scholar
8. 8.
R. Karp, U. V. Vazirani, and V.V. Vazirani: An Optimal Algorithm for Online Bipartite Matching. Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing. (1990) 352–358Google Scholar
9. 9.
M.S. Manasse, L.A. McGeoch, and D.D. Sleator: Competitive algorithms for on-line problems. Twentieth ACM Annual Symposium on Theory of Computing. (1988) 322–333Google Scholar
10. 10.
C.H. Papadimitriou and M. Yannakakis: Linear Programming without the Matrix. STOC 25 (1993) 121–129Google Scholar
11. 11.
P. Raghavan: Lecture Notes on Randomized Algorithms. IRM Research Report. (1990) RC15340(#68237)1/9/90.Google Scholar
12. 12.
D.D. Sleator, and R.E. Tarjan: Amortized efficiency of list update and paging rules. Communications of the ACM 28(2). (1985) 202–208Google Scholar