Combinatorica

, Volume 16, Issue 4, pp 453–464

# Distributed near-optimal matching

• Xiaotie Deng
Article

## Abstract

In this paper, we consider the following distributed bipartite matching problem: LetG=(L,R;E) be a bipartite graph in which boys are partL (left nodes), and girls are partR (right nodes.) There is an edge(l i ,r j )∈E iff boyl i is interested in girlr j . Every boyl i will propose to a girlr j among all those he is interested in, i.e., his adjacent right nodes in the bipartite graphG. 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 isM, then no deterministic algorithm can produce a matching of size not bounded by a constant, but a randomized algorithm achieves$$\sqrt M -$$ — and this is shown optimal by an adversary argument. If we allow many rounds in matching, with the senders learning from their failures, then, for deterministic algorithms, the ratio (of the optimal solution to the solution of the algorithm) is unbounded when the number of rounds is smaller than Δ(G), and becomes bounded (two) at the Δ(G)-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.

## Mathematics Subject Classification (1991)

68 Q 22 68 R 10 05 C 70

## References

1. [1]
N. Alon, andJ. H. Spencer:Probabilistic Method, Viley-Interscience Publication, John Wiley & Sons, Inc. Toronto, 1992.Google Scholar
2. [2]
A. Blum, andP. Chalasani: An Online Algorithm for Improving Performance in Navigation,FOCS93, (1993), 2–11.Google Scholar
3. [3]
4. [4]
X. Deng, andC. Papadimitriou: Competitive Distributed Decision-Making,Proc. 12th IFIP Congress, Madrid, September 1993. And also to appear in Algorithmica.Google Scholar
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), 12–21.Google Scholar
7. [7]
E. Koutsoupias,and C. H. Papadimitriou: On thek-server conjecture,Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, (1994) 507–511.Google Scholar
8. [8]
R. Karp, U. V. Vazirani,and V.V. Varirani: An Optimal Algorithm for Online Bipartite Matching,Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, (1990), 352–358.Google Scholar
9. [9]
M. S. Manasse, L. A. McGeoch,and D.D. Sleator: Competitive algorithms for on-line problems,Journal of Algorithms,11 (1990), 208–230.Google Scholar
10. [10]
C. H. Papadimitriou,and M. Yannakakis: Linear Programming without the Matrix,STOC,25 (1993), 121–129.Google 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), 202–208.Google Scholar