# Greedy Matching: Guarantees and Limitations

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## Abstract

Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In contrast, only few analytical results are known. We show that MinGreedy cannot improve on the trivial approximation ratio of \(\frac{1}{2}\) whp., even for bipartite graphs. Our hard inputs seem to require a small number of high-degree nodes. This motivates an investigation of greedy algorithms on graphs with maximum degree \(\varDelta \): we show that MinGreedy achieves a \({\frac{{\varDelta }-1}{2{\varDelta }-3}} \)-approximation for graphs with \({\varDelta } {=} 3\) and for \(\varDelta \)-regular graphs, and a guarantee of \({\frac{{\varDelta }-1/2}{2{\varDelta }-2}} \) for graphs with maximum degree \({\varDelta } \). Interestingly, our bounds even hold for the deterministic MinGreedy that breaks all ties arbitrarily. Moreover, we investigate the limitations of the greedy paradigm, using the model of *priority algorithms* introduced by Borodin, Nielsen, and Rackoff. We study deterministic priority algorithms and prove a \({\frac{{\varDelta }-1}{2{\varDelta }-3}}\)-inapproximability result for graphs with maximum degree \({\varDelta } \); thus, these greedy algorithms do not achieve a \(\frac{1}{2} {+} \varepsilon \)-approximation and in particular the \(\frac{2}{3}\)-approximation obtained by the deterministic MinGreedy for \({\varDelta } {=} 3\) is optimal in this class. For *k*-uniform hypergraphs we show a tight \(\frac{1}{k}\)-inapproximability bound. We also study fully randomized priority algorithms and give a \(\frac{5}{6}\)-inapproximability bound. Thus, they cannot compete with matching algorithms of other paradigms.

### Keywords

Maximum matching Greedy algorithms Approximation algorithms Priority algorithms Randomized algorithms## Notes

### Acknowledgments

The authors would like to thank Georg Schnitger for many helpful discussions, and Allan Borodin and Nicolas Peña for their valuable comments on an early draft of the paper.

### References

- 1.Angelopoulos, S., Borodin, A.: Randomized priority algorithms. Theor. Comput. Sci.
**411**(26–28), 2542–2558 (2010)MathSciNetCrossRefMATHGoogle Scholar - 2.Aronson, J., Dyer, M.E., Frieze, A.M., Suen, S.: Randomized greedy matching II. Random Struct. Algorithms
**6**(1), 55–74 (1995)MathSciNetCrossRefMATHGoogle Scholar - 3.Aronson, J., Frieze, A.M., Pittel, B.: Maximum matchings in sparse random graphs: Karp–Sipser revisited. Random Struct. Algorithms
**12**(2), 111–177 (1998)MathSciNetCrossRefMATHGoogle Scholar - 4.Bennett, P., Bohman, T.: A Natural Barrier in Random Greedy Hypergraph Matching. CoRR (2012).arXiv:1210.3581
- 5.Berger, B., Singh, R., Xu, J.: Graph algorithms for biological systems analysis. In: Proceedings of the Nineteenth Annual ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 142–151 (2008)Google Scholar
- 6.Besser, B., Werth, B.: On the Approximation Performance of Degree Heuristics for Matching. CoRR (2015).arXiv:1504.05830
- 7.Borodin, A., Boyar, J., Larsen, K.S., Mirmohammadi, N.: Priority algorithms for graph optimization problems. Theor. Comput. Sci.
**411**(1), 239–258 (2010)MathSciNetCrossRefMATHGoogle Scholar - 8.Borodin, A., Ivan, I., Ye, Y., Zimny, B.: On sum coloring and sum multi-coloring for restricted families of graphs. Theor. Comput. Sci.
**418**, 1–13 (2012)MathSciNetCrossRefMATHGoogle Scholar - 9.Borodin, A., Nielsen, M.N., Rackoff, C.: (Incremental) priority algorithms. Algorithmica
**37**(4), 295–326 (2003)MathSciNetCrossRefMATHGoogle Scholar - 10.Chan, T.H.H., Chen, F., Wu, X., Zhao, Z.: Ranking on arbitrary graphs: rematch via continuous LP with monotone and boundary condition constraints. In: Proceedings of the Twenty-Fifth Annual ACM–SIAM Symposium on Discrete Algorithms (SODA), pp. 1112–1122 (2014)Google Scholar
- 11.Chan, Y., Lau, L.: On linear and semidefinite programming relaxations for hypergraph matching. Math. Program.
**135**(1–2), 123–148 (2012)MathSciNetCrossRefMATHGoogle Scholar - 12.Cheng, Y.Q., Wu, V., Collins, R.T., Hanson, A.R., Riseman, E.M.: Maximum-weight bipartite matching technique and its application in image feature matching. In: Proceedings of the SPIE Visual Communications and Image Processing (1996)Google Scholar
- 13.Cygan, M.: Improved approximation for 3-dimensional matching via bounded pathwidth local search. In: Proceedings of the 54th Annual Symposium on Foundations of Computer Science (FOCS), pp. 509–518 (2013)Google Scholar
- 14.Davis, S., Impagliazzo, R.: Models of greedy algorithms for graph problems. Algorithmica
**54**(3), 269–317 (2009)MathSciNetCrossRefMATHGoogle Scholar - 15.Dubhashi, D.P., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
- 16.Dyer, M.E., Frieze, A.M.: Randomized greedy matching. Random Struct. Algorithms
**2**(1), 29–46 (1991)MathSciNetCrossRefMATHGoogle Scholar - 17.Edmonds, J.: Paths, trees, and flowers. Can. J. Math.
**17**, 449–467 (1965)MathSciNetCrossRefMATHGoogle Scholar - 18.Edmonds, J., Fulkerson, D.R.: Transversals and matroid partition. J. Res. Natl. Bur. Stand
**69B**(3), 147–153 (1965)Google Scholar - 19.Frieze, A.M., Radcliffe, A.J., Suen, S.: Analysis of a simple greedy matching algorithm on random cubic graphs. Comb. Probab. Comput.
**4**, 47–66 (1995)MathSciNetCrossRefMATHGoogle Scholar - 20.Gabow, H.N.: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. J. ACM
**23**(2), 221–234 (1976)MathSciNetCrossRefMATHGoogle Scholar - 21.Gabow, H.N.: Set-Merging for the Matching Algorithm of Micali and Vazirani. CoRR (2014). arXiv:1501.00212
- 22.Geelen, J.F.: An algebraic matching algorithm. Combinatorica
**20**(1), 61–70 (2000)MathSciNetCrossRefMATHGoogle Scholar - 23.Goel, G., Tripathi, P.: Matching with our eyes closed. In: Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 718–727 (2012)Google Scholar
- 24.Goldberg, A.V., Karzanov, A.V.: Maximum skew-symmetric flows and matchings. Math. Program.
**100**(3), 537–568 (2004)MathSciNetCrossRefMATHGoogle Scholar - 25.Harvey, N.J.A.: Algebraic algorithms for matching and matroid problems. SIAM J. Comput.
**39**(2), 679–702 (2009)MathSciNetCrossRefMATHGoogle Scholar - 26.Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex.
**15**(1), 20–39 (2006)MathSciNetCrossRefMATHGoogle Scholar - 27.Hosaagrahara, M., Sethu, H.: Degree-sequenced matching algorithms for input-queued switches. Telecommun. Syst.
**34**(1–2), 37–49 (2007)CrossRefGoogle Scholar - 28.Hougardy, S.: Linear time approximation algorithms for degree constrained subgraph problems. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 185–200 (2009)Google Scholar
- 29.Huang, N., Borodin, A.: Bounds on double-sided myopic algorithms for unconstrained non-monotone submodular maximization. In: Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC), pp. 528–539 (2014)Google Scholar
- 30.Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Math.
**2**(1), 68–72 (1989)MathSciNetCrossRefMATHGoogle Scholar - 31.Karande, C., Mehta, A., Tripathi, P.: Online bipartite matching with unknown distributions. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pp. 587–596 (2011)Google Scholar
- 32.Karp, R.M., Sipser, M.: Maximum matchings in sparse random graphs. In: Proceedings of the 22nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 364–375 (1981)Google Scholar
- 33.Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC), pp. 352–358 (1990)Google Scholar
- 34.Korte, B., Hausmann, D.: An analysis of the greedy algorithm for independence systems. Ann. Discrete Math.
**2**, 65–74 (1978)MathSciNetCrossRefMATHGoogle Scholar - 35.Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)MATHGoogle Scholar
- 36.Magun, J.: Greedy matching algorithms: an experimental study. ACM J. Exp. Algorithmics
**3**, 6 (1998)MathSciNetCrossRefMATHGoogle Scholar - 37.Mahdian, M., Yan, Q.: Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pp. 597–606 (2011)Google Scholar
- 38.Miller, Z., Pritikin, D.: On randomized greedy matchings. Random Struct. Algorithms
**10**(3), 353–383 (1997)MathSciNetCrossRefMATHGoogle Scholar - 39.Mucha, M., Sankowski, P.: Maximum matchings via gaussian elimination. In: Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 248–255 (2004)Google Scholar
- 40.Poloczek, M.: Bounds on greedy algorithms for MAX SAT. In: Proceedings of the 19th Annual European Symposium on Algorithms (ESA), pp. 37–48 (2011)Google Scholar
- 41.Poloczek, M., Szegedy, M.: Randomized greedy algorithms for the maximum matching problem with new analysis. In: Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 708–717 (2012)Google Scholar
- 42.Roth, A.E., Sönmez, T., Ünver, M.U.: Pairwise kidney exchange. J. Econ. Theory
**125**(2), 151–188 (2005)MathSciNetCrossRefMATHGoogle Scholar - 43.Tinhofer, G.: A probabilistic analysis of some greedy cardinality matching algorithms. Ann. Oper. Res.
**1**, 239–254 (1984)CrossRefMATHGoogle Scholar - 44.Tripathi, P.: Allocation Problems with Partial Information. Ph.D. thesis, Georgia Institute of Technology (2012)Google Scholar
- 45.Vazirani, V.V.: A Simplification of the MV Matching Algorithm and Its Proof. CoRR (2013).arXiv:1210.4594
- 46.Yao, A.C.C.: Lower bounds by probabilistic arguments. In: Proceedings of the 24th Annual Symposium on Foundations of Computer Science (FOCS), pp. 420–428. IEEE (1983)Google Scholar