Skip to main content
SpringerLink
Log in

Approximate Generalized Matching: f-Matchings and f-Edge Covers

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We present almost linear time approximation schemes for several generalized matching problems on nonbipartite graphs. Our results include \(O_\epsilon (m\alpha (m, n))\)-time algorithms for \((1-\epsilon )\)-maximum weight f-matching and \((1+\epsilon )\)-approximate minimum weight f-edge cover. As a byproduct, we also obtain direct algorithms for the exact cardinality versions of these problems running in \(O(m\alpha (m, n)\sqrt{f(V)})\) time, where f(V) is the sum of degree constraint on the entire vertex set. The technical contributions of this work include an efficient method for maintaining relaxed complementary slackness in generalized matching problems and approximation-preserving reductions between the f-matching and f-edge cover problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. The b-matching problem can be regarded as an f-matching problem on a multigraph in which there is implicitly an infinite supply of each edge.

  2. Using relaxed complementary slackness, matched and unmatched edges have different eligibility criteria (to be included in augmenting paths and blossoms) whereas b-matching blossoms require that all copies of an edge—matched and unmatched alike—are all eligible or all ineligible.

  3. These issues only arise when finding augmenting paths in batches, not one-at-a-time [11], and when the problem is f-matching, not matching.

  4. We use \(yz_F\) and \(yz_C\) to denote the aggregated dual yz for f-matching and f-edge cover respectively. We will omit the subscript if it is clear from the context.

  5. In an actual implementation, the inner/outer labelling can be computed in the search in Blossom Formation step. The labelling continues to be valid after contracting a maximal set of blossoms.

  6. Of course, if \(B_v\) is inner and reachable in \(\widehat{G}\), this only implies that \(\beta (B_v)\) is reachable from an unsaturated vertex in G; other vertices in \(B_v\) may not be reachable in G.

  7. This is because multiple augmenting walks in the underlying graph can intersect a single blossom in \(\Omega \) before we contract the blossom, while after contracting a blossom, any augmenting walk or alternating cycle going through the blossom will forbid the other walks and cycle to use the same blossom again (as it must go through the base edge).

  8. It is still possible that later we discover some path from a descendant of C back to the root that circumvents C

  9. Getting to the active walk does not automatically imply that you can then get to the root, since the ancestral path might not alternate at the vertex when it first reaches the active walk

References

  1. Choromanski, K., Jebara, T., Tang, K.: Adaptive anonymity via \(b\)-matching. In: Advances in Neural Information Processing Systems 27: 27th Annual Conference on Neural Information Processing System, NIPS 2013, pp. 3192–3200 (2013)

  2. Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)

    Article  MathSciNet  Google Scholar 

  3. Drake, D.E., Hougardy, S.: A simple approximation algorithm for the weighted matching problem. Inf. Process. Lett. 85, 211–213 (2003)

    Article  MathSciNet  Google Scholar 

  4. Duan, R., Pettie, S.: Linear-time approximation for maximum weight matching. J. ACM 61, 1:1-1:23 (2014)

    Article  MathSciNet  Google Scholar 

  5. Duan, R., Pettie, S., Su, H.: Scaling algorithms for weighted matching in general graphs. ACM Trans. Algor. 14, 8:1-8:35 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bureau Stand.: Sect. B Math. Math. Phys. 69B, 125–130 (1965)

    Article  MathSciNet  Google Scholar 

  7. Edmonds, J.: Paths, trees and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MathSciNet  Google Scholar 

  8. Edmonds, J., Johnson, E.L.: Matching: A well-solved class of integer linear programs. In: Combinatorial Optimization - Eureka, You Shrink!, Papers Dedicated to Jack Edmonds, 5th International Workshop, pp. 27–30 (2001)

  9. Gabow, H.N.: An efficient implementation of Edmonds’ algorithm for maximum matching on graphs. J. ACM 23(2), 221–234 (1976)

    Article  MathSciNet  Google Scholar 

  10. Gabow, H.N.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing, STOC 1983, pp. 448–456 (1983)

  11. Gabow, H.N.: Data structures for weighted matching and extensions to \(b\)-matching and \(f\)-factors. ACM Trans. Algor. 14, 39:1-39:80 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Gabow, H.N., Sankowski, P.: Algebraic algorithms for \(b\)-matching, shortest undirected paths, and \(f\)-factors. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 137–146 (2013)

  13. Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30, 209–221 (1985)

    Article  MathSciNet  Google Scholar 

  14. Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM J. Comput. 18, 1013–1036 (1989)

    Article  MathSciNet  Google Scholar 

  15. Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for general graph-matching problems. J. ACM 38, 815–853 (1991)

    Article  MathSciNet  Google Scholar 

  16. Håstad, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Math. 182(1), 105–142 (1999)

    Article  MathSciNet  Google Scholar 

  17. Khan, A., Choromanski, K., Pothen, A., Ferdous, S.M., Halappanavar, M., Tumeo, A.: Adaptive anonymization of data using \(b\)-edge cover. In: Proceedings of the International Conference for High Performance Computing,Networking, Storage, and Analysis, SC, pp. 59:1–59:11 (2018)

  18. Khan, A.M., Pothen, A.: A new \(3/2\)-approximation algorithm for the \(b\)-edge cover problem. In: 2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing, CSC, pp. 52–61 (2016)

  19. Lawler, E.: Combinatorial Optimization: Networks and Matroids. Dover Books on Mathematics Series (2001)

  20. Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|}|E|)\) algorithm for finding maximum matching in general graphs. In: 21st Annual Symposium on Foundations of Computer Science, FOCS, pp. 17–27 (1980)

  21. Pettie, S., Sanders, P.: A simpler linear time \(2/3-\epsilon \) approximation for maximum weight matching. Inf. Process. Lett. 91, 271–276 (2004)

    Article  MathSciNet  Google Scholar 

  22. Pulleyblank, W.: Faces of matching polyhedra. PhD thesis, University of Waterloo, Ontario, Canada (1973)

  23. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Algorithms and Combinatorics (2002)

  24. Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. J. ACM 22, 215–225 (1975)

    Article  MathSciNet  Google Scholar 

  25. Tutte, W.T.: On the problem of decomposing a graph into \(n\) connected factors. J. Lond. Math. Soc. s1–36, 221–230 (1961)

    Article  MathSciNet  Google Scholar 

  26. Vazirani, V.V.: A theory of alternating paths and blossoms for proving correctness of the \(O(\sqrt{|V|}|E|)\) general graph maximum matching algorithm. Combinatorica 14(1), 71–109 (1994)

    Article  MathSciNet  Google Scholar 

  27. Vazirani, V.V.: An improved definition of blossoms and a simpler proof of the MV matching algorithm. CoRR, arXiv:1210.4594, (2012)

  28. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Supported by NSF grants CCF-1217338, CNS-1318294, CCF-1514383, CCF-1637546, and CCF-1815316.

Author information

Authors and Affiliations

  1. EECS Department, University of Michigan, 2260 Hayward St., Ann Arbor, MI, 48109, USA

    Dawei Huang & Seth Pettie

Authors
  1. Dawei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seth Pettie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seth Pettie.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, D., Pettie, S. Approximate Generalized Matching: f-Matchings and f-Edge Covers. Algorithmica 84, 1952–1992 (2022). https://doi.org/10.1007/s00453-022-00949-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-022-00949-5

Keywords

Search

Navigation