Skip to main content

On linear and semidefinite programming relaxations for hypergraph matching

Mathematical Programming volume 135, pages 123–148 (2012)Cite this article

Abstract

The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a well-studied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following:

  • We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly \({k-1+\frac{1}{k}}\) for k-uniform hypergraphs, and is exactly k − 1 for k-partite hypergraphs. This yields an improved approximation algorithm for the weighted 3-dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a new way to round linear programming solutions for packing problems.

  • We study the strengthening of the standard LP relaxation by local constraints. We show that, even after linear number of rounds of the Sherali-Adams lift-and-project procedure on the standard LP relaxation, there are k-uniform hypergraphs with integrality gap at least k − 2. On the other hand, we prove that for every constant k, there is a strengthening of the standard LP relaxation by only a polynomial number of constraints, with integrality gap at most \({\frac{k+1}{2}}\) for k-uniform hypergraphs. The construction uses a result in extremal combinatorics.

  • We consider the standard semidefinite programming relaxation of the problem. We prove that the Lovász \({\vartheta}\) -function provides an SDP relaxation with integrality gap at most \({\frac{k+1}{2}}\) . The proof gives an indirect way (not by a rounding algorithm) to bound the ratio between any local optimal solution and any optimal SDP solution. This shows a new connection between local search and linear and semidefinite programming relaxations.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Finland)

References

  1. Aharoni R.: Ryser’s conjecture for tripartite 3-graphs. Combinatorica 21(1), 1–4 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aharoni R., Haxell P.: Hall’s theorem for hypergraphs. J. Graph Theory 35, 83–88 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon N., Kahale N.: Approximating the independence number via the \({\vartheta}\) -function. Math. Program. 80(3), 253–264 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arkin E.M., Hassin R.: On local search for weighted k-Set packing. Math. Oper. Res. 23(3), 640–648 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arora, S., Bollobás, B., Lovász, L.: Proving integrality gaps without knowing the linear program. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 313–322 (2002)

  6. Asadpour, A., Feige, U., Saberi, A.: Santa claus meets hypergraph matchings. In: Proceedings of APPROX 2008, pp. 10–20 (2008)

  7. Bar-Yehuda R., Bendel K., Freund A., Rawitz D.: Local ratio: a unified framework for approximation algorithms. In memoriam: Shimon Even 1935–2004. ACM Comput. Surv. 36(4), 422–463 (2004)

    Article  Google Scholar 

  8. Bar-Yehuda R., Halldórsson M.M., Naor J.(S.), Shachnai H., Shapira I.: Scheduling split intervals. SIAM J. Comput. 36(1), 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bansal, N., Sviridenko, M.: The santa claus problem. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pp. 31–40 (2006)

  10. Berman, P.: A d/2 approximation for maximum weight independent set in d-claw free graphs. In: Proceedings of the 7th Scandinavian Workshop on Algorithms Theory (SWAT). Lecture Notes in Computer Science, vol. 1851, pp. 31–40, Springer, Berlin (2000)

  11. Berman, P., Fürer, M.: Approximating maximum independent set in bounded degree graphs. In: Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 365–371 (1994)

  12. Berman, P., Karpinski, M.: Improved approximation lower bound on small occurrence optimization. Electronic Colloquium on Computational Complexity, Report No. 8 (2003)

  13. Berman, P., Krysta, P.: Optimizing misdirection. In: Proceedings of the 14th Annual Symposium on Discrete Algorithms (SODA), pp. 192–201 (2003)

  14. Bienstock D., Ozbay N.: Tree-width and the Sherali-Adams operator. Discret. Optim. 1, 13–21 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Całczyńska-Karłowicz M.: Theorem on families of finite sets. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 12, 87–89 (1964)

    MATH  Google Scholar 

  16. Chandra, B., Halldórsson, M.M.: Greedy local improvement and weighted set packing approximation. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 169–176 (1999)

  17. Charikar, M., Makarychev, K., Makarychev, Y.: Local global tradeoffs in metric embeddings. In: Proceedings of the 48th Annual Symposium on Foundations of Computer Science (FOCS), pp. 713–723 (2007)

  18. Charikar, M., Makarychev, K., Makarychev, Y.: Integrality gaps for Sherali-Adams relaxations. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 283–292 (2009)

  19. Chataigner F., Manic G., Wakabayashi Y., Yuster R.: Approximation algorithms and hardness results for the clique packing problem. Discret. Appl. Math. 157(7), 1396–1406 (2008)

    Article  MathSciNet  Google Scholar 

  20. Coja-Oghlan A.: The Lovász number of random graphs. Comb. Probab. Comput. 14(4), 439–465 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Erdös, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Hajnal, A., Rado, R, Sós, V.T. (eds.) Proceedings of Colloquia Mathematica Societatis János Bolyai, Infinite and Finite Sets. vol. 10, pp. 609–627. Keszthely, Hungary (1973)

  22. Feige, U.: Randomized graph products, chromatic numbers, and the Lovász \({\vartheta}\) -function. In: Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC), pp. 635–640 (1995)

  23. Feige U., Krauthgamer R.: The probable value of the Lovász–Schrijver relaxations for maximum independent set. SIAM J. Comput. 32(2), 345–370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Füredi Z.: Maximum degree and fractional matchings in uniform hypergraphs. Combinatorica 1(2), 155–162 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Füredi Z., Kahn J., Seymour P.: On the fractional matching polytope of a hypergraph. Combinatorica 13(2), 167–180 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gomes, C.P., Regis, R.G., Shmoys, D.B.: An improved approximation algorithm for the partial latin square problem. In: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 832–833 (2003)

  27. Grötschel M., Lovász L., Schrijver A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorics 1, 169–197 (1981)

    Article  MATH  Google Scholar 

  28. Grötschel M., Lovász L., Schrijver A.: Polynomial algorithms for perfect graphs. Ann. Discret. Math. 21, 325–356 (1984)

    Google Scholar 

  29. Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1988)

    Book  MATH  Google Scholar 

  30. Halldórsson, M.H.: Approximating discrete collections via local improvments. In: Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 160–169 (1995)

  31. Halldórsson M.M., Radhakrishnan J.: Greed is good: approximating independent sets in sparse and bounded-degree graphs. Algorithmica 18(1), 145–163 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hajirasouliha, I., Jowhari, H., Kumar, R., Sundaram, R.: On completing latin squares. In: Proceedings of the 24th International Symposium on Theoretical Aspects of Computer Science (STACS). Lecture Notes in Computer Science, vol. 4393, pp. 524–535. Springer, Berlin (2007)

  33. Hassin R., Rubinstein S.: An approximation algorithm for maximum triangle packing. Discret. Appl. Math. 154(6), 971–979 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Hazan, E., Safra, M., Schwartz, O.: On the complexity of approximating k-dimensional matching. In: Proceedings of APPROX 2003, Lecture Notes in Computer Science, vol. 2764, pp. 59–70. Springer, Berlin (2003)

  36. Hazan E., Safra M., Schwartz O.: On the complexity of approximating k-set packing. Comput. Complex. 15(1), 20–39 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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. Discret. Math. 2, 68–72 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Jain K.: A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica 21(1), 39–60 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kann V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Inf. Process. Lett. 37(1), 27–35 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  40. Knuth D.E.: The sandwich theorem. Electron. J. Comb. 1, 1–49 (1994)

    Google Scholar 

  41. Laurent M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0-1 programming. Math. Oper. Res. 28(3), 470–496 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lovász L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)

    Article  MATH  Google Scholar 

  43. Lovász L., Schrijver A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  44. Magen, A., Moharrami, M.: Robust algorithms for maximum independent set on minor-free graphs based on the Sherali-Adams hierarchy. In: Proceedings of APPROX 2009, pp. 258–271 (2009)

  45. Mathieu, C., Sinclair, A.: Sherali-Adams relaxations of the matching polytope. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 293–302 (2009)

  46. Matoušek J., Nešeřril J.: An Invitation to Discrete Mathematics. Clarendon Press, Oxford (2008)

    Google Scholar 

  47. Padberg M.W.: On the facial structure of set packing polyhedra. Math. Program. 5(1), 199–215 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  48. Schoenebeck, G.: Linear level lasserre lower bounds for certain k-CSPs. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 593–602 (2008)

  49. Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 453–461 (2001)

  50. Tulsiani, M.: CSP gaps and reductions in the lasserre hierarchy. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), pp. 303–312 (2009)

  51. Tuza Zs.: Critical hypergraphs and intersecting set-pair systems. J. Comb. Theory (B) 39, 134–145 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tuza Zs.: On two intersecting set systems and k-continuous boolean functions. Discret. Appl. Math. 16, 183–185 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

    Yuk Hei Chan & Lap Chi Lau

Authors
  1. Yuk Hei Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lap Chi Lau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lap Chi Lau.

Additional information

Lap Chi Lau was supported by GRF project 413609. Part of this work was done while visiting Microsoft Research New England.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chan, Y.H., Lau, L.C. On linear and semidefinite programming relaxations for hypergraph matching. Math. Program. 135, 123–148 (2012). https://doi.org/10.1007/s10107-011-0451-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-011-0451-5

Keywords

  • Linear programming
  • Semidefinite programming
  • Hypergraph matching
  • Rounding algorithm

Mathematics Subject Classification (2000)

  • 90C05
  • 90C22
  • 90C27
  • 68W25
  • 05C65