We introduce the maximum graph homomorphism (MGH) problem: given a graph G, and a target graph H, find a mapping ϕ:V G V H that maximizes the number of edges of G that are mapped to edges of H. This problem encodes various fundamental NP-hard problems including Maxcut and Max-k-cut. We also consider the multiway uncut problem. We are given a graph G and a set of terminals T ⊆ V G . We want to partition V G into |T| parts, each containing exactly one terminal, so as to maximize the number of edges in E G having both endpoints in the same part. Multiway uncut can be viewed as a special case of prelabeled MGH where one is also given a prelabeling \({\ensuremath{\varphi}}':U\mapsto V_H,\ U{\subseteq} V_G\), and the output has to be an extension of ϕ′.

Both MGH and multiway uncut have a trivial 0.5-approximation algorithm. We present a 0.8535-approximation algorithm for multiway uncut based on a natural linear programming relaxation. This relaxation has an integrality gap of \(\frac{6}{7}\simeq 0.8571\), showing that our guarantee is almost tight. For maximum graph homomorphism, we show that a \(\bigl({\ensuremath{\frac{1}{2}}}+{\ensuremath{\varepsilon}}_0)\)-approximation algorithm, for any constant ε 0 > 0, implies an algorithm for distinguishing between certain average-case instances of the subgraph isomorphism problem that appear to be hard. Complementing this, we give a \(\bigl({\ensuremath{\frac{1}{2}}}+\Omega(\frac{1}{|H|\log{|H|}})\bigr)\)-approximation algorithm.


Approximation Algorithm Random Graph Linear Programming Relaxation Label Graph Graph Homomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aggarwal, G., Feder, T., Motwani, R., Zhu, A.: Channel assignment in wireless networks and classification of minimum graph homomorphism. In: ECCC: TR06-040 (2006)Google Scholar
  2. 2.
    Alekhnovich, M.: More on average case vs approximation complexity. In: Proceedings, 44th FOCS, pp. 298–307 (2003)Google Scholar
  3. 3.
    Arora, S., Karger, D., Karpinski, M.: Polynomial time approximation schemes for dense instances of NP-hard problems. J. Comput. Syst. Sci. 58, 193–210 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Babai, L., Erdös, P., Selkow, S.: Random graph isomorphism. SICOMP 9, 628–635 (1980)zbMATHGoogle Scholar
  5. 5.
    Calinescu, G., Karloff, H., Rabani, Y.: An improved approximation algorithm for multiway cut. Journal of Computer and System Sciences 60, 564–574 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Charikar, M., Wirth, A.: Maximizing quadratic programs: Extending Grothendieck’s inequality. In: Proceedings, 45th FOCS, pp. 54–60 (2004)Google Scholar
  7. 7.
    Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: A maximal tractable class of soft constraints. Journal of Artificial Intelligence Research 22, 1–22 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cooper, C., Dyer, M., Frieze, A.: On Markov chains for randomly H-coloring a graph. Journal of Algorithms 39, 117–134 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dahlhaus, E., Johnson, D., Papadimitriou, C., Seymour, P., Yannakakis, M.: The complexity of multiterminal cuts. SICOMP 23, 864–894 (1994)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Demaine, E.D., Feige, U., Hajiaghayi, M.T., Salavatipour, M.: Combination can be hard: Approximability of the unique coverage problem. In: Proceedings, 17th SODA, pp. 162–171 (2006)Google Scholar
  11. 11.
    Díaz, J., Serna, M.J., Thilikos, D.M.: The complexity of restrictive H-coloring. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 126–137. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Dyer, M.E., Greenhill, C.S.: The complexity of counting graph homomorphisms. Random Structures and Algorithms 25, 346–352 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Feder, T., Hell, P.: List homomorphisms to reflexive graphs. Journal of Combinatorial Theory, Series B 72, 236–250 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings, 34th STOC, pp. 534–543 (2002)Google Scholar
  15. 15.
    Feige, U., Langberg, M.: The RPR2̂ rounding technique for semidefinite programs. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 213–224. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Freund, A., Karloff, H.: A lower bound of \(8/(7+\frac{1}{k-1})\) on the integrality ratio of the Calinescu-Karloff-Rabani relaxation for multiway cut. Information Processing Letters 75, 43–50 (2000)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Frieze, A., McDiarmid, C.: Algorithmic theory of random graphs. Random Structures and Algorithms 10, 5–42 (1997)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Gutin, G., Rafiey, A., Yeo, A., Tso, M.: Level of repair analysis and minimum cost homomorphisms of graphs. Discrete Applied Mathematics 154, 881–889 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford Univ. Press, Oxford (2004)zbMATHCrossRefGoogle Scholar
  22. 22.
    Kann, V.: On the approximability of the maximum common subgraph problem. In: Proceedings, 9th STACS, pp. 377–388 (1992)Google Scholar
  23. 23.
    Karger, D., Klein, P., Stein, C., Thorup, M., Young, N.: Rounding algorithms for a geometric embedding of minimum multiway cut. M. of OR 29, 436–461 (2004)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Kleinberg, J., Tardos, É.: Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields. Journal of the ACM 49, 616–639 (2002)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Turán, P.: On an extremal problem in graph theory. Mat. Fiz. Lapok 48, 436–452 (1941)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Michael Langberg
    • 1
  • Yuval Rabani
    • 2
  • Chaitanya Swamy
    • 3
  1. 1.Dept. of Computer ScienceCaltechPasadenaUSA
  2. 2.Computer Science Dept.Technion — Israel Institute of TechnologyHaifaIsrael
  3. 3.Center for the Mathematics of InformationCaltechPasadenaUSA

Personalised recommendations