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Approximation Algorithms for Connected Maximum Cut and Related Problems

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9294)

Abstract

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V,E) and the goal is to find a subset of vertices S ⊆ V that maximizes the number of edges in the cut δ(S) such that the induced graph G[S] is connected. We present the first non-trivial \(\Omega(\frac{1}{\log n})\) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connected maximum cut problem remains NP-hard even on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus.

Keywords

  • Approximation Algorithm
  • Polynomial Time
  • Planar Graph
  • Facility Location Problem
  • Connected Subgraph

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.

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References

  1. Badanidiyuru, A., Vondrák, J.: Fast algorithms for maximizing submodular functions. In: SODA, pp. 1497–1514 (2014)

    Google Scholar 

  2. Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: A tight linear time (1/2)-approximation for unconstrained submodular maximization. In: FOCS, pp. 649–658 (2012)

    Google Scholar 

  3. Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: Submodular maximization with cardinality constraints. In: SODA, pp. 1433–1452 (2014)

    Google Scholar 

  4. Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint. In: IPCO, pp. 182–196 (2007)

    Google Scholar 

  5. Censor-Hillel, K., Ghaffari, M., Giakkoupis, G., Haeupler, B., Kuhn, F.: Tight bounds on vertex connectivity under vertex sampling. In: SODA (2015)

    Google Scholar 

  6. Censor-Hillel, K., Ghaffari, M., Kuhn, F.: A new perspective on vertex connectivity. In: SODA, pp. 546–561 (2014)

    Google Scholar 

  7. Chekuri, C., Ene, A.: Submodular Cost Allocation Problem and Applications. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 354–366. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  8. Cygan, M.: Deterministic parameterized connected vertex cover. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 95–106. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  9. Das, B., Bharghavan, V.: Routing in ad-hoc networks using minimum connected dominating sets. In: ICC, vol. 1, pp. 376–380 (1997)

    Google Scholar 

  10. de Berg, M., Khosravi, A.: Finding perfect auto-partitions is NP-hard. In: EuroCG 2008, pp. 255–258 (2008)

    Google Scholar 

  11. Demaine, E.D., Hajiaghayi, M., Kawarabayashi, K.-I.: Contraction decomposition in H-minor-free graphs and algorithmic applications. In: STOC, pp. 441–450 (2011)

    Google Scholar 

  12. Du, D.Z., Wan, P.J.: Connected dominating set: theory and applications. Springer optimization and its applications (2013)

    Google Scholar 

  13. Eisenbrand, F., Grandoni, F., Rothvoß, T., Schäfer, G.: Approximating connected facility location problems via random facility sampling and core detouring. In: SODA, pp. 1174–1183 (2008)

    Google Scholar 

  14. Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: STOC, pp. 448–455 (2003)

    Google Scholar 

  15. Feige, U., Mirrokni, V.S., Vondrak, J.: Maximizing non-monotone submodular functions. SIAM Journal on Computing 40(4), 1133–1153 (2011)

    MathSciNet  CrossRef  MATH  Google Scholar 

  16. Felzenszwalb, P.F., Huttenlocher, D.P.: Efficient graph-based image segmentation. International Journal of Computer Vision 59(2), 167–181 (2004)

    CrossRef  Google Scholar 

  17. Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group Steiner tree problem. In: SODA, pp. 253–259 (1998)

    Google Scholar 

  18. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42(6), 1115–1145 (1995)

    MathSciNet  CrossRef  MATH  Google Scholar 

  19. Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. Hadlock, F.: Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing 4(3), 221–225 (1975)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Haglin, D.J., Venkatesan, S.M.: Approximation and intractability results for the maximum cut problem and its variants. IEEE Transactions on Computers 40(1), 110–113 (1991)

    MathSciNet  CrossRef  Google Scholar 

  22. Hajiaghayi, M.T., Kortsarz, G., MacDavid, R., Purohit, M., Sarpatwar, K.: Approximation algorithms for connected maximum cut and related problems. CoRR (2015), http://arxiv.org/abs/1507.00648

  23. Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing 37(1), 319–357 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  24. Khuller, S., Purohit, M., Sarpatwar, K.K.: Analyzing the optimal neighborhood: Algorithms for budgeted and partial connected dominating set problems. In: SODA, pp. 1702–1713 (2014)

    Google Scholar 

  25. Kuo, T.-W., Lin, K.C.-J., Tsai, M.-J.: Maximizing submodular set function with connectivity constraint: theory and application to networks. In: INFOCOM, pp. 1977–1985 (2013)

    Google Scholar 

  26. Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press (1995)

    Google Scholar 

  27. Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions-I. Mathematical Programming 14(1), 265–294 (1978)

    MathSciNet  CrossRef  MATH  Google Scholar 

  28. Slav Petrov. Image segmentation with maximum cuts (2005)

    Google Scholar 

  29. Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: STOC, pp. 255–264 (2008)

    Google Scholar 

  30. Solis-Oba, R.: 2-Approximation algorithm for finding a spanning tree with maximum number of leaves. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 441–452. Springer, Heidelberg (1998)

    Google Scholar 

  31. Swamy, C., Kumar, A.: Primal–dual algorithms for connected facility location problems. Algorithmica 40(4), 245–269 (2004)

    MathSciNet  CrossRef  MATH  Google Scholar 

  32. Vicente, S., Kolmogorov, V., Rother, C.: Graph cut based image segmentation with connectivity priors. In: CVPR, pp. 1–8 (2008)

    Google Scholar 

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Correspondence to Mohammad Taghi Hajiaghayi .

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Hajiaghayi, M.T., Kortsarz, G., MacDavid, R., Purohit, M., Sarpatwar, K. (2015). Approximation Algorithms for Connected Maximum Cut and Related Problems. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_58

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  • DOI: https://doi.org/10.1007/978-3-662-48350-3_58

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