Advertisement

Vertex Cover in Graphs with Locally Few Colors

  • Fabian Kuhn
  • Monaldo Mastrolilli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

In [13], Erdős et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any Δ ≥ 2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than Δ different colors (bounded local colorability), and (ii) adjacent vertices from two color classes induce a complete bipartite graph (biclique coloring).

We investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given. This generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number. Assuming the Unique Game Conjecture, we provide a tight characterization. We prove that it is UGC-hard to improve the approximation ratio of 2 − 2/(Δ + 1) if the given local coloring is not a biclique coloring. A matching upper bound is also provided. Vice versa, when properties (i) and (ii) hold, we present a randomized algorithm with approximation ratio of \(2- \Omega(1)\frac{\ln \ln \Delta}{\ln \Delta}\). This matches known inapproximability results for the special case of bounded degree graphs.

Moreover, we show that the obtained result finds a natural application in a classical scheduling problem, namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time. In a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs G P of incomparable pairs defined in the dimension theory of partial orders. We show that G P satisfies properties (i) and (ii) where Δ − 1 is the maximum number of predecessors (or successors) of each job.

Keywords

approximation local coloring scheduling vertex cover 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ambühl, C., Mastrolilli, M.: Single machine precedence constrained scheduling is a vertex cover problem. Algorithmica 53(4), 488–503 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambühl, C., Mastrolilli, M., Mutsanas, N., Svensson, O.: Scheduling with precedence constraints of low fractional dimension. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 130–144. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Ambühl, C., Mastrolilli, M., Svensson, O.: Approximating precedence-constrained single machine scheduling by coloring. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 15–26. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Ambühl, C., Mastrolilli, M., Svensson, O.: Inapproximability results for sparsest cut, optimal linear arrangement, and precedence constrained scheduling. In: FOCS, pp. 329–337 (2007)Google Scholar
  5. 5.
    Austrin, P., Khot, S., Safra, M.: Inapproximability of vertex cover and independent set in bounded degree graphs. In: IEEE Conference on Computational Complexity, pp. 74–80 (2009)Google Scholar
  6. 6.
    Bansal, N., Khot, S.: Optimal Long-Code test with one free bit. In: Foundations of Computer Science (FOCS), pp. 453–462 (2009)Google Scholar
  7. 7.
    Brightwell, G.R., Scheinerman, E.R.: Fractional dimension of partial orders. Order 9, 139–158 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chekuri, C., Motwani, R.: Precedence constrained scheduling to minimize sum of weighted completion times on a single machine. Discrete Applied Mathematics 98(1-2), 29–38 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chudak, F.A., Hochbaum, D.S.: A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine. Operations Research Letters 25, 199–204 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Correa, J.R., Schulz, A.S.: Single machine scheduling with precedence constraints. Mathematics of Operations Research 30(4), 1005–1021 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162(1), 439–485 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dushnik, B., Miller, E.: Partially ordered sets. American Journal of Mathematics 63, 600–610 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Erdös, P., Füredi, Z., Hajnal, A., Komjáth, P., Rödl, V., Seress, Á.: Coloring graphs with locally few colors. Discrete Mathematics 59(1-2), 21–34 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Felsner, Trotter: On the fractional dimension of partially ordered sets. DMATH: Discrete Mathematics 136, 101–117 (1994)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Felsner, S., Trotter, W.T.: Dimension, graph and hypergraph coloring. Order 17(2), 167–177 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Graham, R., Lawler, E., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, vol. 5, pp. 287–326. North-Holland, Amsterdam (1979)zbMATHGoogle Scholar
  17. 17.
    Hall, L.A., Schulz, A.S., Shmoys, D.B., Wein, J.: Scheduling to minimize average completion time: off-line and on-line algorithms. Mathematics of Operations Research 22, 513–544 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Halldórsson, M.M., Radhakrishnan, J.: Greed is good: Approximating independent sets in sparse and bounded-degree graphs. Algorithmica 18(1), 145–163 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Halperin, E.: Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. SIAM J. Comput. 31(5), 1608–1623 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6, 243–254 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Karakostas, G.: A better approximation ratio for the vertex cover problem. ACM Transactions on Algorithms 5(4) (2009)Google Scholar
  22. 22.
    Karger, D.R., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. J. ACM 45(2), 246–265 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Khot, S.: On the power of unique 2-prover 1-round games. In: STOC, pp. 767–775 (2002)Google Scholar
  24. 24.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci. 74(3), 335–349 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kleinberg, J.M., Goemans, M.X.: The lovász theta function and a semidefinite programming relaxation of vertex cover. SIAM J. Discrete Math. 11(2), 196–204 (1998)CrossRefzbMATHGoogle Scholar
  26. 26.
    Körner, J., Pilotto, C., Simonyi, G.: Local chromatic number and Sperner capacity. Journal on Combinatorial Theory, Series B 95(1), 101–117 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lawler, E.L.: Sequencing jobs to minimize total weighted completion time subject to precedence constraints. Annals of Discrete Mathematics 2, 75–90 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lenstra, J.K., Rinnooy Kan, A.H.G.: The complexity of scheduling under precedence constraints. Operations Research 26, 22–35 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Margot, F., Queyranne, M., Wang, Y.: Decompositions, network flows and a precedence constrained single machine scheduling problem. Operations Research 51(6), 981–992 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nemhauser, G.L., Trotter, L.E.: Vertex packings: Structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Potts, C.N.: An algorithm for the single machine sequencing problem with precedence constraints. Mathematical Programming Study 13, 78–87 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Schulz, A.S.: Scheduling to minimize total weighted completion time: Performance guarantees of LP-based heuristics and lower bounds. In: Cunningham, W.H., Queyranne, M., McCormick, S.T. (eds.) IPCO 1996. LNCS, vol. 1084, pp. 301–315. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  33. 33.
    Schuurman, P., Woeginger, G.J.: Polynomial time approximation algorithms for machine scheduling: ten open problems. Journal of Scheduling 2(5), 203–213 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Fabian Kuhn
    • 1
  • Monaldo Mastrolilli
    • 2
  1. 1.Faculty of InformaticsUniversity of Lugano (USI)LuganoSwitzerland
  2. 2.Dalle Molle Institute for Artificial Intelligence (IDSIA)MannoSwitzerland

Personalised recommendations