Greedy Algorithms for Minimisation Problems in Random Regular Graphs
In this paper we introduce a general strategy for approximating the solution to minimisation problems in random regular graphs. We describe how the approach can be applied to the minimum vertex cover (MVC), minimum independent dominating set (MIDS) and minimum edge dominating set (MEDS) problems. In almost all cases we are able to improve the best known results for these problems. Results for the MVC problem translate immediately to results for the maximum independent set problem. We also derive lower bounds on the size of an optimal MIDS.
KeywordsMinimisation Problem Greedy Algorithm Random Graph Regular Graph Vertex Cover
Unable to display preview. Download preview PDF.
- 1.P. Alimonti, T. Calamoneri. Improved approximations of independent dominating set in bounded degree graphs. In Proc. 22nd WG, pp 2–16, LNCS 1197, Springer-Verlag, 1997.Google Scholar
- 2.P. Alimonti, V. Kann. Hardness of approximating problems on cubic graphs. In Proc. 3rd CIAC, pp 288–298. LNCS 1203, Springer-Verlag, 1997.Google Scholar
- 4.G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, M. Protasi. Complexity and Approximation. Springer-Verlag, 1999.Google Scholar
- 5.P. Berman, T. Fujito. Approximating independent sets in degree 3 graphs. In Proc. WADS’95, pp 449–460. LNCS 955, Springer-Verlag, 1995.Google Scholar
- 6.P. Berman, M. Karpinski. On some tighter inapproximability results. Technical Report TR98-29, ECCC, 1998.Google Scholar
- 8.R. L. Burden, J. D. Faires, A. C. Reynolds. Numerical Analysis. Wadsworth Int., 1981.Google Scholar
- 9.P. Crescenzi, V. Kann. A compendium of NP optimization problems. Available at http://www.nada.kth.se/~viggo/wwwcompendium/, 2000.
- 10.W. Duckworth, N. C. Wormald. Linear programming and the worst case analysis of greedy algorithms for cubic graphs. To be submitted. Preprint available from the authors.Google Scholar
- 11.W. Duckworth, N. C. Wormald. Minimum independent dominating sets of random cubic graphs. Submitted to RSA, 2000.Google Scholar
- 14.A. M. Frieze, T. Łuczak. On the independence and chromatic number of random regular graphs. J. Comb. Theory, B 54:123–132, 1992.Google Scholar
- 15.M. R. Garey, D. S. Johnson. Computer and Intractability, a Guide to the Theory of NP-Completeness. Freeman & Company, 1979.Google Scholar
- 19.S. Janson, T. Łuczak, A. Rucýnski. Random Graphs. John Wiley & Sons, 2000.Google Scholar
- 20.V. Kann. On the Approximability of NP-complete Optimization Problems. PhD thesis, Royal Institute of Technology, Stockholm, 1992.Google Scholar
- 23.T. Nierhoff. The k-Center Problem and r-Independent Sets. PhD thesis, Humboldt-Universität zu Berlin, Berlin, 1999.Google Scholar
- 26.N. C. Wormald. The differential equation method for random graph processes and greedy algorithms. In M. Karoński, H. J. Pr:omel, editors, Lectures on Approximation and Randomized Algorithms, pages 73–155. PWN, Warsaw, 1999.Google Scholar
- 27.M. Yannakakis, F. Gavril. Edge dominating sets in graphs. SIAM J. Applied Math., 38(3):364–372, June 1980.Google Scholar
- 28.M. Zito. Randomised Techniques in Combinatorial Algorithmics. PhD thesis, University of Warwick, 1999.Google Scholar