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.

[1]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[2]  Mihalis Yannakakis,et al.  Edge Dominating Sets in Graphs , 1980 .

[3]  Alan M. Frieze,et al.  On the independence number of random graphs , 1990, Discret. Math..

[4]  J. Miller.  Numerical Analysis , 1966, Nature.

[5]  B. Bollobás,et al.  Cliques in random graphs , 1976, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  Alan M. Frieze,et al.  On the independence and chromatic numbers of random regular graphs , 1992, J. Comb. Theory, Ser. B.

[7]  N. Wormald.  Differential Equations for Random Processes and Random Graphs , 1995 .

[8]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[9]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[10]  Pierluigi Crescenzi,et al.  A compendium of NP optimization problems , 1994, WWW Spring 1994.

[11]  P. Berman,et al.  On Some Tighter Inapproximability Results , 1998, Electron. Colloquium Comput. Complex..

[12]  David S. Johnson,et al.  Some Simplified NP-Complete Graph Problems , 1976, Theor. Comput. Sci..

[13]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[14]  Nicholas C. Wormald,et al.  Almost All Regular Graphs Are Hamiltonian , 1994, Random Struct. Algorithms.

[15]  Tiziana Calamoneri,et al.  Improved Approximations of Independent Dominating Set in Bounded Degree Graphs , 1996, WG.

[16]  B. Pittel,et al.  Maximum matchings in sparse random graphs: Karp-Sipser revisited , 1998 .

[17]  D. Hochbaum.  Easy Solutions for the K–Center Problem or the Dominating Set Problem on Random Graphs , 1985 .

[18]  Michele A. A. Zito,et al.  Randomised techniques in combinatorial algorithmics , 1999 .

[19]  Piotr Berman,et al.  On the Approximation Properties of Independent Set Problem in Degree 3 Graphs , 1999, WADS.

[20]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[21]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[22]  William Duckworth,et al.  Minimum independent dominating sets of random cubic graphs , 2002, Random Struct. Algorithms.

[23]  B. Korte,et al.  An Analysis of the Greedy Heuristic for Independence Systems , 1978 .

[24]  G. Grimmett,et al.  On colouring random graphs , 1975 .

[25]  Viggo Kann,et al.  Hardness of Approximating Problems on Cubic Graphs , 1997, CIAC.

[26]  Till Nierhoff.  The k-center problem and r-independent sets: a study in probabilistic analysis , 1999 .

[27]  N. Wormald.  The differential equation method for random graph processes and greedy algorithms , 1999 .

[28]  Alan M. Frieze,et al.  Algorithmic theory of random graphs , 1997, Random Struct. Algorithms.

[29]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[30]  Michele Zito.  Small Maximal Matchings in Random Graphs , 2000, LATIN.