The Greedy Independent Set in a Random Graph with Given Degrees

We analyse the size of an independent set in a random graph on n vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as n→∞, expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges to the jamming constant as n→∞. The results hold under weaker assumptions in a random multigraph with given degrees constructed via the configuration model.

[1]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[2]  Steven R. Finch,et al.  Mathematical constants , 2005, Encyclopedia of mathematics and its applications.

[3]  James W. Evans,et al.  Random and cooperative sequential adsorption , 1993 .

[4]  David Gamarnik,et al.  Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections , 2008, Combinatorics, Probability and Computing.

[5]  Nicholas C. Wormald,et al.  Large independent sets in regular graphs of large girth , 2007, J. Comb. Theory, Ser. B.

[6]  E. S. Page.  The Distribution of Vacancies on a Line , 1959 .

[7]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[8]  Paul J. Flory,et al.  Intramolecular Reaction between Neighboring Substituents of Vinyl Polymers , 1939 .

[9]  N. Wormald.  Models of random regular graphs , 2010 .

[10]  Svante Janson,et al.  Law of large numbers for the SIR epidemic on a random graph with given degrees , 2013, Random Struct. Algorithms.

[11]  Alan M. Frieze,et al.  On the Independence Number of Random Cubic Graphs , 1994, Random Struct. Algorithms.

[12]  Colin McDiarmid,et al.  Topics in Chromatic Graph Theory: Colouring random graphs , 2015 .

[13]  Svante Janson,et al.  A simple solution to the k-core problem , 2007, Random Struct. Algorithms.

[14]  P. Bermolen,et al.  The jamming constant of uniform random graphs , 2013, 1310.8475.

[15]  Svante Janson,et al.  A new approach to the giant component problem , 2007, Random Struct. Algorithms.

[16]  Svante Janson.  The probability that a random multigraph is simple. II , 2014, J. Appl. Probab..

[17]  W. Feller,et al.  An Introduction to Probability Theory and Its Application. , 1951 .

[18]  M. Penrose.  Random Parking, Sequential Adsorption,¶and the Jamming Limit , 2001 .

[19]  Béla Bollobás,et al.  The independence ratio of regular graphs , 1981 .

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

[21]  Lucas Gerin.  The Page-Rényi Parking Process , 2015, Electron. J. Comb..

[22]  John P. Mullooly.  A one dimensional random space-filling problem , 1968 .

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

[24]  David Gamarnik,et al.  Combinatorial approach to the interpolation method and scaling limits in sparse random graphs , 2010, STOC '10.

[25]  Peter J. Diggle,et al.  Statistical Analysis of Spatial and Spatio-Temporal Point Patterns , 2013 .

[26]  A. Cadilhe,et al.  Random sequential adsorption: from continuum to lattice and pre-patterned substrates , 2006, cond-mat/0611413.

[27]  Our Molecular Physics Correspondent,et al.  What is Random Packing? , 1972, Nature.

[28]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[29]  Svante Janson.  The Probability That a Random Multigraph is Simple , 2009, Comb. Probab. Comput..

[30]  Svante Janson.  The probability that a random multigraph is simple. II , 2014, Journal of Applied Probability.

[31]  Allan Sly,et al.  Maximum independent sets on random regular graphs , 2013, 1310.4787.

[32]  The Page-R enyi parking process , 2015 .

[33]  P. Bermolen,et al.  The Jamming Constant of Random Graphs , 2013 .