Greed is good: Approximating independent sets in sparse and bounded-degree graphs

AbstractTheminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2)/3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turán’s bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved $$(2\bar d + 3)/5$$ performance ratio on graphs with average degree $$\bar d$$ , improving on the previous best $$(\bar d + 1)/2$$ of Hochbaum. Finally, we present an efficient parallel and distributed algorithm attaining the performance guarantees of Greedy.

[1]  Colin McDiarmid,et al.  Small transversals in hypergraphs , 1992, Comb..

[2]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[3]  L. Lovász.  Three short proofs in graph theory , 1975 .

[4]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[5]  Rajeev Motwani,et al.  On syntactic versus computational views of approximability , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[6]  US M. Halld.  Improved Approximations of Independent Sets in Bounded-Degree Graphs via Subgraph Removal MAGN , 1994 .

[7]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[8]  Andrew V. Goldberg,et al.  Parallel symmetry-breaking in sparse graphs , 1987, STOC.

[9]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[10]  Piotr Berman,et al.  Approximating maximum independent set in bounded degree graphs , 1994, SODA '94.

[11]  Hans Ulrich Simon,et al.  On Approximate Solutions for Combinatorial Optimization Problems , 1990, SIAM J. Discret. Math..

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

[13]  Magnús M. Halldórsson,et al.  Greedy Approximations of Independent Sets in Low Degree Graphs , 1995, ISAAC.

[14]  Noga Alon,et al.  Derandomized graph products , 1995, computational complexity.

[15]  Dionysios Kountanis,et al.  Approximation Algorithms for the Chromatic Sum , 1989, Great Lakes Computer Science Conference.

[16]  R. Steele.  Optimization , 2005 .

[17]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[18]  P. Erdös.  On an extremal problem in graph theory , 1970 .

[19]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[20]  R. L. Brooks.  On Colouring the Nodes of a Network , 1941 .

[21]  Jaikumar Radhakrishnan,et al.  Improved Approximations of Independent Sets in Bounded-Degree Graphs via Subgraph Removal , 1994, Nord. J. Comput..

[22]  Richard M. Karp,et al.  A Survey of Parallel Algorithms for Shared-Memory Machines , 1988 .

[23]  Jerrold R. Griggs.  Lower bounds on the independence number in terms of the degrees , 1983, J. Comb. Theory, Ser. B.

[24]  James B. Shearer,et al.  A note on the independence number of triangle-free graphs , 1983, Discret. Math..

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

[26]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[27]  Dorit S. Hochbaum,et al.  Efficient bounds for the stable set, vertex cover and set packing problems , 1983, Discret. Appl. Math..