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

@article{Halldrsson2006GreedIG,
title={Greed is good: Approximating independent sets in sparse and bounded-degree graphs},
journal={Algorithmica},
year={2006},
volume={18},
pages={145-163}
}
• Published 2006
• Mathematics, Computer Science
• Algorithmica
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… Expand
267 Citations
Independent Sets in Bounded-Degree Hypergraphs
• Mathematics, Computer Science
• 2007
A general technique is proposed that reduces the worst case analysis of certain algorithms to their performance in the case of ordinary graphs and shows that the greedy algorithm that corresponds to the classical greedy set cover algorithm has a performance ratio of (Δ + 1)/2. Expand
Simple and Local Independent Set Approximation
• Mathematics, Computer Science
• SIROCCO
• 2018
A randomized approach of Boppana forms a simple 1-round distributed algorithm, as well as a streaming and preemptive online algorithm, which is shown to give a tight $(Delta+1)/2$-approximation in unweighted graphs of maximum degree $\Delta$, which is best possible for 1- round distributed algorithms. Expand
Ultimate greedy approximation of independent sets in subcubic graphs
• Computer Science, Mathematics
• SODA
• 2020
The main contribution is a new mathematical theory for the design of such greedy algorithms with efficiently computable advice and for the analysis of their approximation ratios, which achieves the ultimate approximation ratio of 5/4 for greedy on graphs with maximum degree 3. Expand
A note on the Greedy algorithm for finding independent sets of Ck-free graphs
• Mathematics, Computer Science
• Inf. Process. Lett.
• 2009
This paper shows that if the input graph is assumed to be C"k-free then the greedy algorithm obtains an independent set of size at least 2n/(d@?+1+k), and proves that the LP-based algorithm has the performance ratio (d@?]+1-k)/4. Expand
Randomized Greedy Algorithms for Independent Sets and Matchings in Regular Graphs: Exact Results and Finite Girth Corrections
• Computer Science, Mathematics
• Combinatorics, Probability and Computing
• 2009
The results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings, and show that the cardinality of independent sets and matchings produced by the greedy algorithm in arbitrary bounded-degree graphs is concentrated around the mean. Expand
Independent sets in bounded-degree hypergraphs
• Computer Science, Mathematics
• Discret. Appl. Math.
• 2009
A general technique is proposed that reduces the worst case analysis of certain algorithms on hypergraphs to their analysis on ordinary graphs and shows that the greedy algorithm for MIS that corresponds to the classical greedy set cover algorithm has a performance ratio of (@D+1)/2. Expand
Improved (In-)Approximability Bounds for d-Scattered Set
• Computer Science, Mathematics
• WAOA
• 2019
A lower bound of $\Delta^{\lfloor d/2\rfloor-\epsilon}$ on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree $\Delta$ and an improved upper bound of $O(\Delta^{n^{1-\ epsilon}}{\rho d}\rho(d+\rho)}}$ are shown. Expand
Sequential Algorithms and Independent Sets Discovering on Large Sparse Random Graphs
• Computer Science, Mathematics
• ArXiv
• 2020
It is shown that a low complexity degree-greedy exploration is actually asymptotically optimal on a large class of sparse random graphs, and two variants of sequential exploration algorithms are presented: static and dynamic degree-aware explorations, and hydrodynamic limits are derived for both. Expand
The max quasi-independent set Problem
• Computer Science, Mathematics
• CSR
• 2010
This paper deals with the problem of finding quasi-independent sets in graphs, formally defined in three versions, which are shown to be polynomially equivalent, and shows an exact solution method that runs within time \$O*(2-27/23}{d+1}n}) on graphs of average degree bounded by d. Expand
Experimental Evaluation of the Greedy and Random Algorithms for Finding Independent Sets in Random Graphs
• Mathematics, Computer Science
• WEA
• 2005
The results of an experimental investigation of the comparative performance of several efficient heuristics for constructing maximal independent sets are presented, suggesting that the lower bound is asymptotically tight. Expand

References

SHOWING 1-10 OF 46 REFERENCES
Improved Approximations of Independent Sets in Bounded-Degree Graphs via Subgraph Removal
• Mathematics, Computer Science
• Nord. J. Comput.
• 1994
An algorithm schema for improving the approximation of algorithms for this problem, which is based on preprocessing the input by removing cliques is introduced, and an implementation of a theorem on the independence number of clique-free graphs is given, leading to the first o(Δ) ratio for the independent set problem. Expand
Improved Approximations of Independent Sets in Bounded-Degree Graphs via Subgraph Removal
Finding maximum independent sets in graphs with bounded maximum degree is a well-studied NP -complete problem. We introduce an algorithm schema for improving the approximation of algorithms for thisExpand
Lower bounds on the independence number in terms of the degrees
• J. R. Griggs
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 1983
It is proved here that if G is a connected triangle-free graph on n ≥ 3 vertices and ifG is neither an odd cycle nor an odd path, then the bound above can be increased by nΔ(Δ + 1), where Δ is the maximum degree. Expand
Derandomized graph products
• Mathematics, Computer Science
• computational complexity
• 2005
This paper proves a lower bound for the probability that all steps of a random walk stay within a specified set of vertices of a graph, which extends also to the case where different sets of Vertices are specified for different time steps of the walk. Expand
On the Approximation Properties of Independent Set Problem in Degree 3 Graphs
• Mathematics, Computer Science
• 1995
This paper shows that the Independent Set problem for bounded degree graphs remains MAX SNP-complete when the maximum degree is bounded by 3, and studies better poly-time approximation of the problem for degree 3 graphs, and improves the previously best ratio. Expand
Locality in Distributed Graph Algorithms
• N. Linial
• Mathematics, Computer Science
• SIAM J. Comput.
• 1992
This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data. Expand
Greedy Approximations of Independent Sets in Low Degree Graphs
• Mathematics, Computer Science
• ISAAC
• 1995
This work investigates the power of a family of greedy algorithms for the independent set problem graphs of maximum degree three, and presents two such algorithms that run in linear time, and shows their performance ratios to be 3/2 and 9/7 ≈ 1.28. Expand
Parallel symmetry-breaking in sparse graphs
• Computer Science, Mathematics
• STOC
• 1987
Efficient deterministic techniques for breaking symmetry in parallel are described and applied to construct fast linear processor algorithms for several problems, including (&Dgr; + 1)-coloring constant-degree graphs, 5-coloring planar graphs, and finding depth-first-search trees inPlanar graphs. Expand
On Approximate Solutions for Combinatorial Optimization Problems
• H. Simon
• Mathematics, Computer Science
• SIAM J. Discret. Math.
• 1990
In this paper, continuous reductions are used for the analysis of several basic combinatorial problems including graph coloring, consistent deterministic finite automaton, covering by cliques, covered by complete bipartite subgraphs, independent set, set packing, and others. Expand
Approximation Algorithms for the Chromatic Sum
• Mathematics, Computer Science
• Great Lakes Computer Science Conference
• 1989
It is proved that a simple greedy algorithm applied to sparse graphs gives a "good" approximation of the chromatic sum, the smallest total among all proper colorings of G using natural numbers. Expand