A Simple Parallel Algorithm for the Maximal Independent Set Problem
Two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set (MIS) problem. The first strategy consists of assigning identical copies o...
A note on greedy algorithms for the maximum weighted independent set problem
In this paper, we consider three simple and natural greedy algorithms for the maximum weighted independent set problem. We show that two of them output an independent set of weight at least Σv∈V(G) W(v)/[d(v) + 1] and the third algorithm outputs an independent set of weight at least Σv∈V(G) W(v)2/[Σu∈NG+(v) W(u)].These results are generalization of theorem of Caro and Wei.
A fast parallel algorithm for the maximal independent set problem
A parallel algorithm is presented that accepts as input a graph <italic>G</italic> and produces a maximal independent set of vertices in <italic>G</italic>. On a P-RAM without the concurrent write or concurrent read features, the algorithm executes in <italic>O</italic>((log <italic>n</italic>)<supscrpt>4</supscrpt>) time and uses <italic>O</italic>((<italic>n</italic>/(log <italic>n</italic>))<supscrpt>3</supscrpt>) processors, where <italic>n</italic> is the number of vertices in <italic>G</italic>. The algorithm has several novel features that may find other applications. These include the use of balanced incomplete block designs to replace random sampling by deterministic sampling, and the use of a “dynamic pigeonhole principle” that generalizes the conventional pigeonhole principle.
A fast parallel algorithm for the maximal independent set problem
A parallel algorithm is presented which accepts as input a graph <italic>G</italic> and produces a maximal independent set of vertices in <italic>G.</italic> On a P-RAM without the concurrent write or concurrent read features, the algorithm executes in <italic>O((log n)</italic><supscrpt>4</supscrpt>) time and uses <italic>O((n/log n)</italic><supscrpt>3</supscrpt>) processors, where <italic>n</italic> is the number of vertices in <italic>G.</italic> The algorithm has several novel features that may find other applications. These include the use of balanced incomplete block designs to replace random sampling by deterministic sampling, and the use of a “dynamic pigeonhole principle” that generalizes the conventional pigeonhole principle.
On the Complexity of Approximating the Independent Set Problem
We show that for some positive constant c it is not feasible to approximate Independent Set (for graphs of n nodes) within a factor of n c , provided Maximum 2-Satisfiability does not have a randomized polynomial time approximation scheme. We also study reductions preserving the quality of approximations and exhibit complete problems.
Journal of Graph Algorithms and Applications Approximations of Weighted Independent Set and Hereditary Subset Problems
The focus of this study is to clarify the approximability of the important versions of the maximum independent set problem, and to apply, where possible, the technique to related hereditary subgraph and subset problem. We report improved performance ratios for the Independent Set problem in weighted general graphs, weighted bounded-degree graphs, and in sparse graphs. Other problems with better than previously reported ratios include Weighted Set Packing, Longest Subsequence, Maximum Independent Sequence, and Independent Set in hypergraphs.
Heuristics for finding large independent sets, with applications to coloring semi-random graphs
We study a semi-random graph model for finding independent sets. For /spl alpha/>0, an n-vertex graph with an independent set S of site /spl alpha/n is constructed by blending random and adversarial decisions. Randomly and independently with probability p, each pair of vertices, such that one is in S and the other is not, is connected by an edge. An adversary can then add edges arbitrarily (provided that S remains an independent set). The smaller p is, the larger the control the adversary has over the semi-random graph. We design heuristics that with high probability recover S when p>(1+/spl epsiv/)ln n/|S|, for any constant /spl epsiv/>0. We show that when p<(1-/spl epsiv/) In n/|S|, an independent set of size |S| cannot be recovered, unless NP/spl sube/BPP. We use our remits to obtain greatly improved coloring algorithms for the model of k-colorable semi-random graphs introduced by A. Blum and J. Spencer (1995).
A fast parallel algorithm for the maximal independent set problem
A parallel algorithm is presented that accepts as input a graph <italic>G</italic> and produces a maximal independent set of vertices in <italic>G</italic>. On a P-RAM without the concurrent write or concurrent read features, the algorithm executes in <italic>O</italic>((log <italic>n</italic>)<supscrpt>4</supscrpt>) time and uses <italic>O</italic>((<italic>n</italic>/(log <italic>n</italic>))<supscrpt>3</supscrpt>) processors, where <italic>n</italic> is the number of vertices in <italic>G</italic>. The algorithm has several novel features that may find other applications. These include the use of balanced incomplete block designs to replace random sampling by deterministic sampling, and the use of a “dynamic pigeonhole principle” that generalizes the conventional pigeonhole principle.
On the Approximation Properties of Independent Set Problem in Degree 3 Graphs
The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNP-complete when the maximum degree is bounded by 3. Some related problems are also shown to be MAX SNP-complete at the lowest possible degree bounds. Next we study better poly-time approximation of the problem for degree 3 graphs, and improve the previously best ratio, 5/4, to arbitrarily close to 6/5. This result also provides improved poly-time approximation ratios, B+3/5+e, for odd degree B.
A survey of approximately optimal solutions to some covering and packing problems
We survey approximation algorithms for some well-known and very natural combinatorial optimization problems, the minimum set covering, the minimum vertex covering, the maximum set packing, and maximum independent set problems; we discuss their approximation performance and their complexity. For already known results, any time we have conceived simpler proofs than those already published, we give these proofs, and, for the rest, we cite the simpler published ones. Finally, we discuss how one can relate the approximability behavior (from both a positive and a negative point of view) of vertex covering to the approximability behavior of a restricted class of independent set problems.
Optimized Crossover for the Independent Set Problem
We propose a knowledge-based crossover mechanism for genetic algorithms that exploits the structure of the solution rather than its coding. More generally, we suggest broad guidelines for constructing the knowledge-based crossover mechanisms. This technique uses an optimized crossover mechanism, in which the one of the two children is constructed in such a way as to have the best objective function value from the feasible set of children, while the other is constructed so as to maintain the diversity of the search space. We implement our approach on a classical combinatorial problem, called the independent set problem. The resulting genetic algorithm dominates all other genetic algorithms for the problem and yields one of the best heuristics for the independent set problem in terms of robustness and time performance. The primary purpose of this paper is to demonstrate the power of knowledge based mechanisms in genetic algorithms.
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