Large independent sets on random d-regular graphs with d small

Theoretical Computer Science, 2013

We present an O * (1.0836 n )-time algorithm for finding a maximum independent set in an n-vertex graph with degree bounded by 3, which improves all previous running time bounds for this problem. Our approach has the following two features. Without increasing the number of reduction/branching rules to get an improved time bound, we first successfully extract the essence from the previously known reduction rules such as domination, which can be used to get simple algorithms. More formally, we introduce a procedure for computing "confining sets," which unifies several known reducible subgraphs and covers new reducible subgraphs. Second we identify those instances that generate the worst recurrence among all recurrences of our branching rules as "bottleneck instances" and prove that bottleneck instances cannot appear consecutively after each branching operation.

Theoretical Computer Science, 2008

We investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs. In particular, (a) we propose a new model for random intersection graphs (G n,m, p) which includes the model of [M. Karoński, E.R. Scheinerman, K.B. Singer-Cohen, On random intersection graphs: The subgraph problem, Combinatorics, Probability and Computing journal 8 (1999), 131-159] (the ''uniform'' random intersection graph models) as an important special case. We also define an interesting variation of the model of random intersection graphs, similar in spirit to random regular graphs. (b) For this model we derive exact formulae for the mean and variance of the number of independent sets of size k (for any k) in the graph. (c) We then propose and analyse three algorithms for the efficient construction of large independent sets in this model. The first two are variations of the greedy technique while the third is a totally new algorithm. Our algorithms are analysed for the special case of uniform random intersection graphs. Our analyses show that these algorithms succeed in finding close to optimal independent sets for an interesting range of graph parameters.

Algorithm theory--SWAT'94: 4th Scandinavian …, 1994

Finding maximum independent sets in graphs with bounded maximum degree ∆ is a well-studied N P -complete problem. We introduce an algorithm schema for improving the approximation of algorithms for this problem, which is based on preprocessing the input by removing cliques.

BIT Numerical Mathematics, 1992

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to O(n=(log n) 2 ). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee for the clique and coloring problems.

Encyclopedia of Algorithms, 2008

We investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs. In particular, (a) we propose a new model for random intersection graphs (G n,m, p ) which includes the model of [10] (the "uniform" random intersection graphs model) as an important special case. We also define an interesting variation of the model of random intersection graphs, similar in spirit to random regular graphs. (b) For this model we derive exact formulae for the mean and variance of the number of independent sets of size k (for any k) in the graph. (c) We then propose and analyse three algorithms for the efficient construction of large independent sets in this model. The first two are variations of the greedy technique while the third is a totally new algorithm. Our algorithms are analysed for the special case of uniform random intersection graphs.

Computing Research Repository, 2007

We develop an experimental algorithm for the exact solving of the maximum independent set problem. The algorithm consecutively finds the maximal independent sets of vertices in an arbitrary undirected graph such that the next such set contains more elements than the preceding one. For this purpose, we use a technique, developed by Ford and Fulkerson for the finite partially ordered sets, in particular, their method for partition of a poset into the minimum number of chains with finding the maximum antichain. In the process of solving, a special digraph is constructed, and a conjecture is formulated concerning properties of such digraph. This allows to offer of the solution algorithm. Its theoretical estimation of running time equals to is O(n 8 ), where n is the number of graph vertices. The offered algorithm was tested by a program on random graphs. The testing the confirms correctness of the algorithm.

Information Processing Letters, 1992

Paschos, V.Th., A (A/2)-approximation algorithm for the maximum independent set problem, Information Processing Letters 43 (1992) 11-13. We show how we can improve the approximation ratio for the independent set problem on a graph. We obtain thus an approximation ratio asymptotically equal to A/2, in place of A-1, where A is the maximum degree of the graph.

Lecture Notes in Computer Science, 2011

We present a simple algorithm for the maximum independent set problem in an n-vertex graph with degree bounded by 4, which runs in O * (1.1455 n ) time and improves all previous algorithms for this problem. In this paper, we use the "Measure and Conquer method" to analyze the running time bound, and use some good reduction and branching rules with a new idea on setting weights to obtain the improved time bound.

Random Struct. Algorithms, 2021

In this paper, we consider a randomized greedy algorithm for independent sets in $r$-uniform $d$-regular hypergraphs $G$ on $n$ vertices with girth $g$. By analyzing the expected size of the independent sets generated by this algorithm, we show that $\alpha(G)\geq (f(d,r)-\epsilon(g,d,r))n$, where $\epsilon(g,d,r)$ converges to $0$ as $g\rightarrow\infty$ for fixed $d$ and $r$, and $f(d,r)$ is determined by a differential equation. This extends earlier results of Gamarnik and Goldberg for graphs. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the independent sets generated by this algorithm concentrate around the mean asymptotically almost surely.

Journal of the ACM, 1985

A parallel algorithm is presented that accepts as input a graph G and produces a maximal independent set of vertices in G. On a P-RAM without the concurrent write or concurrent read features, the algorithm executes in G((10gn)~) time and uses 0((n/(logn))3) processors, where n is the number of vertices in G. 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.