Parallel algorithms for fractional and maximal independent sets in planar graphs

Maximal independent sets in bipartite graphs obtained from Boolean lattices

A locally-optimal structure is a combinatorial structure that cannot be improved by certain (greedy) local moves, even though it may not be globally optimal. An example is a maximal independent set in a graph. It is trivial to construct an independent set in a graph. It is easy to (greedily) construct a maximal independent set. However, it is NP-hard to construct a globally-optimal (maximum) independent set.This situation is typical. Constructing a locally-optimal structure is somewhat more difficult than constructing an arbitrary structure, and constructing a globally-optimal structure is more difficult than constructing a locally-optimal structure. The same situation arises with listing. The differences between the problems become obscured when we move from listing to counting because nearly everything is \(\#\text {P} \)-complete. However, we highlight an interesting phenomenon that arises in approximate counting, where approximately counting locally-optimal structures is apparently more difficult than approximately counting globally-optimal structures. Specifically, we show that counting maximal independent sets is complete for \(\#\text {P} \) with respect to approximation-preserving reductions, whereas counting all independent sets, or counting maximum independent sets is complete for an apparently smaller class, #RH\(\varPi _1\) which has a prominent role in the complexity of approximate counting. Motivated by the difficulty of approximately counting maximal independent sets in bipartite graphs, we also study counting problems involving minimal separators and minimal edge separators (which are also locally-optimal structures). Minimal separators have applications via fixed-parameter-tractable algorithms for constructing triangulations and phylogenetic trees. Although exact (exponential-time) algorithms exist for listing these structures, we show that the counting problems are as hard as they could possibly be. All of the exact counting problems are \(\#\text {P} \)-complete, and all of the approximation problems are complete for \(\#\text {P} \) with respect to approximation-preserving reductions. A full version [14] containing detailed proofs is available at http://arxiv.org/abs/1411.6829. Theorem-numbering here matches the full version.

Maximal independent sets in grid graphs

The main result of this paper is an algorithm counting maximal independent sets in graphs with maximum degree at most 3 in time O *(1.2570n) and polynomial space.KeywordsMaximum DegreeRecursive CallInternal VertexPrimal GraphSparse GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

J.Y.-T. Leung (J. Algorithms, no.5, (1984)) presented algorithms for generating all the maximal independent sets in interval graphs and circular-arc graphs. The algorithms take O(n/sup 2/+ beta ) steps, where beta is the sum of the number of nodes in all maximal independent sets. The authors use a new technique to give fast and efficient algorithms for finding all the maximum weight independent sets in interval graphs and circular-arc graphs. The algorithms take O(max(n/sup 2/, beta )) steps in O(n/sup 2/) space, where beta is the sum of the number of nodes in all maximum weight independent sets. The algorithms can be directly applied for finding a maximum weight independent set in these graphs in O(n/sup 2/) steps. Thus, the result is an improvement over the best known result of O(n/sup 2/ log n) for finding the maximum weight independent set in circular-arc graphs. >

In this paper, we give a class of graphs that do not admit disjoint maximum and maximal independent (MMI) sets. The concept of inverse independence was introduced by Bhat and Bhat in [Inverse independence number of a graph, Int. J. Comput. Appl. 42(5) (2012) 9–13]. Let [Formula: see text] be a [Formula: see text]-set in [Formula: see text]. An independent set [Formula: see text] is called an inverse independent set with respect to [Formula: see text]. The inverse independence number [Formula: see text] is the size of the largest inverse independent set in [Formula: see text]. Bhat and Bhat gave few bounds on the independence number of a graph, we continue the study by giving some new bounds and exact value for particular classes of graphs: spider tree, the rooted product and Cartesian product of two particular graphs.

Let mi(G) be the number of maximal independent sets in a graph G. A graph G is mi-minimal if mi(H) 2. Hence the extremal problem of calculating m(k) = max{IV(G)1: G is a mi-minimal graph with mi(G) = k} has a solution for any k ~ 1 We show that 2(k -1) ~ m(k) ~ k(k -1) for any k ~ andconjecture that m(k) = 2(k - 1). We also prove NP-completeness of some related problems.

In this lecture and the next we develop a probabilistic NC algorithm of Luby for finding a maximal independent set in an undirected graph. Recall that a set of vertices of a graph is independent if the induced subgraph on those vertices has no edges. A maximal independent set is one contained in no larger independent set. A maximal independent set need not be of maximum cardinality among all independent sets in the graph.

A parallel algorithm for finding a near-maximum independent set in a circle graph is presented. An independent set in a graph is a set of vertices, no two of which are adjacent. A maximum independent set is an independent set whose cardinality is the largest among all independent sets of a graph. The algorithm is modified for predicting the secondary structure in ribonucleic acids (RNA). The proposed system, composed of an n neural network array (where n is the number of edges in the circle graph of the number of possible base pairs), not only generates a near-maximum independent set but also predicts the secondary structure of ribonucleic acids within several hundred iteration steps. The simulator discovered several solutions which are more stable structures, in a sequence of 359 bases from the potato spindle tuber viroid, than previously proposed structures.

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.

Theminimum-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 Turan'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.

Counting independent sets in graphs

It is well known [9] that finding a maximal independent set in a graph is in class NC and [10] that finding a maximal independent set in a hypergraph with fixed dimension is in RNC. It is not known whether this latter problem remains in NC when the dimension is part of the input. We will study the problem when the problem instances are randomly chosen. It was shown in [6] that the expected running time of a simple parallel algorithm for finding the lexicographically first maximal independent set (Ifmis) in a random simple graph is logarithmic in the input size. In this paper, we will prove a generalization of this result. We show that if a random k-uniform hypergraph has vertex set {1, 2, …, n} and its edges are chosen independently with probability p from the set of (nk) possible edges, then our algorithm finds the Ifmis in O() expected time. The hidden constant is independent of k, p. © 1996 John Wiley & Sons, Inc. Random Struct. Alg., 9, 359–377 (1996)

Finding maximum independent sets in graphs with bounded maximum degree Δ is a well-studied NP-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.We give an implementation of a theorem on the independence number of clique-free graphs, and use it to obtain an O(Δ/log log Δ) performance ratio with our schema. This is the first o(Δ) ratio for the independent set problem. We also obtain an efficient method with a Δ/6(1 + o(1)) performance ratio, improving on the best performance ratio known for intermediate values of Δ.

Let F:0,1n⟶0,1n be a parallel dynamical system over an undirected graph with a Boolean maxterm or minterm function as a global evolution operator. It is well known that every periodic point has at most two periods. Actually, periodic points of different periods cannot coexist, and a fixed point theorem is also known. In addition, an upper bound for the number of periodic points of F has been given. In this paper, we complete the study, solving the minimum number of periodic points’ problem for this kind of dynamical systems which has been usually considered from the point of view of complexity. In order to do this, we use methods based on the notions of minimal dominating sets and maximal independent sets in graphs, respectively. More specifically, we find a lower bound for the number of fixed points and a lower bound for the number of 2-periodic points of F. In addition, we provide a formula that allows us to calculate the exact number of fixed points. Furthermore, we provide some conditions under which these lower bounds are attained, thus generalizing the fixed-point theorem and the 2-period theorem for these systems.