Approximately Counting Locally-Optimal Structures

Maximal independent sets in bipartite graphs obtained from Boolean lattices

Parallel algorithms for fractional and maximal independent sets in planar graphs

A grid graph is the Cartesian product of two path graphs. Enumerating all maximal independent sets in a graph is a well‐known combinatorial problem. For a general graph, it is . In this work, we provide a polynomial‐time algorithm to generate the whole family of maximal independent sets (mis) of complete grid graphs with two rows. The same algorithm is used in two particular cases: chordless paths and cycles. We apply this result to characterize the independent graph (intersection graph of maximal independent sets) of these three classes of graphs. We also present an alternative proof of Euler's result for grid graphs with three rows that can be used for enumerating the family of mis.

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.

Linear-time algorithms for counting independent sets in bipartite permutation 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.

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

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 → ∞.

Understanding the complexity of approximately counting the number of weighted or unweighted independent sets in a bipartite graph (#BIS) is a central open problem in the field of approximate counting. Here we consider a subclass of this problem and give an FPTAS for approximating the partition function of the hard-core model for bipartite graphs when there is sufficient imbalance in the degrees or fugacities between the sides (L, R) of the bipartition. This includes, among others, the biregular case when λ = 1 (approximating the number of independent sets of G) and ΔR ≥ 7ΔL log(ΔL). Our approximation algorithm is based on truncating the cluster expansion of a polymer model partition function that expresses the hard-core partition function in terms of deviations from independent sets that are empty on one side of the bipartition. Further consequences of this method for unbalanced bipartite graphs include an efficient sampling algorithm for the hard-core model and zero-freeness results for the partition function with complex fugacities. By utilizing connections between the cluster expansion and joint cumulants of certain random variables, we go beyond previous algorithmic applications of the cluster expansion to prove that the hard-core model exhibits exponential decay of correlations for all graphs and fugacities satisfying our conditions. This illustrates the applicability of statistical mechanics tools to algorithmic problems and refines our understanding of the connections between different methods of approximate counting.

Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well studied for various classes of graphs. When it comes to random graphs, the classic Erdős–Rényi–Gilbert random graph [Formula: see text] has been analyzed and shown to have the largest independent sets of size [Formula: see text] with high probability (w.h.p.) This classic model does not capture any dependency structure between edges that can appear in real-world networks. We define random graphs [Formula: see text] whose existence of edges is determined by a Markov process that is also governed by a decay parameter [Formula: see text]. We prove that w.h.p. [Formula: see text] has independent sets of size [Formula: see text] for arbitrary [Formula: see text]. This is derived using bounds on the terms of a harmonic series, a Turán bound on a stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Because [Formula: see text] collapses to [Formula: see text] when there is no decay, it follows that having even the slightest bit of dependency (any [Formula: see text]) in the random graph construction leads to the presence of large independent sets, and thus, our random model has a phase transition at its boundary value of r = 1. This implies that there are large matchings in the line graph of [Formula: see text], which is a Markov random field. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most [Formula: see text] w.h.p. when the lowest degree vertex is picked at each iteration and also show that, under any other permutation of vertices, the algorithm outputs a set of size [Formula: see text], where [Formula: see text] and, hence, has a performance ratio of [Formula: see text]. Funding: The initial phase of this research was supported by the National Science Foundation [Grant DMS-1913294].