Limits of local algorithms over sparse random graphs

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.

This paper presents a self-stabilizing algorithm for computing multiple disjoint maximal independent sets in a graph. The first set is a maximal independent set in the entire graph, the second set is a maximal independent set in the subgraph formed by removing the nodes in the first independent set, and each subsequent set is computed similarly by excluding the nodes from all previous sets. The algorithm introduces a novel method by using a single local integer variable for each node to determine its membership in a specific independent set, offering a more efficient solution compared to earlier methods. Additionally, this algorithm provides an upper bound on the chromatic number required for the graph coloring problem. The solution is developed under the central daemon model and guarantees termination after a finite number of moves.

Finding maximum independent sets in graphs with bounded maximum degree is a well-studied NP-complete problem. We study two approaches for finding approximate solutions, and obtain several improved performance ratios.The first is a subgraph removal schema introduced in our previous paper. Using better component algorithms, we obtain an efficient method with a Δ/6(1+o(1)) performance ratio. We then produce an implementation of a theorem of Ajtai et al. on the independence number of clique-free graphs, and use it to obtain a O(Δ/loglogΔ) performance ratio with our schema. This is the first o(Δ) ratio.The second is a local search method of Berman and Fürer for which they proved a fine performance ratio but by using extreme amounts of time. We show how to substantially decrease the computing requirements while maintaining the same performance ratios of roughly (Δ+3)/5 for graphs with maximum degree Δ. We then show that a scaled-down version of their algorithm yields a (Δ+3)/4 performance, improving on previous bounds for reasonably efficient methods.KeywordsLocal SearchMaximum DegreePerformance RatioFree GraphIndependence NumberThese 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.

Graphs with large minimum degree containing no copy of a clique on $r$ vertices ($K_r$) must contain relatively large independent sets. A classical result of Andrásfai, Erdős, and Sós implies that $K_r$-free graphs $G$ with degree larger than $((3r-7)/(3r-4))|V(G)|$ must be $(r-1)$-partite. An obvious consequence of this result is that the same degree threshold implies an independent set of order $(1/(r-1))|V(G)|$. The following paper provides improved bounds on the minimum degree which would imply the same conclusion. This problem was first considered by Brandt, and we provide improvements over these initial results for $r > 5$.

The maximum independent set problem is a classic and fundamental combinatorial challenge, where the objective is to find the largest subset of vertices in a graph such that no two vertices are adjacent. In this paper, we introduce a novel linear prioritized local algorithm tailored to address this problem on random d-regular graphs with a small and fixed degree d. Through exhaustive numerical simulations, we empirically investigated the independence ratio, i.e., the ratio between the cardinality of the independent set found and the order of the graph, which was achieved by our algorithm across random d-regular graphs with degree d ranging from 5 to 100. Remarkably, for every d within this range, our results surpassed the existing lower bounds determined by theoretical methods. Consequently, our findings suggest new conjectured lower bounds for the MIS problem on such graph structures. This finding has been obtained using a prioritized local algorithm. This algorithm is termed ‘prioritized’ because it strategically assigns priority in vertex selection, thereby iteratively adding them to the independent set.

We give a survey of the enumeration of independent sets in graphs and of some related problems.

Attainable estimates of the number of independent sets in graphs with a given size of the maximal independent set are obtained. Three graph classes—trees, forests, and the class of all graphs—are considered. Extremal graphs are described.

Let k be the asymptotic value of the independence number of the random graph G(n, p). We prove that if the edge probability p(n) satisfies p(n) ≫ n−2/5ln6/5n then the probability that G(n, p) does not contain an independent set of size k − c, for some absolute constant c > 0, is at most exp{−cn2/(k4p)}. We also show that the obtained exponent is tight up to logarithmic factors, and apply our result to obtain new bounds on the choice number of random graphs. We also discuss a general setting where our approach can be applied to provide an exponential bound on the probability of certain events in product probability spaces. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 1–14, 2003

In this paper we derive nonasymptotic upper bounds for the size of reachable sets in random graphs. These bounds are subject to a phase transition phenomenon triggered by the spectral radius of the hazard matrix, a reweighted version of the adjacency matrix. Such bounds are valid for a large class of random graphs, called local positive correlation (LPC) random graphs, displaying local positive correlation. In particular, in our main result we state that the size of reachable sets in the subcritical regime for LPC random graphs is at most of order O(√n), where n is the size of the network, and of order O(n2/3) in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the hazard matrix with respect to 1. As a corollary, we also show that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the independent cascade model.

This work presents some random graphs with their dominating sets found by us. After giving an overview of the literature related to the topic and the route of scientific development leading to the definition of the random graphs, the logical process leading to the dominating sets of random graphs has been presented.

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

We give the first treatment of the classic independent set problem in graphs and hypergraphs in the streaming setting. The objective is to find space-efficient algorithms that output independent sets that are "combinatorially optimal", that is, with size guarantee in terms of the degree sequence alone. Our main result is a randomized algorithm that achieves this using space in bits that is linear in the number of vertices. We use this to examine assumptions about the streaming model, and advocate the study of output-efficient algorithms that measure space usage relative to the size of the output solution. In that sense, our main algorithm uses space linear in the output size. We also examine algorithms that use little or no space in addition to the bits storing the output. Our algorithms fall also into an online streaming model, where output-changes can go only in one direction. In particular a feasible solution must be maintained at all times, and items that are removed from the solution can never reenter. We obtain tight bounds on deterministic algorithms for independent sets in graphs in that model.

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random d-regular graph on n vertices, and let \lambda be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon [86'] that a random d-regular graph is almost Ramanujan, in the following sense: for every e>0, \lambda<2\sqrt{d-1} + e asymptotically almost surely. Friedman famously presented a proof of this conjecture in [08']. Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random d-regular graph satisfies \lambda < 2\sqrt{d-1} + 1 a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For d even, a d-regular graph on n vertices is an n-covering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph H, we study the spectrum of G, a random n-covering of H. Let \lambda be the largest absolute value of a non-trivial eigenvalue of G. Extending Alon's conjecture to this more general model, Friedman [03'] conjectured that for every e>0, a.a.s. \lambda < \rho+e, where \rho is the spectral radius of the universal cover of H. When H is regular we get a bound of \rho+0.84, and for an arbitrary H, we prove a nearly optimal upper bound of \sqrt{3}\rho. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph G, each pair of vertices are joined by an edge with a probability p, where p is a constant between 0 and 1. We show that a maximum independent set in a random graph that contains n vertices can be computed in expected computation time . In addition, we show that, with high probability, the parameterized independent set problem is fixed parameter tractable in random graphs and the maximum independent set in a random graph in n vertices can be approximated within a ratio of in expected polynomial time.

An abstract simplicial complex C is said to be k-connected if for each $-1\leq d\leq k$ and each continuous map f from the sphere $S^d$ to ||C|| (the body of the geometric realization of C), the map f can be extended to a continuous map from the ball $B^{d+1}$ to ||C||. In 2000 a link was discovered between the topological connectedness of the independence complex of a graph and various other important graph parameters to do with colouring and partitioning. When the graph represents some other combinatorial structure, for example when it is the line graph of a hypergraph H, this link can be exploited to obtain information such as lower bounds on the matching number of H. Since its discovery there have been many other applications of this phenomenon to combinatorial problems. The aim of this article is to outline this general method and to describe some of its applications.