Independent Set Algorithm Research Articles

Analysis of the Malatya Centrality-Based Clique Method on DIMACS Benchmarks and Random Graphs

The Maximum Clique Problem (MCP) is an NP-complete problem that aims to find the largest clique (complete subgraph) within a given graph. This problem has widespread applications in fields such as social network analysis, bioinformatics, chemical information systems, and communication networks. Although various solution methods have been proposed in the literature, an effective method that provides a general solution across different graph types is not yet available. In this study, we demonstrate that our previously proposed maximum clique method provides an effective solution for MCP on DIMACS benchmark graphs and random graphs generated using different models such as Erdos-Renyi, Forest Fire, Watts-Strogatz, and Regular Random models. The proposed method begins by computing the complement of the original graph. In the complement graph, the maximum independent set is determined using the Malatya Independent Set Algorithm, an efficient algorithm well-recognized in the literature. The maximum independent set identified in the complement graph is then used to determine the subgraph corresponding to the maximum clique in the original graph. The effectiveness of the proposed method was validated through tests on widely recognized DIMACS benchmark graphs. Additionally, to evaluate its performance and robustness on unpredictable graphs, tests were conducted on random graphs of varying complexities generated using Erdos-Renyi, Forest Fire, Watts-Strogatz, and Regular Random models. The results of these tests demonstrated that the method produced optimal or near-optimal MCP solutions for both DIMACS and random graphs. The successful outcomes further highlight the efficiency of the proposed method and its potential applicability in MCP-related domains.

Iterative quantum algorithms for maximum independent set

Iterative quantum algorithms for maximum independent set

A complete shortest independent loop set algorithm for model structure analysis

AbstractThis article looks at the well‐known shortest independent loop set algorithm, which is used to identify an independent loop set in system dynamics models. It addresses the cases where the algorithm fails to find a complete set due to its limitation to geodetic cycles and introduces a modification that has the potential to identify a complete independent loop set in every case while maintaining the efficiency of the original algorithm. The modified algorithm uses second‐shortest paths to complete the loop set. Special attention is given to the creation of the second‐shortest path matrix required for the modified algorithm, and to the application of the algorithm to a model where the independent loop set created by the shortest independent loop set algorithm was previously shown to be incomplete. © 2024 System Dynamics Society.

Air volume reconstruction and sensor optimization distribution in building intelligent ventilation network

Ensuring the accuracy and reliability of ventilation parameter monitoring is pivotal for the development of intelligent ventilation systems. To attain a visual representation of airflow, solving the challenge of airflow reconstruction necessitates the strategic use of a limited number of sensors. In addressing these concerns, this article introduces an optimization approach for ventilation airflow leveraging the Breadth-First Search (BFS) algorithm. Additionally, it proposes an optimization distribution method for mine ventilation sensors, grounded in the Independent Cut Set algorithm. Research has found that compared to the traditional PSO algorithm, the BFS algorithm produces a higher optimal air volume solution when optimizing the air volume; Comparatively, the proposed algorithm exhibits significantly shorter average running times than the Particle Swarm Optimization (PSO) algorithm. It boasts the highest average convergence rate, ensuring superior accuracy, and possesses a notable capability to escape local minima, facilitating the acquisition of optimal solutions. Leveraging the independent cut set algorithm optimizes the calculation process through matrix operations. Exploiting the properties of matrices allows for a more rapid and intuitive resolution of sensor localization problems.

Computing Solution Space Properties of Combinatorial Optimization Problems Via Generic Tensor Networks

We introduce a unified framework to compute the solution space properties of a broad class of combinatorial optimization problems. These properties include finding one of the optimum solutions, counting the number of solutions of a given size, and enumeration and sampling of solutions of a given size. Using the independent set problem as an example, we show how all these solution space properties can be computed in the unified approach of generic tensor networks. We demonstrate the versatility of this computational tool by applying it to several examples, including computing the entropy constant for hardcore lattice gases, studying the overlap gap properties, and analyzing the performance of quantum and classical algorithms for finding maximum independent sets.

Grid induced minor theorem for graphs of small degree

A graph H is an induced minor of a graph G if H can be obtained from G by vertex deletions and edge contractions. We show that there is a function f(k,d)=O(k10+2d5) so that if a graph has treewidth at least f(k,d) and maximum degree at most d, then it contains a k×k-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon (2021) [1] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph H, there is a subexponential time algorithm for maximum weight independent set on H-induced-minor-free graphs.

An Asynchronous Maximum Independent Set Algorithm By Myopic Luminous Robots On Grids

Abstract We consider the problem of constructing a maximum independent set with mobile myopic luminous robots on a grid network whose size is finite but unknown to the robots. In this setting, the robots enter the grid network one by one from a corner of the grid, and they eventually have to be disseminated on the grid nodes so that the occupied positions form a maximum independent set of the network. We assume that robots are asynchronous, anonymous, silent and they execute the same distributed algorithm. In this paper, we propose two algorithms: The first one assumes that the number of light colors of each robot is three and the visible range is two, but uses the additional assumption that a local edge-labeling exists for each node. To remove this assumption, the second one assumes that the number of light colors of each robot is seven, and that the visible range is three. In both algorithms, the number of movements is $O(n(L+l))$ steps, where $n$ is the number of nodes and $L$ and $l$ are the grid dimensions.

Faster Graph Coloring in Polynomial Space

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.

Loosely-stabilizing maximal independent set algorithms with unreliable communications

Self-stabilization is a promising paradigm for designing highly adaptive distributed systems. However, it cannot be realized when the communication between processes is unreliable with some constant probability. To circumvent such impossibility, this paper adopts the concept of loose-stabilization for the first time, which is a practical alternative of self-stabilization, and proposes three systematic approaches to realize loose-stabilization in the atomic-state model with probabilistically erroneous communications, namely, the redundant-state approach, the step-up approach, and the repetition approach. Further, we apply these approaches to design three corresponding loosely-stabilizing algorithms for the maximal independent set problem.

Distributed minimum vertex coloring and maximum independent set in chordal graphs

We give deterministic distributed (1+ϵ)-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in O(1ϵlog⁡n) rounds, and our independent set algorithm has a runtime of O(1ϵlog⁡(1ϵ)log⁎⁡n) rounds. For coloring, existing lower bounds imply that the dependencies on 1ϵ and log⁡n are best possible. For independent set, we prove that Ω(1ϵ) rounds are necessary.Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O(log⁡n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first O(log⁡1ϵ) layers are required as they already contain a large enough independent set. We develop a (1+ϵ)-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers.While tree decompositions have only played a minor role in distributed computing, our work demonstrates their potential for designing efficient distributed algorithms.

Verification of Karci Algorithm’s Efficiency for Maximum Independent Set Problem in Graph Theory

The maximum independent set problem is an NP-complete problem in graph theory. The Karci Algorithm is based on fundamental cut-sets of given graph, and node with minimum independence values are selected for maximum independent set. In this study, the analytical verification of this algorithm for some special graphs was analysed, and the obtained results were explained. The verification of Karci’s Algorithm for maximum independent set was handled in partial.

Constructing benchmark test sets for biological sequence analysis using independent set algorithms.

Biological sequence families contain many sequences that are very similar to each other because they are related by evolution, so the strategy for splitting data into separate training and test sets is a nontrivial choice in benchmarking sequence analysis methods. A random split is insufficient because it will yield test sequences that are closely related or even identical to training sequences. Adapting ideas from independent set graph algorithms, we describe two new methods for splitting sequence data into dissimilar training and test sets. These algorithms input a sequence family and produce a split in which each test sequence is less than p% identical to any individual training sequence. These algorithms successfully split more families than a previous approach, enabling construction of more diverse benchmark datasets.

A faster algorithm for maximum independent set on interval filament graphs

We provide an algorithm requiring only $O(N^2)$ time to compute the maximum weight independent set in an $N$-vertex interval filament graph. This implies an $O(N^4)$-time algorithm to compute the maximum weight induced matching in such graphs. Both algorithms significantly improve upon the previous best complexities for these problems. Previously, the maximum weight independent set and maximum weight induced matching problems required $O(N^3)$ and $O(N^6)$ time respectively.

Independent sets in hypergraphs omitting an intersection

AbstractA k‐uniform hypergraph with n vertices is an ‐omitting system if it has no two edges with intersection size . If in addition it has no two edges with intersection size greater than , then it is an ‐system. Rödl and Šiňajová proved a sharp lower bound for the independence number of ‐systems. We consider the same question for ‐omitting systems. Our proofs use adaptations of the random greedy independent set algorithm, and pseudorandom graphs. We also prove related results where we forbid more than two edges with a prescribed common intersection size leading to some applications in Ramsey theory. For example, we obtain good bounds for the Ramsey number , where F is the k‐uniform Fan. The behavior is quite different than the case which is the classical Ramsey number .

Large deviations of the greedy independent set algorithm on sparse random graphs

AbstractWe study the greedy independent set algorithm on sparse Erdős–Rényi random graphs . A large deviation principle was recently established by Bermolen et al. in 2020, however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel in 1982. Using discrete calculus, we identify the optimal trajectory realizing a given large deviation and obtain the rate function in a simple closed form. In particular, we show that Pittel's bounds are sharp. The proof is brief and elementary. We think the methods presented here will be useful in analyzing the tail behavior of other random processes.

On approximating MIS over B1-VPG graphs*

In this paper, we present an approximation algorithm for the maximum independent set (MIS) problem over the class of [Formula: see text]-VPG graphs when the input is specified by a [Formula: see text]-VPG representation. We obtain a [Formula: see text]-approximation algorithm running in [Formula: see text] time. This is an improvement over the previously best [Formula: see text]-approximation algorithm [J. Fox and J. Pach, Computing the independence number of intersection graphs, in Proc. Twenty-Second Annual ACM-SIAM Symp. Discrete Algorithms (SODA 2011), 2011, pp. 1161–1165, doi:10.1137/1.9781611973082.87] (for some fixed [Formula: see text]) designed for some subclasses of string graphs, on [Formula: see text]-VPG graphs.

Computing inductive vertex orderings

Inductive vertex orderings or edge orientations of graphs, such as inductive k-independence and k-simpliciality orderings, and k-perfect orientation, arise naturally in geometric intersection graphs and have been systematically studied in Ye and Borodin (2012) [4] and Kammer and Tholey (2014) [5]. In general, those orderings and orientations are hard to compute, even approximately. In this note, we show that an inductive k-independence ordering can be computed in k-simplicial graphs, and a k-approximation (that is, a solution with approximation ratio at most k) to the optimal inductive independence ordering can be computed in k-perfectly orientable graphs. As a result, we obtain a k-approximation (2k-approximation) algorithm for maximum weight independent sets in k-simplicial (k-perfectly orientable, respectively) graphs that runs in polynomial time independent of k without prior knowledge of the ordering.

A constant amortized time enumeration algorithm for independent sets in graphs with bounded clique number

In this study, we address the independent set enumeration problem. Although several efficient enumeration algorithms and careful analyses have been proposed for maximal independent sets, no fine-grained analysis has been given for the non-maximal variant. As the main result, we propose an enumeration algorithm for the non-maximal variant that runs in O(q) amortized time and linear space, where q is the clique number, i.e., the maximum size of a clique in an input graph. Note that the proposed algorithm works correctly even if the exact value of q is unknown. It is optimal for graphs with a bounded clique number, such as, triangle-free graphs, bipartite graphs, planar graphs, bounded degenerate graphs, nowhere dense graphs, and F-free graphs for any fixed graph F, where a F-free graph is a graph that has no copy of F as a subgraph. Furthermore, with a slight modification of our proposed algorithm, we can enumerate independent sets with the size at most k in the same time and space complexity. This problem is a generalization of the original problem since this is equal to the original problem if k=n.

Approximation algorithms for maximum weight k-coverings of graphs by packings

Let [Formula: see text] be a family of graphs and let [Formula: see text] be a set of connected graphs, each with at most [Formula: see text] vertices, [Formula: see text] fixed. A [Formula: see text]-packing of a graph GA is a vertex induced subgraph of GA with every connected component isomorphic to a member of [Formula: see text]. A maximum weight [Formula: see text]-covering of a graph GA by [Formula: see text]-packings, is a maximum weight subgraph of GA exactly covered by [Formula: see text] vertex disjoint [Formula: see text]-packings. For a graph [Formula: see text] let [Formula: see text](GA) be a graph, every vertex [Formula: see text] of which corresponds to a vertex subgraph [Formula: see text] of GA isomorphic to a member of [Formula: see text], two vertices [Formula: see text] of [Formula: see text](GA) being adjacent if and only if [Formula: see text] and [Formula: see text] have common vertices or interconnecting edges. The closed neighborhoods containment graph [Formula: see text] of a graph [Formula: see text], is the graph with vertex set [Formula: see text] and edges directed from vertices [Formula: see text] to [Formula: see text] if and only if they are adjacent in GA and the closed neighborhood of [Formula: see text] is contained in the closed neighborhood of [Formula: see text]. A graph [Formula: see text] is a [Formula: see text] reduced graph if it can be obtained from a graph [Formula: see text] by deleting the edges of a transitive subgraph [Formula: see text] of CNCG(GA). We describe 1.582-approximation algorithms for maximum weight [Formula: see text]-coverings by [Formula: see text]-packings of [Formula: see text] and [Formula: see text] reduced graphs when [Formula: see text] is vertex hereditary, has an algorithm for maximum weight independent set and [Formula: see text]. These algorithms can be applied to families of interval filament, subtree filament, weakly chordal, AT-free and circle graphs, to find 1.582 approximate maximum weight [Formula: see text]-coverings by vertex disjoint induced matchings, dissociation sets, forests whose subtrees have at most [Formula: see text] vertices, etc.

Randomized greedy algorithm for independent sets in regular uniform hypergraphs with large girth

AbstractIn 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 , where converges to 0 as g → ∞ for fixed d and r, and f(d, r) is determined by a differential equation. This extends earlier results of Garmarnik and Goldberg for graphs [8]. 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.