Maximum Clique Algorithm Research Articles

Group-k consistent measurement set maximization via maximum clique over k-uniform hypergraphs for robust multi-robot map merging

This paper unifies the theory of consistent-set maximization for robust outlier detection in a simultaneous localization and mapping framework. We first describe the notion of pairwise consistency before discussing how a consistency graph can be formed by evaluating pairs of measurements for consistency. Finding the largest set of consistent measurements is transformed into an instance of the maximum clique problem and can be solved relatively quickly using existing maximum-clique solvers. We then generalize our algorithm to check consistency on a group- k basis by using a generalized notion of consistency and using generalized graphs. We also present modified maximum clique algorithms that function over generalized graphs to find the set of measurements that is internally group- k consistent. We address the exponential nature of group- k consistency and present methods that can substantially decrease the number of necessary checks performed when evaluating consistency. We extend our prior work to perform data association, and to multi-agent systems in both simulation and hardware, and provide a comparison with other state-of-the-art methods.

Cellular goore game with application to finding maximum clique in social networks

Abstract The goore game (GG) is a model for collective decision making under uncertainty, which can be used as a tool for stochastic optimization of a discrete variable function. The GG has a fascinating property that can be resolved in a distributed manner with no intercommunication between the players. The game has found applications in many network applications, including sensor networks, quality-of-service routing, and social networks. In this paper, we introduce an extension of GG called cellular goore game (CGG) for the first time. The CGG is a network of GGs. In this network, each node (or subset of nodes in the network) plays the rule of referees, each of which participates in a GG with its neighboring players (voters) at any time. Like in GG, each player independently selects its optimal action between two available actions based on their gains and losses received from its adjacent referee. Players in CGG know nothing about how other players are playing or even how/why they are rewarded/penalized. The potential of the CGG is shown by providing an algorithm for finding a maximum clique in social networks. Our research provides the first-time study of the CGG for finding a maximum clique in graphs. The performance of the CGG-based algorithm for finding maximum clique is studied on the standard clique benchmark called DIMACS by several experiments. The obtained result shows that the CGG-based algorithm is superior to the existing algorithms in terms of finding maximum clique size and time.

Hybrid Maximum Clique Algorithm Using Parallel Integer Programming for Uniform Test Assembly

Educational assessments often require uniform test forms, for which each test form has equivalent measurement accuracy but with a different set of items. For uniform test assembly, an important issue is the increase of the number of assembled uniform tests. Although many automatic uniform test assembly methods exist, the maximum clique algorithm (MCA)-based method is known to assemble the greatest number of uniform tests with the highest measurement accuracy based on the item response theory. In that method, the graph is constructed by sequentially adding a randomly formed test as a vertex without considering the graph structure. However, an important difficulty is its high space complexity, which interrupts search cliques with more than a hundred thousand vertices. To overcome this difficulty, this article proposes a new uniform test assembly algorithm: hybrid maximum clique algorithm using parallel integer programming. The first step searches a maximum clique that is as large as possible up to computer memory limitations using a state-of-the-art MCA with low time complexity but with high space complexity. The second step repeatedly searches a vertex connected with all vertices of the current maximum clique from the remaining vertices using integer programming with low space complexity but with high time complexity. The proposed method constructs a larger number of tests than the traditional methods do. Finally, we use simulation and actual data experiments to demonstrate the effectiveness of the proposed method. Results show that our method assembles a 1.5–2.7 times greater number of uniform tests than traditional methods can.

Exact Maximum Clique Algorithm for Different Graph Types Using Machine Learning

Finding a maximum clique is important in research areas such as computational chemistry, social network analysis, and bioinformatics. It is possible to compare the maximum clique size between protein graphs to determine their similarity and function. In this paper, improvements based on machine learning (ML) are added to a dynamic algorithm for finding the maximum clique in a protein graph, Maximum Clique Dynamic (MaxCliqueDyn; short: MCQD). This algorithm was published in 2007 and has been widely used in bioinformatics since then. It uses an empirically determined parameter, Tlimit, that determines the algorithm’s flow. We have extended the MCQD algorithm with an initial phase of a machine learning-based prediction of the Tlimit parameter that is best suited for each input graph. Such adaptability to graph types based on state-of-the-art machine learning is a novel approach that has not been used in most graph-theoretic algorithms. We show empirically that the resulting new algorithm MCQD-ML improves search speed on certain types of graphs, in particular molecular docking graphs used in drug design where they determine energetically favorable conformations of small molecules in a protein binding site. In such cases, the speed-up is twofold.

A new branch-and-filter exact algorithm for binary constraint satisfaction problems

A binary constraint satisfaction problem (BCSP) consists in determining an assignment of values to variables that is compatible with a set of constraints. The problem is called binary because the constraints involve only pairs of variables. The BCSP is a cornerstone problem in Constraint Programming (CP), appearing in a very wide range of real-world applications. In this work, we develop a new exact algorithm which effectively solves the BCSP by reformulating it as a k-clique problem on the underlying microstructure graph representation. Our new algorithm exploits the cutting-edge branching scheme of the state-of-the-art maximum clique algorithms combined with two filtering phases in which the domains of the variables are reduced. Our filtering phases are based on colouring techniques and on heuristically solving an associated boolean satisfiability (SAT) problem. In addition, the algorithm initialization phase performs a reordering of the microstructure graph vertices that produces an often easier reformulation to solve. We carry out an extensive computational campaign on a benchmark of almost 2000 instances, encompassing numerous real and synthetic problems from the literature. The performance of the new algorithm is compared against four SAT-based solvers and three general purpose CP solvers. Our tests reveal that the new algorithm significantly outperforms all the others in several classes of BCSP instances.

Finding Maximum k-Cliques Faster Using Lazy Global Domination

A clique in a graph is a set of vertices, each of which is adjacent to every other vertex in this set. A k-clique relaxes this requirement, requiring vertices to be within a distance k of each other, rather than directly adjacent. In theory, a maximum clique algorithm can easily be adapted to solve the maximum k-clique problem, although large sparse k-clique graphs reduce to large dense clique graphs, which can be computationally challenging. We adapt a state of the art maximum clique algorithm to show that this reduction is in fact useful in practice, and introduce a lazy global domination rule which sometimes vastly reduces the search space. We include experimental results for a range of real-world and benchmark graphs, and a detailed look at random graphs. We also use thread-parallel search to solve some harder instances.

A local core number based algorithm for the maximum clique problem

‎The maximum clique problem (MCP) is to determine a complete subgraph of maximum cardinality in a graph‎. ‎MCP is a fundamental problem in combinatorial optimization and is noticeable for its wide range of applications‎. ‎In this paper‎, ‎we present two branch-and-bound exact algorithms for finding a maximum clique in an undirected graph‎. ‎Many efficient exact branch and bound maximum clique algorithms use approximate coloring to compute an upper bound on the clique number but‎, ‎as a new pruning strategy‎, ‎we show that local core number is more efficient‎. ‎Moreover‎, ‎instead of neighbors set of a vertex‎, ‎our search area is restricted to a subset of the set in each subproblem which speeds up clique finding process‎. ‎This subset is based on the core of the vertices of a given graph‎. ‎We improved the MCQ and MaxCliqueDyn algorithms with respect to the new pruning strategy and search area restriction‎. ‎Experimental results demonstrate that the improved algorithms outperform the previous well-known algorithms for many instances when applied to DIMACS benchmark and random graphs‎.

Clique Is Hard on Average for Regular Resolution

We prove that for k ≪ 4√ n regular resolution requires length n Ω( k ) to establish that an Erdős–Rényi graph with appropriately chosen edge density does not contain a k -clique. This lower bound is optimal up to the multiplicative constant in the exponent and also implies unconditional n Ω( k ) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

AN EMPIRICAL RESEARCH OF CRITICAL INCIDENT OF EARTHQUAKE DISASTER BASEDON CREDIBLE ASSOCIATION DATA MINING

Earthquake disasters usually cause panic in the community affected areas, so it is necessary to be analyzed to deal with earthquake events in the future. This paper analyzes data from 9 major earthquakes in Indonesia over the past 4 years and determines 14 critical events. The analysis is based on credible association rules (CAR), data mining, and the maximum clique algorithm. To verify the accuracy of the association relationship and CAR effectiveness, it is performed using a maximum clique algorithm. Based on the results of data mining, that earthquakes have a credible association relationship and have a probability of critical events in various regions in Indonesia. Thus, these results can be used for prediction, early warning, and logistic distribution planning.

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

Combining fragment docking with graph theory to improve ligand docking for homology model structures

Computational protein–ligand docking is well-known to be prone to inaccuracies in input receptor structures, and it is challenging to obtain good docking results with computationally predicted receptor structures (e.g. through homology modeling). Here we introduce a fragment-based docking method and test if it reduces requirements on the accuracy of an input receptor structures relative to non-fragment docking approaches. In this method, small rigid fragments are docked first using AutoDock Vina to generate a large number of favorably docked poses spanning the receptor binding pocket. Then a graph theory maximum clique algorithm is applied to find combined sets of docked poses of different fragment types onto which the complete ligand can be properly aligned. On the basis of these alignments, possible binding poses of complete ligand are determined. This docking method is first tested for bound docking on a series of Cytochrome P450 (CYP450) enzyme–substrate complexes, in which experimentally determined receptor structures are used. For all complexes tested, ligand poses of less than 1 Å root mean square deviations (RMSD) from the actual binding positions can be recovered. Then the method is tested for unbound docking with modeled receptor structures for a number of protein–ligand complexes from different families including the very recent severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) protease. For all complexes, poses with RMSD less than 3 Å from actual binding positions can be recovered. Our results suggest that for docking with approximately modeled receptor structures, fragment-based methods can be more effective than common complete ligand docking approaches.Electronic supplementary materialThe online version of this article (10.1007/s10822-020-00345-7) contains supplementary material, which is available to authorized users.

The Use of an Exact Algorithm within a Tabu Search Maximum Clique Algorithm

Let G=(V,E) be an undirected graph with vertex set V and edge set E. A clique C of G is a subset of the vertices of V with every pair of vertices of C adjacent. A maximum clique is a clique with the maximum number of vertices. A tabu search algorithm for the maximum clique problem that uses an exact algorithm on subproblems is presented. The exact algorithm uses a graph coloring upper bound for pruning, and the best such algorithm to use in this context is considered. The final tabu search algorithm successfully finds the optimal or best known solution for all standard benchmarks considered. It is compared with a state-of-the-art algorithm that does not use exact search. It is slower to find the known optimal solution for most instances but is faster for five instances and finds a larger clique for two instances.

NK-MaxClique and MMCQ: Tow New Exact Branch and Bound Algorithms for the Maximum Clique Problem

The maximum clique problem (MCP) is a fundamental problem in combinatorial optimization which finds important applications in real-word. This paper describes two new efficient branch-and-bound maximum clique algorithms NK-MaxClique and MMCQ, designed for solving MCP. We define some pruning conditions based on core numbers and vertex ordering to efficiently remove many of the search space. With respect to this ordering, the algorithms consider the vertices respectively to find the corresponding maximum clique in subproblems. Simulation results demonstrate that the algorithms outperform the previous well-known algorithms for many instances when applied to DIMACS benchmark and random graphs.

Computational study of separation algorithms for clique inequalities

Clique inequalities appear in linear descriptions of many combinatorial optimisation problems. In general, they form an exponential family and, in addition, the associated separation problem is strongly NP-hard, being equivalent to a maximum weight clique problem. Therefore, most of the known (both exact and heuristic) separation procedures follow the decomposition scheme of a maximum clique algorithm. We introduce a new heuristic, aimed at constructing a collection of (violated) clique inequalities covering all the edges of the underlying graph. We present an extensive computational experience showing that this closely approximates the results of an exact separation oracle while being faster than standard heuristics.

Approximate Maximum Clique Algorithm (AMCA): A Clever Technique for Solving the Maximum Clique Problem through Near Optimal Algorithm for Minimum Vertex Cover Problem

Background and Objective: The process of solving the Maximum Clique (MC) problem through approximation algorithms is harder, however, the Maximum Vertex Cover (MVC) problem can easily be solved using approximation algorithms. In this paper, a technique has been proposed to use the approximation algorithms of Minimum Vertex Cover (MVC) for the solution of the Maximum Cliques (MC) problem. Material and methods: To test the proposed technique, selected approximation algorithms have been developed to small graph instances. The algorithms that were used for experiments are Maximum Degree Greedy (MDG). Vertex support Algorithm (VSA). Mean of Neighbors of Minimum Degree Algorithm (MNMA), Modified Vertex Support Algorithm (MVSA), Maximum Adjacent Minimum Degree Algorithm (MAMA), and Clever Steady Strategy Algorithms (CSSA). Results: The development of an efficient approximation algorithm for the Maximum Clique (MC) problem is very difficult due to its complex nature. The only way left is to use the approximation algorithm of Minimum Vertex Cover (MVC) for the solution of the Maximum Clique (MC) problem. The experimental analysis of the proposed algorithm has revealed that the Maximum Clique (MC) problem can be efficiently solved with approximation algorithms of Minimum Vertex Cover (MVC). The proposed algorithm has efficiently solved the MC problem within the reduced time limit. conclusions: It is a difficult task to directly solve the MC problem through approximation algorithms. The proposed method provides a platform to efficiently solve the MC problem by using approximation algorithm of Minimum Vertex Cover (MVC).

An enhanced bitstring encoding for exact maximum clique search in sparse graphs

This paper describes BBMCW, a new efficient exact maximum clique algorithm tailored for large sparse graphs which can be bit-encoded directly into memory without a heavy performance penalty. These graphs occur in real-life problems when some form of locality may be exploited to reduce their scale. One such example is correspondence graphs derived from data association problems. The new algorithm is based on the bit-parallel kernel used by the BBMC family of published exact algorithms. BBMCW employs a new bitstring encoding that we denote ‘watched’, because it is reminiscent of the ‘watched literal’ technique used in satisfiability and other constraint problems. The new encoding reduces the number of spurious operations computed by the BBMC bit-parallel kernel in large sparse graphs. Moreover, BBMCW also improves on bound computation proposed in the literature for bit-parallel solvers. Experimental results show that the new algorithm performs better than prior algorithms over data sets of both real and synthetic sparse graphs. In the real data sets, the improvement in performance averages more than two orders of magnitude with respect to the state-of-the-art exact solver IncMaxCLQ.

A parallel maximum clique algorithm for large and massive sparse graphs

This paper describes BBMCPara, a new parallel exact maximum clique algorithm tailored for large and massive sparse graphs. The paper first presents a sequential algorithm BBMCSP, which builds on ideas from a leading bit-parallel published algorithm for middle-size graphs. It employs heavy pre-processing and a new sparse bitset encoding to outperform other state-of-the-art algorithms by up to several orders of magnitude over a set of real networks. BBMCPara parallelizes BBMCSP by splitting according to a preprocessing step of the latter. On a 20-core computer, it averages speedups close to an order of magnitude over real graphs of up to 3 million vertices. According to the reported results, BBMCPara appears to be the current fastest algorithm for large and massive real networks to the best of our knowledge.

Energy Efficiency of Parallel Multicore Programs

The increasing energy consumption of large-scale high performance resources raises technical and economical concerns. A reduction of consumed energy in multicore systems is possible to some extent with an optimized usage of computing and memory resources that is tailored to specific HPC applications. The essential step towards more sustainable consumption of energy is its reliable measurements for each component of the system and selection of optimally configured resources for specific applications. This paper briefly surveys the current approaches for measuring and profiling power consumption in large scale systems. Then, a practical case study of a real-time power measurement of multicore computing system is presented on two real HPC applications: maximum clique algorithm and numerical weather prediction model. We assume that the computing resources are allocated in a HPC cloud on a pay-per-use basis. The measurements demonstrate that the minimal energy is consumed when all available cores (up to the scaling limit for a particular application) are used on their maximal frequencies and with threads binded to the cores.

A new exact maximum clique algorithm for large and massive sparse graphs

This paper describes a new very efficient branch-and-bound exact maximum clique algorithm BBMCSP, designed for large and massive sparse graphs which appear frequently in real life problems from different fields.State-of-the-art exact maximum clique algorithms encode the adjacency matrix in full but when dealing with sparse graphs some form of compression is required. The new algorithm is based on a leading bit-parallel non-sparse solver but employs a novel sparse encoding for the adjacency matrix. Moreover, it also improves on recent optimizations proposed in literature for the sparse case such as core-based bounds.Reported results show that it is several orders of magnitude better than state-of-the-art. Moreover, a number of real networks with many millions of nodes are solved in a few seconds.

LiSiCA: A Software for Ligand-Based Virtual Screening and Its Application for the Discovery of Butyrylcholinesterase Inhibitors

We developed LiSiCA (ligand similarity using clique algorithm)--ligand-based virtual screening software that uses a fast maximum clique algorithm to find two- and three-dimensional similarities between pairs of molecules and applied it to the discovery of novel potent butyrylcholinesterase inhibitors. LiSiCA, which runs in parallel on multiple processor cores, was successfully tested on the Database of Useful Decoys-Enhanced, to evaluate its ability to discriminate active molecules from decoys. We then applied LiSiCA for the discovery of novel inhibitors of human butyrylcholinesterase, a promising anti-Alzheimer target, using a known inhibitor as the reference compound. We demonstrated that LiSiCA is capable of finding potent nanomolar inhibitors, whose scaffolds differed from the reference compound, thus proving its ability for scaffold hopping and usefulness in drug discovery.