Efficient Maximum Clique Detection via Grover’s Algorithm with Real-time Global Size Tracking

In this paper, we give a comprehensive overview of the development of clique algorithms and their use for drug design based on the search for cliques in protein product graphs. The maximum clique problem is a computational problem of finding largest subsets of vertices in a graph that are all pairwise adjacent. A related problem is the maximum weight clique problem and the highest weight k-clique problem, which both extend the algorithm to weighted graphs. The review covers our developed algorithms, starting with our improved branch-and-bound algorithm for finding maximum cliques in undirected graphs from 2007 up to the recent developments of algorithms for weighted graphs in 2024. We show the application of these algorithms to early stages of drug discovery, in particular to protein binding site detection based on protein similarity search in large protein databases and to protein-ligand molecular docking.

Optimal parallel time bounds for the maximum clique problem on intervals

In communication networks such as social networks, wireless networks and biology networks, it is of importance to find all cliques which can help understand the network topology. The clique structures can also be utilized in facilitating message forwarding in wireless networks. For instance, using a set of cliques of maximal and disjoint, one of the nodes in each clique can be elected to forward messages, by which the duplicate message transmissions can be efficiently reduced. Further, it is well known that the maximum clique problem (MCP) is closely related to the maximum independent set and the vertex cover problems. In recent years, the fundamental problem of finding maximal cliques or maximum cliques has attracted lots of attentions. However, few of these works focus on distributed solutions in wireless networks. In this paper, we pay our attention to this missing corner of research. Specifically, we first give a distributed algorithm which can compute all maximal cliques in a wireless network represented by a graph. The algorithm takes O(n) time and uses O(mn) messages, where n is the number of nodes and m the number of edges. Then, with the proposed algorithms MCP (maximum clique problem) and UMCP (unique MCP), we show that a unique maximum clique can be selected from all maximal cliques in O(n) rounds and using O(mn) messages. To the best of our knowledge, our algorithms are the first deterministic distributed solutions for MCP in wireless networks.

Improving the performance of stochastic local search for maximum vertex weight clique problem using programming by optimization

The intent of maximum clique problem is to discover a clique of largest size or a complete subgraph in a graph; for a graph clique is a subgraph which is complete i.e., every vertex of that subgraph are adjacent to each other. It belongs to the category of the NP-hard problems, for which many solutions exist and many more are continuously being proposed. We are recapitulating the core idea and search strategies of few of the branch and bound based recently proposed algorithms. Intent of this work is to give a comparative study of newly proposed solutions to maximum clique problem which will be helpful to decide which algorithm is efficient for a given graph. It also instigates to design more efficient solutions for maximum clique problem.

In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, we want to decide if a given k-partite graph contains a clique with k nodes. The maximum clique problem asks for finding a maximum clique in a given finite simple graph. The problem of deciding if the given graph contains a clique with k vertices is called the k-clique problem. The first problem is NP-hard and the second one is NP-complete. The special clique search problem, we propose, is still an NP-complete problem. We will show that the k-clique problem in the special case of k-partite graphs is more tractable than in the general case. In order to illustrate the possible practical utility of this restricted type clique search problem we will show that the job shop scheduling problem can be reduced to such a clique search problem in a suitable constructed graph. We carry out numerical experiments to assess the efficiency of the approach. It is a common practice that before one embarks on a large scale clique search typically one attempts to simplify and tidy up the given graph. This procedure is commonly referred as preconditioning or kernelization of the given graph. Of course, the preconditioning or kernelization is meant with respect to the given type of clique search problem. The other main topic of the paper is to describe a number of kernelization methods tailored particularly to the proposed special k-clique problem. Some of these techniques works in connection with the generic k-clique problem. In these situations, we will see that they are more efficient in the case of k-partite graphs. Some other preconditioning methods applicable only to k-partite graphs. We illustrate how expedient these preconditioning methods can be by solving non-trivial scheduling problems to optimality employing only kernelization techniques dispensing with exhaustive clique search algorithms altogether.

The maximum clique problem ( <i>MCP</i> ) has various applications to reveal the structure and function of graphs. Graphs are constantly updated in the real life. However, no algorithm is specifically designed for dynamic graph. Although <i>MCP</i> in dynamic graphs can be solved by simply invoking a state-of-the-art static approach, such as <i>PMC</i> , when the graph is updated, such an approach of simply re-calculating from scratch is inefficient. The key issue with <i>MCP</i> algorithm is to find a large clique, namely a <i>seed</i> , as fast as possible. Thus, search space can be pruned based on the seed. Size of the seed greedily found by <i>PMC</i> cannot be guaranteed, as it fluctuates considerably. Moreover, the time required to find a seed under <i>PMC</i> is up to <inline-formula><tex-math notation="LaTeX">$O(| E| \cdot \Delta (G))$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\Delta (G)$</tex-math></inline-formula> is the highest degree in <i>G</i> . In this article, we intend to find a sizable seed by updating the previous maximum clique with the incident vertices of the inserted/deleted edge. Size of the seed now is guaranteed to be no less than <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})\; - \;1$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> is the size of the maximum clique on the updated graph. Moreover, the seed can be found in a time complexity of <inline-formula><tex-math notation="LaTeX">$O(\Delta (G)^{2})$</tex-math></inline-formula> . Two other crucial issues related to the <i>MCP</i> in dynamic graphs are refreshing rate and refreshing overhead. After a tight upper bound is imposed on <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> , the necessity of refreshing is evaluated by comparing the seed with its largest challenger, then unnecessary refreshing is wiped out effectively. The size of the largest challenger is judiciously estimated using a lazy growth strategy. Subsequently, the search space in refreshing is confined on a much smaller subgraph using a local refreshing strategy. Extensive experiments indicate that the proposed approach outperforms the baseline algorithm by approximately one order of magnitude.

Given an undirected graph G=(V,E) with vertex set V={1,?,n} and edge set E?V×V. The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and a long term memory based restart strategy. Our proposed algorithm is evaluated on the whole set of 80 DIMACS challenge benchmarks and compared with five state-of-the-art algorithms. Computational results show that our proposed algorithm attains the largest known clique for 79 benchmarks.

This paper considers a family of cutting planes, recently developed for mixed 0–1 polynomial programs and shows that they define facets for the maximum edge-weighted clique problem. There exists a polynomial time exact separation algorithm for these inequalities. The result of this paper may contribute to the development of more efficient algorithms for the maximum edge-weighted clique problem that use cutting planes.

We propose a simpler and more efficient scheme for the implementation of the\nmulti-valued Grover's quantum search. The multi-valued search generalizes the\noriginal Grover's search by replacing qubits with qudits --- quantum systems of\n$d$ discrete states. The qudit database is exponentially larger than the qubit\ndatabase and thus it requires fewer particles to control. The Hadamard gate,\nwhich is the key elementary gate in the qubit implementation of Grover's\nsearch, is replaced by a $d$-dimensional (complex-valued) unitary matrix $F$,\nthe only condition for which is to have a column of equal moduli elements\nirrespective of their phases; it can be realized through any physical\ninteraction, which achieves an equal-weight superposition state. An example of\nsuch a transformation is the $d$-dimensional discrete Fourier transform, used\nin earlier proposals; however, its construction is much more costly than that\nof the far simpler matrix $F$. We present examples of how such a transform $F$\ncan be constructed in realistic qudit systems in a \\emph{single} interaction\nstep.\n

Graph pattern mining is an important part of the emerging social network science, and the research of the maximum clique problem is one of the most important research branches. In the big data environment, the mass of nodes and complexity of edges in the graph set a higher requirement on the speed and accuracy of the maximum clique (MCP) study. In this paper, we present a parallel graph partitioning framework for solving the maximum clique problem(PPMC) based on Hadoop. Firstly, This paper presents a new graph partition method based on degree sorting(GPD), the subgraph scale is greatly reduced. In order to balance the workload of solving the maximum clique problems. Secondly, the improved ant algorithm proposed by our team is used to solve the maximum clique of each subgraph. Finally, the framework is deployed on Hadoop distributed cloud computing platform and tested by Stanford large-scale datasets. The experimental results show that GPD algorithm can greatly reduce the subgraph scale, the subgraph number, and the complexity of time and space of solving the maximum clique problem of large-scale instances. And the efficiency and scalability of the Hadoop based parallel graph partitioning framework for solving the maximum clique problems has been further proved.

The Maximum Clique Problem (MCP) is one of the most important combinatorial optimization problems that has many practical applications such as community search in social networks. Since the MCP is known to be NP-hard, much effort has been devoted to the development of metaheuristic algorithms to find a high quality clique (solution) within reasonable running times. The Multi-start k-opt Local Search incorporating k-opt local search (MKLS) is well known as a simple and effective metaheuristic for MCP. However it takes long time to search the high-quality solution for difficult massive graphs such as real world social networks, because the search space is too large. In the case of applying metaheuristic algorithms for massive sparse graphs, adequate process such as reduction process is necessary to focus on promising search space. In this paper, we present a Multi-start k-opt Local Search with graph Reduction process (MKLS-R), for solving the maximum clique problem on massive graphs. MKLS-R is evaluated on difficult massive graphs of Network-Repository graphs. The experimental results showed that the graph reduction process in MKLS-R contributes to the improvement of the search performance of MKLS for the difficult massive graphs.

Quantum annealers, such as the device built by D-Wave Systems, Inc., offer a way to compute solutions of NP-hard problems that can be expressed in Ising or quadratic unconstrained binary optimization (QUBO) form. Although such solutions are typically of very high quality, problem instances are usually not solved to optimality due to imperfections of the current generations quantum annealers. In this contribution, we aim to understand some of the factors contributing to the hardness of a problem instance, and to use machine learning models to predict the accuracy of the D-Wave 2000Q annealer for solving specific problems. We focus on the maximum clique problem, a classic NP-hard problem with important applications in network analysis, bioinformatics, and computational chemistry. By training a machine learning classification model on basic problem characteristics such as the number of edges in the graph, or annealing parameters, such as the D-Wave’s chain strength, we are able to rank certain features in the order of their contribution to the solution hardness, and present a simple decision tree which allows to predict whether a problem will be solvable to optimality with the D-Wave 2000Q. We extend these results by training a machine learning regression model that predicts the clique size found by D-Wave.

The maximum clique problem involves finding the largest set of pairwise adjacent vertices in a graph. The problem is classic but still attracts much attention because of its hardness and its prominent applications. Our work is based on the existence of an order of all the vertices whereby those belonging to a maximum clique stay close enough to each other. Such an order can be identified via the extraction of a particular subgraph from the original graph. The problem can consequently be seen as a permutation problem that can be addressed efficiently by metaheuristics. We first design a memetic algorithm (MA) for this purpose. Computational experiments conducted on the DIMACS benchmark instances clearly show that our MA not only outperforms existing genetic approaches, but it also compares very well to state-of-the-art algorithms regarding the maximal clique size found after different runs. Furthermore, we show that a hybridization of MA with an iterated local search (ILS) improves the stability of the algorithm. This hybridization (MA-ILS) permits to find two distinct maximal cliques of size 79 and one of size 80 for the C2000.9 instance of the DIMACS benchmark.

How to decrease the volume in DNA computers is of a great significance in the research of DNA computing.For the objective to decrease the DNA volume of the maximum clique problem which is a famous NP-complete problem,the pruning strategy is introduced into the DNA supercomputing and a new DNA algorithm is proposed.The new algorithm consists of a Degree Searcher,a Clique Generator,a Dense Parallel Searcher,a Sparse Parallel Searcher and a Maximum Clique Searcher.In a computer simulation,the DNA strands of maximum number required is O(3~(1/2)~n)on the condition of not varying the time complexity where n is the number of the vertex in a graph,while O(2n)DNA strands are needed in present molecular algorithms for the maximum clique problems.Furthermore,this algorithm is highly space-efficient and error-tolerant compared to conventional brute-force searching.