Improved Volume Molecular Solutions for the Maximum Clique Problem on DNA-Based Supercomputing

Optimal parallel time bounds for the maximum clique problem on intervals

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.

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

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.

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.

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 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.

<title>Abstract</title> The maximum clique problem(MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an improved algorithm that dynamically tracks the maximum clique size by encoding prior constraints on the vertex count—derived from Turán's theorem and complete graph properties—into global variables through quantum circuit pre-detection. The algorithm further synergizes with Grover's search to optimize the solution space. Our auxiliary-qubit encoding scheme dynamically tracks clique sizes during quantum search, eliminating iterative measurements, achieving MCP solution with $O\left ( \sqrt{2^{n } } \right ) $ Grover iterations and $O\left ( 1 \right ) $ measurements. This represents an $\boldsymbol{n}$-fold improvement over state-of-the-art Grover-based methods, which require $O(n\sqrt{2^n})$ iterations and $O(n)$ measurements for $n$-vertex graphs. We validate algorithmic correctness through simulations on IBM's Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.

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 ( <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.

A new upper bound for the maximum weight clique problem

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.

Computing maximum k-defective cliques in massive graphs

In the last years many algorithms have been proposed for solving the maximum clique problem. Most of these algorithms have been tested on randomly generated graphs. In this paper we present different test problem generators that arise from a variety of practical applications, as well as graphs with known maximum cliques. In addition, we provide computational experience with two exact algorithms using the generated test problems.

Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane