Optimal parallel time bounds for the maximum clique problem on intervals

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

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.

A new upper bound for the maximum weight clique problem

This paper explores techniques to quickly solve the maximum weight clique problem (MWCP) in very large scale sparse graphs. Due to their size, and the hardness of MWCP, it is infeasible to solve many of these graphs with exact algorithms. Although recent heuristic algorithms make progress in solving MWCP in large graphs, they still need considerable time to get a high-quality solution. In this work, we focus on solving MWCP for large sparse graphs within a short time limit. We propose a new method for MWCP which interleaves clique finding with data reduction rules. We propose novel ideas to make this process efficient, and develop an algorithm called FastWClq. Experiments on a broad range of large sparse graphs show that FastWClq finds better solutions than state-of-the-art algorithms while the running time of FastWClq is much shorter than the competitors for most instances. Further, FastWClq proves the optimality of its solutions for roughly half of the graphs, all with at least 105 vertices, with an average time of 21 seconds.

The faces in a 2D fine drawing of an object provide important information for the reconstruction of its 3D geometry. In this paper, a graph-based optimization method is proposed for identifying the faces is a line drawing. The face identification is formulated as a maximum weight clique problem. This formulation is proven to be equivalent to the formulation proposed by Shpitalni and Upson (1996). The advantage of our formulation is that it enables one to develop a much faster algorithm to find the faces in a drawing. The significant improvement in speed is derived from two algorithms provided: the depth-first graph search for quickly generating possible faces from a drawing; and the maximum weight clique finding for obtaining the optimal face configurations of the drawing. The experimental results shown that our algorithm generates the same results of face identification as Shpitalni and Lipson's method, but is much faster when dealing with objects of more than 20 faces.

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.

A Lagrangian relaxation approach to combinatorial exchange in freight logistics

The radio-frequency (RF) technology is a scalable solution for the backhaul planning. However, its performance is limited in terms of data rate and latency. Free Space Optical (FSO) backhaul, on the other hand, offers a higher data rate but is sensitive to weather conditions. To combine the advantages of RF and FSO backhauls, this paper proposes a cost-efficient backhaul network using the hybrid RF/FSO technology. To ensure a resilient backhaul, the paper imposes a given degree of redundancy by connecting each node through K link-disjoint paths so as to cope with potential link failures. Hence, the network planning problem considered in this paper is the one of minimizing the total deployment cost by choosing the appropriate link type, i.e., either hybrid RF/FSO or optical fiber (OF), between each couple of base-stations while guaranteeing K link-disjoint connections, a data rate target, and a reliability threshold. The paper solves the problem using graph theory techniques. It reformulates the problem as a maximum weight clique problem in the planning graph, under a specified realistic assumption about the cost of OF and hybrid RF/FSO links. Simulation results show the cost of the different planning and suggest that the proposed heuristic solution has a close-to-optimal performance for a significant gain in computation complexity.

The maximum weighted clique (MWC) problem, as a typical NP-complete problem, is difficult to be solved by the electronic computer algorithm. The aim of the problem is to seek a vertex clique with maximal weight sum in a given undirected graph. It is an extremely important problem in the field of optimal engineering scheme and control with numerous practical applications. From the point of view of practice, we give a parallel biological algorithm to solve the MWC problem. For the maximum weighted clique problem withmedges andnvertices, we use fixed length DNA strands to represent different vertices and edges, fully conduct biochemical reaction, and find the solution to the MVC problem in certain length range withO(n2)time complexity, comparing to the exponential time level by previous computer algorithms. We expand the applied scope of parallel biological computation and reduce computational complexity of practical engineering problems. Meanwhile, we provide a meaningful reference for solving other complex problems.

Maximum weight independent sets and cliques in intersection graphs of filaments

Graphs and Algorithms A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

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.

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.

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.