Clique Finder

The Maximum Clique Problem (MCP) is a classical NP-hard problem that has gained considerable attention due to its numerous real-world applications and theoretical complexity. It is inherently computationally complex, and so exact methods may require prohibitive computing time. Nature-inspired meta-heuristics have proven their utility in solving many NP-hard problems. In this research, we propose a simulated annealing-based algorithm that we call Clique Finder algorithm to solve the MCP. Our algorithm uses a logarithmic cooling schedule and two moves that are selected in an adaptive manner. The objective (error) function is the total number of missing links in the clique, which is to be minimized. The proposed algorithm was evaluated using benchmark graphs from the open-source library DIMACS, and results show that the proposed algorithm had a high success rate.

Clique Finder

The Maximum Clique Problem (MCP) is a classical NP-hard problem that has gained considerable attention due to its numerous real-world applications and theoretical complexity. It is inherently computationally complex, and so exact methods may require prohibitive computing time. Nature-inspired meta-heuristics have proven their utility in solving many NP-hard problems. In this research, we propose a simulated annealing-based algorithm that we call Clique Finder algorithm to solve the MCP. Our algorithm uses a logarithmic cooling schedule and two moves that are selected in an adaptive manner. The objective (error) function is the total number of missing links in the clique, which is to be minimized. The proposed algorithm was evaluated using benchmark graphs from the open-source library DIMACS, and results show that the proposed algorithm had a high success rate.

Determination of the Most Interconnected Sections of Main Gas Pipelines Using the Maximum Clique Method

This article is devoted to the definition of the most important combinations of objects in critical network infrastructures. This study was carried out using the example of the Russian gas transmission network. Since natural gas is widely used in the energy sector, the gas transmission network can be exposed to terrorist threats, and the actions of intruders can be directed at both gas fields and gas pipelines. A defender–attacker model was proposed to simulate attacks. In this model, the defender solves the maximum flow problem to satisfy the needs of gas consumers. By excluding gas pipelines, the attacker tries to minimize the maximum flow in the gas transmission network. Russian and European gas transmission networks are territorially very extensive and have a significant number of mutual intersections and redundant pipelines. Therefore, one of the approaches to inflicting maximum damage on the system is modeled as an attack on a clique. A clique in this study is several interconnected objects. The article presents the list of the most interconnected sections of main gas pipelines, the failure of which can cause the greatest damage to the system in the form of a gas shortage among consumers. Conclusions were drawn about the applicability of the maximum clique method for identifying the most important objects in network critical infrastructures.

Conceptual configuration design of line-foldable deployable space truss structures employing graph theory

From the perspective of the truss as a whole, this research investigates the conceptual configuration design for deployable space truss structures that are line-foldable with the help of graph theory. First, the bijection between a truss and its graph model is established. Therefore, operations can be performed based on graph models. Second, by introducing Maxwell’s rule, maximum clique, and chordless cycle, the principle of conceptual configuration synthesis is analyzed. A corresponding procedure is formed and it is verified by a truss with seven nodes. Third, assisted by some theorems of graph theory, the simplified double-color topological graph of deployable space truss structures is acquired and it also displays the procedure with a case. Finally, based on the above analysis, it obtains the optimal conceptual configurations. This novel research lays the foundation for kinematic synthesis and geometric dimension designs.

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.

Making the Most of Parallel Composition in Differential Privacy

We show that the ‘optimal’ use of the parallel composition theorem corresponds to finding the size of the largest subset of queries that ‘overlap’ on the data domain, a quantity we call the maximum overlap of the queries. It has previously been shown that a certain instance of this problem, formulated in terms of determining the sensitivity of the queries, is NP-hard, but also that it is possible to use graph-theoretic algorithms, such as finding the maximum clique, to approximate query sensitivity. In this paper, we consider a significant generalization of the aforementioned instance which encompasses both a wider range of differentially private mechanisms and a broader class of queries. We show that for a particular class of predicate queries, determining if they are disjoint can be done in time polynomial in the number of attributes. For this class, we show that the maximum overlap problem remains NP-hard as a function of the number of queries. However, we show that efficient approximate solutions exist by relating maximum overlap to the clique and chromatic numbers of a certain graph determined by the queries. The link to chromatic number allows us to use more efficient approximate algorithms, which cannot be done for the clique number as it may underestimate the privacy budget. Our approach is defined in the general setting of f-differential privacy, which subsumes standard pure differential privacy and Gaussian differential privacy. We prove the parallel composition theorem for f-differential privacy. We evaluate our approach on synthetic and real-world data sets of queries. We show that the approach can scale to large domain sizes (up to 1020000), and that its application can reduce the noise added to query answers by up to 60%.

Treatment Trajectories Graph Compression Algorithm Based on Cliques

Learning treatment methods and disease progression is significant part of medicine. Graph representation of data provides wide area for visualization and optimization of structure. Present work is dedicated to suggest method of data processing for increasing information interpretability. Graph compression algorithm based on maximum clique search is applied to data set with acute coronary syndrome treatment trajectories. Results of compression are studied using graph entropy measures.

A modified simplex partition algorithm to test copositivity

A real symmetric matrix A is copositive if $$x^\top Ax\ge 0$$ x ⊤ A x ≥ 0 for all $$x\ge 0$$ x ≥ 0 . As A is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015), are based upon the creation of increasingly fine simplicial partitions of simplices, testing for copositivity on each. We present a variant that decomposes a simplex $$\bigtriangleup $$ △ , say with n vertices, into a simplex $$\bigtriangleup _1$$ △ 1 and a polyhedron $$\varOmega _1$$ Ω 1 ; and then partitions $$\varOmega _1$$ Ω 1 into a set of at most $$(n-1)$$ ( n - 1 ) simplices. We show that if A is copositive on $$\varOmega _1$$ Ω 1 then A is copositive on $$\bigtriangleup _1$$ △ 1 , allowing us to remove $$\bigtriangleup _1$$ △ 1 from further consideration. Numerical results from examples that arise from the maximum clique problem show a significant reduction in the time needed to establish copositivity of matrices.