Clique Cover Research Articles

Spectral radius and clique partitions of graphs

We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint t-cliques. The extremal graphs attaining the bounds are exactly the block graphs of Steiner 2-designs and the regular graphs with Kt-decompositions, respectively.

The Inductive Graph Dimension from the Minimum Edge Clique Cover

In this paper we prove that the inductively defined graph dimension has a simple additive property under the join operation. The dimension of the join of two simple graphs is one plus the sum of the dimensions of the component graphs: \(\mathrm {dim\,} (G_1+ G_2) = 1 +\mathrm {dim\,} G_1+ \mathrm {dim\,} G_2\). We use this formula to derive an expression for the inductive dimension of an arbitrary finite simple graph from its minimum edge clique cover. A corollary of the formula is that any arbitrary finite simple graph whose maximal cliques are all of order N has dimension \(N{-}1\). We finish by finding lower and upper bounds on the inductive dimension of a simple graph in terms of its clique number.

Multi-Attribute Community Detection in International Trade Network

Understanding the structure of communities in a network has a great importance in the economic analysis. Communities are indeed characterized by specific properties, that are different from those of both the individual nodes and the whole network, and they can affect various processes on the network. In the International Trade Network, community detection aims to search sets of countries (or of trade sectors) which have a high intra-cluster connectivity and a low inter-cluster connectivity. In general, exchanges among countries occur according to preferential economic relationships ranging over different sectors. In this paper, we combine community detection with specific topological indicators, such as centrality measures. As a result, a new weighted network is constructed from the original one, in which weights are determined taking into account all the topological indicators in a multi-criteria approach. To solve the resulting Clique Partitioning Problem and find homogeneous group of nations, we use a new fast algorithm, based on quick descents to a local optimal solution. The analysis allows to cluster countries by interconnections, economic power and intensity of trade, giving an important overview on the international trade patterns.

Revising the Observation Satellite Scheduling Problem Based on Deep Reinforcement Learning

Earth observation satellite task scheduling research plays a key role in space-based remote sensing services. An effective task scheduling strategy can maximize the utilization of satellite resources and obtain larger objective observation profits. In this paper, inspired by the success of deep reinforcement learning in optimization domains, the deep deterministic policy gradient algorithm is adopted to solve a time-continuous satellite task scheduling problem. Moreover, an improved graph-based minimum clique partition algorithm is proposed for preprocessing in the task clustering phase by considering the maximum task priority and the minimum observation slewing angle under constraint conditions. Experimental simulation results demonstrate that the deep reinforcement learning-based task scheduling method is feasible and performs much better than traditional metaheuristic optimization algorithms, especially in large-scale problems.

A branch-and-price procedure for clustering data that are graph connected

This paper studies the Graph-Connected Clique-Partitioning Problem (GCCP), a clustering optimization model in which units are characterized by both individual and relational data. This problem, introduced by Benati, Puerto, and Rodríguez-Chía (2017) under the name of Connected Partitioning Problem, shows that the combination of the two data types improves the clustering quality in comparison with other methodologies. Nevertheless, the resulting optimization problem is difficult to solve; only small-sized instances can be solved exactly, large-sized instances require the application of heuristic algorithms. In this paper we improve the exact and the heuristic algorithms previously proposed. Here, we provide a new Integer Linear Programming (ILP) formulation, that solves larger instances, but at the cost of using an exponential number of variables. In order to limit the number of variables necessary to calculate the optimum, the new ILP formulation is solved implementing a branch-and-price (B&P) algorithm. The resulting pricing problem is itself a new combinatorial model: the Maximum-weighted Graph-Connected Single-Clique problem (MGCSC), that we solve testing various Mixed Integer Linear Programming (MILP) formulations and proposing a new fast “random shrink” heuristic. In this way, we are able to improve the previous algorithms: The B&P method outperforms the computational times of the previous MILP algorithms and the new random shrink heuristic, when applied to GCCP, is both faster and more accurate than the previous heuristic methods. Moreover, the combination of column generation and random shrink is itself a new MILP-relaxed matheuristic that can be applied to large instances too. Its main advantage is that all heuristic local optima are combined together in a restricted MILP, consisting in the application of the exact B&P method but solving heuristically the pricing problem.

Network clique cover approximation to analyze complex contagions through group interactions

Contagion processes have been proven to fundamentally depend on the structural properties of the interaction networks conveying them. Many real networked systems are characterized by clustered substructures representing either collections of all-to-all pair-wise interactions (cliques) and/or group interactions, involving many of their members at once. In this work, focusing on interaction structures represented as simplicial complexes, we present a discrete-time microscopic model of complex contagion for a susceptible-infected-susceptible dynamics. Introducing a particular edge clique cover and a heuristic to find it, the model accounts for the higher-order dynamical correlations among the members of the substructures (cliques/simplices). The analytical computation of the critical point reveals that higher-order correlations are responsible for its dependence on the higher-order couplings. While such dependence eludes any mean-field model, the possibility of a bi-stable region is extended to structured populations.

Eigenvalues and clique partitions of graphs

A clique partition ε of graph G is a set of cliques such that each edge of G belongs to exactly one clique, and the total size of ε is the sum of cardinalities of all elements in ε. The ε-degree of a vertex u is the number of cliques in ε containing u. We say that ε is a k-restricted clique partition if each vertex has ε-degree at least k. The (k-restricted) clique partition number of G is the smallest cardinality of a (k-restricted) clique partition of G. In this paper, we obtain eigenvalue bounds for ε-degrees, clique partition number and restricted clique partition number of a graph. As applications, we derive the De Bruijn-Erdős Theorem from our eigenvalue bounds, obtain accurate estimation of the 2-restricted clique partition number of line graphs, and give spectral lower bounds for the minimum total size of clique partitions of a graph.

A clique-based discrete bat algorithm for influence maximization in identifying top-k influential nodes of social networks

The problem of identifying the top-k influential node is still an open and deeply felt issue. The development of a stable and efficient algorithm to deal with such identification is still a challenging research hot spot. Although conventional centrality-based and greedy-based methods show high performance, they are not very efficient when dealing with large-scale social networks. Recently, algorithms based on swarm intelligence are applied to solve the problems mentioned above, and the existing researches show that such algorithms can obtain the optimal global solution. In particular, the discrete bat algorithm (DBA) has been proved to have excellent performance, but the evolution mechanism based on a random selection strategy leads to the optimal solution's instability. To solve this problem, in this paper, we propose a clique-DBA algorithm. The proposed algorithm is based on the clique partition of a network and enhances the initial DBA algorithm's stability. The experimental results show that the proposed clique-DBA algorithm converges to a determined local influence estimation (LIE) value in each run, eliminating the phenomenon of large fluctuation of LIE fitness value generated by the original DBA algorithm. Finally, the simulated results achieved under the independent cascade model show that the clique-DBA algorithm has a comparable performance of influence spreading compared with the algorithms proposed in the state of the art.

A Hybrid Evolutionary Algorithm for the Clique Partitioning Problem

The clique partitioning problem (CPP) of an edge-weighted complete graph is to partition the vertex set V into k disjoint subsets such that the sum of the edge weights within all cliques induced by the subsets is as large as possible. The problem has a number of practical applications in areas, such as data mining, engineering, and bioinformatics, and is, however, computationally challenging. To solve this NP-hard problem, we propose the first evolutionary algorithm that combines a dedicated merge-divide crossover operator to generate offspring solutions and an effective simulated annealing-based local optimization procedure to find high-quality local optima. The extensive experiments on three sets of 94 benchmark instances (including two sets of 63 classical benchmark instances and one new set of 31 large benchmark) show a remarkable performance of the proposed approach compared to the state-of-the-art methods. We analyze the key algorithmic ingredients to shed light on their impacts on the performance of the algorithm. The algorithm and its available source code can benefit people working on practical problems related to CPP.

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators. Tame graph classes have good algorithmic properties, which follow, for example, from an algorithmic metatheorem of Fomin, Todinca, and Villanger from 2015. We show that a hereditary graph class $\mathcal{G}$ is tame if and only if the subclass consisting of graphs in $\mathcal{G}$ without clique cutsets is tame. This result and Ramsey's theorem lead to several types of sufficient conditions for a graph class to be tame. In particular, we show that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, where a complete prism is the Cartesian product of a complete graph with a $K_2$. We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.

Pseudo orthogonal Latin squares

Abstract Two Latin squares A, B of order n are called pseudo orthogonal if for any 1 ≤ i, j ≤ n there exists a k, 1 ≤ k ≤ n, such that A(i, k) = B(j, k). We prove that the existence of a family of m mutually pseudo orthogonal Latin squares of order n is equivalent to the existence of a family of m mutually orthogonal Latin squares of order n. We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.

Limit theory of combinatorial optimization for random geometric graphs

In the random geometric graph G(n,rn), n vertices are placed randomly in Euclidean d-space and edges are added between any pair of vertices distant at most rn from each other. We establish strong laws of large numbers (LLNs) for a large class of graph parameters, evaluated for G(n,rn) in the thermodynamic limit with nrnd= const., and also in the dense limit with nrnd→∞, rn→0. Examples include domination number, independence number, clique-covering number, eternal domination number and triangle packing number. The general theory is based on certain subadditivity and superadditivity properties, and also yields LLNs for other functionals such as the minimum weight for the traveling salesman, spanning tree, matching, bipartite matching and bipartite traveling salesman problems, for a general class of weight functions with at most polynomial growth of order d−ε, under thermodynamic scaling of the distance parameter.

Edge clique covers in graphs with independence number two

AbstractThe edge clique cover number of a graph is the size of the smallest collection of complete subgraphs whose union covers all edges of . Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if is claw‐free, then is bounded above by its order (denoted ). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in for for such graphs, improving upon the bou nd of due to Javadi, Maleki, and Omoomi. More precisely we prove that is at most the minimum of and , where is the minimum degree of . In the fractional version of the problem, we improve these upper bounds to . We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for ) equals , and when contains no induced subgraph isomorphic to where is any fixed graph of order 4.

Parameterized algorithms for identifying gene co-expression modules via weighted clique decomposition.

We present a new combinatorial model for identifying regulatory modules in gene co-expression data using a decomposition into weighted cliques. To capture complex interaction effects, we generalize the previously-studied weighted edge clique partition problem. As a first step, we restrict ourselves to the noise-free setting, and show that the problem is fixed parameter tractable when parameterized by the number of modules (cliques). We present two new algorithms for finding these decompositions, using linear programming and integer partitioning to determine the clique weights. Further, we implement these algorithms in Python and test them on a biologically-inspired synthetic corpus generated using real-world data from transcription factors and a latent variable analysis of co-expression in varying cell types.

Parameterized algorithms for locating-dominating sets

A locating-dominating set D of a graph G is a dominating set of G where each vertex not in D has a unique neighborhood in D, and the Locating-Dominating Set problem asks if G contains such a dominating set of bounded size. This problem is known to be NP-hard even on restricted graph classes, such as interval graphs, split graphs, and planar bipartite subcubic graphs. On the other hand, it is known to be solvable in polynomial time for some graph classes, such as trees and, more generally, graphs of bounded cliquewidth. While these results have numerous implications on the parameterized complexity of the problem, little is known in terms of kernelization under structural parameterizations. In this work, we begin filling this gap in the literature. Our first result shows that Locating-Dominating Set is W[1]-hard when parameterized by the size of a minimum clique cover. We present an exponential kernel for the distance to cluster parameterization and show that, unless NP ⊆ coNP/poly, no polynomial kernel exists for Locating-Dominating Set when parameterized by vertex cover nor when parameterized by distance to clique. We then turn our attention to parameters not bounded by either of the previous two, and exhibit a linear kernel when parameterizing by the max leaf number; in this context, we leave the parameterization by feedback edge set as the primary open problem in our study.

Monopolar graphs: Complexity of computing classical graph parameters

A graph G=(V,E) is monopolar if V can be partitioned into a stable set and a set inducing the union of vertex-disjoint cliques. Motivated by an application of the clique partitioning problem on monopolar graphs to the cosmetic manufacturing, we study the complexity of computing classical graph parameters on the class of monopolar graphs. We show that computing the clique partitioning, stability and chromatic numbers of monopolar graphs is NP-hard. Conversely, we prove that every monopolar graph has a polynomial number of maximal cliques thus obtaining that a maximum-weight clique can be found in polynomial time on monopolar graphs.

CPG graphs: Some structural and hardness results

In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in Deniz et al. (2018). A CPG graph is a graph for which there exists a collection of pairwise interiorly disjoint paths on a grid in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. If every such path has at most k bends for some k≥0, the graph is said to be Bk-CPG.We first show that, for any k≥0, the class of Bk-CPG graphs is strictly contained in the class of Bk+1-CPG graphs even within the class of planar graphs, thus implying that there exists no k≥0 such that every planar CPG graph is Bk-CPG. The main result of the paper is that recognizing CPG graphs and Bk-CPG graphs with k≥1 is NP-complete. Moreover, we show that the same remains true even within the class of planar graphs in the case k≥3. We then consider several graph problems restricted to CPG graphs and show, in particular, that Independent Set and Clique Cover remain NP-hard for B0-CPG graphs. Finally, we consider the related classes Bk-EPG of edge-intersection graphs of paths with at most k bends on a grid. Although it is possible to optimally color a B0-EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is NP-complete for B1-EPG graphs.

Latest Algorithms on Particular Graph Classes

Many optimization problems such as Maximum Independent Set, Maximum Clique, Minimum Clique Cover and Maximum Induced Matching are NP-hard on general graphs. However, they could be solved in polynomial time when restricted to some particular graph classes such as comparability and co-comparability graph classes. In this paper, we summarize the latest algorithms solving some classical NP-hard problems on some graph classes over the years. Moreover, we apply the -redundant technique to obtain linear time O(j j) algorithms which find a Maximum Induced Matching on interval and circular-arc graphs. Inspired of these results, we have proposed some competitive programming problems for some programming contests in Vietnam in recent years.

On the Complexity of Colouring Antiprismatic Graphs

A graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs. They showed that the class can be divided into two subclasses: the orientable prismatic graphs, and the non-orientable prismatic graphs. We give a polynomial time algorithm that solves the clique cover problem in every non-orientable prismatic graph. It relies on the the structural description and on later work of Javadi and Hajebi. We give a polynomial time algorithm which solves the vertex-disjoint triangles problem for every prismatic graph. It does not rely on the structural description.

Scalable Activation of Rare Triggers in Hardware Trojans by Repeated Maximal Clique Sampling

Hardware Trojans are serious threat to security and reliability of computing systems. It is hard to detect these malicious implants using traditional validation methods since an adversary is likely to hide them under rare trigger conditions. While existing statistical test generation methods are promising for Trojan detection, they are not suitable for activating extremely rare trigger conditions in stealthy Trojans. To address the fundamental challenge of activating rare triggers, we propose a new test generation paradigm for trigger activation by repeated maximal clique sampling (TARMAC). The basic idea is to utilize a satisfiability modulo theories (SMTs) solver to construct a test corresponding to each maximal clique. This article makes three fundamental contributions: 1) it proves that the trigger activation problem can be mapped to clique cover problem, and the test vectors generated by covering maximal cliques are complete and compact; 2) it proposes efficient test generation algorithms to activate trigger conditions by repeated maximal clique sampling; and 3) it outlines an efficient mechanism to run the clique sampling in parallel to significantly improve the scalability of our test generation framework. The experimental results demonstrate that our proposed approach is scalable and it outperforms state-of-the-art approaches by several orders-of-magnitude in detecting stealthy Trojans.