DIMACS Graphs Research Articles

An exact algorithm to find a maximum weight clique in a weighted undirected graph

We introduce a new algorithm MaxCliqueWeight for identifying a maximum weight clique in a weighted graph, and its variant MaxCliqueDynWeight with dynamically varying bounds. This algorithm uses an efficient branch-and-bound approach with a new weighted graph coloring algorithm that efficiently determines upper weight bounds for a maximum weighted clique in a graph. We evaluate our algorithm on random weighted graphs with node counts up to 10,000 and on standard DIMACS benchmark graphs used in a variety of research areas. Our findings reveal a remarkable improvement in computational speed when compared to existing algorithms, particularly evident in the case of high-density random graphs and DIMACS graphs, where our newly developed algorithm outperforms existing alternatives by several orders of magnitude. The newly developed algorithm and its variant are freely available to the broader research community at http://insilab.org/maxcliqueweight, paving the way for transformative applications in various research areas, including drug discovery.

Improving lower bounds for equitable chromatic number

In many practical applications the underlying graph must be as equitable colored as possible. A coloring is called equitable if the sizes of any two color classes differ by at most one, and the least number of colors for which a graph has such a coloring is called its equitable chromatic number.We introduce a new integer linear programming approach for studying the equitable coloring number of a graph and show how to use it for improving lower bounds for this number. The two stage method is based on finding or upper bounding the maximum cardinality of an equitable color class in a valid equitable coloring and, then, sequentially improving the lower bound for the equitable coloring number.The computational experiments were carried out on DIMACS graphs and other graphs from the literature. The lower bounds were improved for more than half of the instances considered.

Testing a Heuristic Algorithm for Finding a Maximum Clique on DIMACS and Facebook Graphs

In this paper we propose a new heuristic algorithm for solving a maximum clique search problem (MCP). While the proposed algorithm (called TrustCLQ) uses a general approach to solving MCP, it is almost independent of the order of vertices and does not exploit a partition of the graph into independent sets. The algorithm was tested on DIMACS library graphs which are often employed for testing MCP solution algorithms. TrustCLQ algorithm was compared with the well-known ILS heuristic algorithm (as well as with a standard algorithm from networkx library) on DIMACS data sets. Moreover, TrustCLQ algorithm has been tested on Facebook social graphs

Parallel Computing Aspects in Improved Edge Cover Based Graph Coloring Algorithm

Objective: To improve the Edge Cover based Graph Coloring Algorithm (ECGCA) using independent set by incorporating parallel computing aspects in algorithm. Finding optimum time complexity is one of the main objectives of this paper. Methods/Statistical Analysis: This paper introduced some modification in ECGCA. Algorithm is implemented and tested using Java programming language. Java multithreading concept is used to achieve parallel computing in algorithm. DIMACS graph instances are used to test algorithm. Finding: Algorithm is tested on more than 75 DIMACS graph instances. To analyze the time complexity, execution time of algorithm in seconds is calculated by program. Algorithm is tested on different DIMACS graph instances. Test data is analyze in this paper and found that proposed algorithm executed in optimum time for large graphs. This paper also compared parallel algorithm and found that proposed parallel algorithm is less time complex than sequential algorithm. Most of the exact graph coloring algorithms are not suitable for large graph (more than 100 vertices) but proposed algorithm is tested on many large graphs and high execution success rate of algorithm is achieved. Application: This paper shows the experimental results of different type of application data. It means this algorithm can be used for maximum types of applications.

Parallel Hybrid Genetic Algorithm for Maximum Clique Problem on OpenCL

The maximum clique problem is to find the maximum sized clique of pairwise adjacent vertices in a given graph, which is a NP-Complete problem. In this paper, an effective parallel hybrid genetic algorithm is proposed, which consists of genetic algorithm and a local optimization strategy for solving maximum clique problem. In this algorithm, selection, crossover, mutation, fitness evaluation and replacement operators are implemented parallel on OpenCL. In addition, we have tested our algorithm by using a set of benchmark instances from the DIMACS graphs. The comparison results shows that the implementation on GPU provide better performance that CPU, even when the benchmark graphs become more large and complicate.

Branch-and-bound algorithm for the maximum triangle packing problem

This work addresses the problem of finding the maximum number of unweighted vertex-disjoint triangles in an undirected graph G. It is a challenging NP-hard combinatorial problem and it is well-known to be APX-hard. A branch-and-bound algorithm which uses a lower bound based on neighborhood degree is presented. A naive upper bound is proposed as well as another one based on a surrogate relaxation of the related integer linear program which is analogous to a multidimensional knapsack problem. Further, a Greedy Search algorithm and a genetic algorithm are described to improve the lower bound. A computational comparison of lower bounds, branch-and-bound algorithm and CPLEX solver is provided using randomly generated benchmarks and well-known DIMACS implementation challenges. The empirical study shows that the branch-and-bound finds the optimal triangle packing solution for small randomly generated MTP instances (up to 100 vertices and 200 triangles) and some DIMACS graphs. For some larger instances and DIMACS challenges graphs, we remark that our lower bound outperforms CPLEX solver regarding the triangle packing solution and the computation time.

An Immunological Algorithm for Combinatorial Optimization: the Fuel Distribution Problem as Case Study

Abstract Routing problems are classical combinatorial optimization tasks that find much applicability in numerous industrial and real-world scenarios. One challenging variant of the routing problem is the Fuel Distribution Problem (FDP) that a transportation company must face in its everyday operations. The main activity of a transportation fuel company is restocking all its stores, i.e. petrol stations, along a geographical map, with the goal to minimizing its overall costs. In this research work we present a hybrid heuristic based on the metaphor of the immune system for solving the FDP, which basically asks to find a set of routes as shorter as possible for a fixed number of company’s vehicles in order to satisfy the several received demands of customers. In particular, the presented immunological algorithm takes inspiration by the clonal selection principle, whose key features are cloning, hyper- mutation, and aging operators. Such algorithm is also characterized, in having a (i) deterministic approach based on the Depth First Search (DFS) algorithm - used in the scheme of assigning a vertex to a vehicle - and (ii) a local search operator, based on the exploration of the neighborhood. The algorithm has been tested on one real data instance, with 82 vertices, and 25 others artificial different instances, taken from DIMACS graph coloring benchmark. The experimental results presented in this work, not only prove the robustness and efficiency of the developed algorithm, but show also the goodness of the local search, and the approach based on the DFS algorithm. Both methodologies help the algorithm to better explore the complex search space.

Finding quasi core with simulated stacked neural networks

Studying networks is promising for diverse applications. We are often interested in exploring significant substructures in different types of real-life networks. Finding cliques, which denote a complete subgraph of a graph, is one such important problem in network analysis. Interestingly, many real-life networks often contain a significant number of almost (quasi) complete subgraphs, which are not entirely complete due to the presence of noise. Considering these networks as weighted adds further challenges to the problem. Finding quasi-complete subgraphs in weighted graphs has never been formally addressed. In this paper, we propose a stacked neural network model for finding out the largest quasi-complete module (core) in weighted graphs. We show the effectiveness of the proposed approach on DIMACS graphs. We also highlight its utility in analyzing scientific collaboration networks, social networks and biological networks.

A branch-and-cut algorithm for the equitable coloring problem using a formulation by representatives

An equitable k-coloring of a graph is defined by a partition of its vertices into k disjoint stable subsets, such that the difference between the cardinalities of any two subsets is at most one. The equitable coloring problem consists of finding the minimum value of k such that a given graph can be equitably k-colored. We present two new integer programming formulations based on representatives for the equitable coloring problem. We propose a primal constructive heuristic, branching strategies, and the first branch-and-cut algorithm in the literature of the equitable coloring problem. The computational experiments were carried out on randomly generated graphs, DIMACS graphs, and other graphs from the literature.

Comments on "The 1993 DIMACS graph coloring Challenge" and "Energy function-based approaches to graph Coloring"

Since all graphs in the 1993 DIMACS graph coloring challenge are undirected, each edge should be only counted once. However, in some files each edge is counted once, whereas in others each edge is counted twice; so a systematical check on the DIMACS challenge is made to eliminate the inconsistencies. Besides, the experimental results of a previous paper by Blas et al. counted each violated edges twice and neglected the inconsistencies in the DIMACS challenge. So the correct experimental results of a previous paper by Blas et al are also given.