An analysis between different algorithms for the graph vertex coloring problem

This research focuses on an analysis of different algorithms for the graph vertex coloring problem. Some approaches to solving the problem are discussed. Moreover, some studies for the problem and several methods for its solution are analyzed as well. An exact algorithm (using the backtracking method) is presented. The complexity analysis of the algorithm is discussed. Determining the average execution time of the exact algorithm is consistent with the multitasking mode of the operating system. This algorithm generates optimal solutions for all studied graphs. In addition, two heuristic algorithms for solving the graph vertex coloring problem are used as well. The results show that the exact algorithm can be used to solve the graph vertex coloring problem for small graphs with 30-35 vertices. For half of the graphs, all three algorithms have found the optimal solutions. The suboptimal solutions generated by the approximate algorithms are identical in terms of the number of colors needed to color the corresponding graphs. The results show that the linear increase in the number of vertices and edges of the analyzed graphs causes a linear increase in the number of colors needed to color these graphs

Determining the expected runtime of an exact graph coloring algorithm

Exact algorithms for graph coloring tend to have high vari- ance in their runtime, posing a signi�cant obstacle to their practical application. The problem could be mitigated by appropriate prediction of the runtime. For this purpose, we devise an algorithm to efficiently compute the expected run-time of an exact graph coloring algorithm as a function of the parameters of the problem instance: the graph's size, edge density, and the number of available colors. Specifically, we investigate the complexity of a typical backtracking algorithm for coloring random graphs with k colors. Using the expected size of the search tree as the measure of complexity, we devise a polynomial-time algorithm for predicting algorithm complexity depending on the parameters of the problem instance. Our method also delivers the expected number of solutions (i.e., number of valid colorings) of the given problem instance, which can help us decide whether the given problem instance is likely to be feasible or not. Based on our algorithm, we also show in accordance with previous results that increasing the number of vertices of the graph does not increase the complexity beyond some com- plexity limit. However, this complexity limit grows rapidly when the number of colors increases

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Algorithmic Applications of Hypergraph and Partition Containers

We present a general method to convert algorithms into faster algorithms for almost-regular input instances. Informally, an almost-regular input is an input in which the maximum degree is larger than the average degree by at most a constant factor. This family of inputs vastly generalizes several families of inputs for which we commonly have improved algorithms, including bounded-degree inputs and random inputs. It also generalizes families of inputs for which we don't usually have faster algorithms, including regular-inputs of arbitrarily high degree and very dense inputs. We apply our method to achieve breakthroughs in exact algorithms for several central NP-Complete problems including $k$-SAT, Graph Coloring, and Maximum Independent Set. Our main tool is the first algorithmic application of the relatively new Hypergraph Container Method (Saxton and Thomason 2015, Balogh, Morris and Samotij 2015). This recent breakthrough, which generalizes an earlier version for graphs (Kleitman and Winston 1982, Sapozhenko 2001), has been used extensively in recent years in extremal combinatorics. An important component of our work is the generalization of (hyper-)graph containers to Partition Containers

Exact Algorithms for the Graph Coloring Problem

The graph coloring problem is the problem of partitioning the vertices of a graph into the smallest possible set of independent sets. Since it is a well-known NP-Hard problem, it is of great interest of the computer science finding results over exact algorithms that solve it. The main algorithms of this kind, though, are scattered through the literature. In this paper, we group and contextualize some of these algorithms, which are based in Dynamic Programming, Branch-and-Bound and Integer Linear Programming. The algorithms for the first group are based in the work of Lawler, which searches maximal independent sets on each subset of vertices of a graph as the base of his algorithm. In the second group, the algorithms are based in the work of Brelaz, which adapted the DSATUR procedure to an exact version, and in the work of Zykov, which introduced the definition of Zykov trees. The third group contains the algorithms based in the work of Mehrotra and Trick, which uses the Column Generation method