In this paper, we revisit SAT encodings of the partial-ordering based ILP model for the graph coloring problem (GCP) and suggest a generalization for the bandwidth coloring problem (BCP). The GCP asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two adjacent vertices get different colors. The BCP is a generalization, where each edge has a weight that enforces a minimal'distance' between the assigned colors, and the goal is to minimize the'largest' color used. For the widely studied GCP, we experimentally compare the partial-ordering based SAT encoding to the state-of-the-art approaches on the DIMACS benchmark set. Our evaluation confirms that this SAT encoding is effective for sparse graphs and even outperforms the state-of-the-art on some DIMACS instances. For the BCP, our theoretical analysis shows that the partial-ordering based SAT and ILP formulations have an asymptotically smaller size than that of the classical assignment-based model. Our practical evaluation confirms not only a dominance compared to the assignment-based encodings but also to the state-of-the-art approaches on a set of benchmark instances. Up to our knowledge, we have solved several open instances of the BCP from the literature for the first time.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clus- ters exactly using advanced techniques from the knowledge compilation literature.

In this paper, we combine a novel Sequential Graph Coloring Heuristic Algorithm (SGCHA) with a non-systematic method based on a cultural algorithm to solve the graph coloring problem (GCP). The GCP involves finding the minimum number of colors for coloring the graph vertices such that adjacent vertices have distinct colors. In our solving approach, we first use an estimator which is implemented with SGCHA to predict the minimum colors. Then, in the non-systematic part which has been designed using cultural algorithms, we improve the prediction. Various components of the cultural algorithm have been implemented to solve the GCP with a self adaptive behavior in an efficient manner. As a result of utilizing the SGCHA and a cultural algorithm, the proposed method is capable of finding the solution in a very efficient running time. The experimental results show that the proposed algorithm has a high performance in time and quality of the solution returned for solving graph coloring instances taken from DIMACS website. The quality of the solution is measured here by comparing the returned solution with the optimal one.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clusters exactlyusing advanced techniques from the knowledge compilation literature.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clusters exactly using advanced techniques from the knowledge compilation literature.

We show that an important and computationally challenging solution space feature of the graph coloring problem (COL), namely the number of clusters of solutions, can be accurately estimated by a technique very similar to one for counting the number of solutions. This cluster counting approach can be naturally written in terms of a new factor graph derived from the factor graph representing the COL instance. Using a variant of the Belief Propagation inference framework, we can efficiently approximate cluster counts in random COL problems over a large range of graph densities. We illustrate the algorithm on instances with up to 100, 000 vertices. Moreover, we supply a methodology for computing the number of clusters exactly using advanced techniques from the knowledge compilation literature.