Complexity of two coloring problems in cubic planar bipartite mixed graphs

In this note we consider two coloring problems in mixed graphs, i.e., graphs containing edges and arcs, which arise from scheduling problems where disjunctive and precedence constraints have to be taken into account. We show that they are both NP-complete in cubic planar bipartite mixed graphs, which strengthens some results of Ries and de Werra (2008) [9].     

A short proof of the NP-completeness of minimum sum interval coloring

In the minimum sum coloring problem we have to assign positive integers to the vertices of a graph in such a way that neighbors receive different numbers and the sum of the numbers is minimized. Szkalicki has shown that minimum sum coloring is NP-hard for interval graphs. Here we present a simpler proof of this result.     

Optimal shift coloring of trees

This paper was motivated by the problem of scheduling the openings of pharmacies during week-ends and holiday periods (shifts). The problem can be modeled as a coloring problem on a graph. In this paper we focus on the special case where the underlying graph is a tree, or, more generally, it is endowed with a tree-metric, and we provide a polynomial-time algorithm. We also provide direct optimal solutions for special trees like stars and paths.     

Complexity results for minimum sum edge coloring

In the Minimum Sum Edge Coloring problem we have to assign positive integers to the edges of a graph such that adjacent edges receive different integers and the sum of the assigned numbers is minimal. We show that the problem is (a) NP-hard for planar bipartite graphs with maximum degree 3, (b) NP-hard for 3-regular planar graphs, (c) NP-hard for partial 2-trees, and (d) APX-hard for bipartite graphs.     

A branch-and-price approach for the partition coloring problem

This work proposes a new integer programming model for the partition coloring problem and a branch-and-price algorithm to solve it. Experiments are reported for random graphs and instances originating from routing and wavelength assignment problems arising in telecommunication network design. We show that our method largely outperforms previously existing approaches.     

An ant-based algorithm for coloring graphs

This paper presents an ant-based algorithm for the graph coloring problem. An important difference that distinguishes this algorithm from previous ant algorithms is the manner in which ants are used in the algorithm. Unlike previous ant algorithms where each ant colors the entire graph, each ant in this algorithm colors just a portion of the graph using only local information. These individual coloring actions by the ants form a coloring of the graph. Even with the lack of pheromone laying capacity by the ants, the algorithm performed well on a set of 119 benchmark graphs. Furthermore, the algorithm produced very consistent results, having very small standard deviations over 50 runs of each graph tested.     

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χ_ρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.     

On local search for the generalized graph coloring problem

Given an edge-weighted graph and an integer k, the generalized graph coloring problem is the problem of partitioning the vertex set into k subsets so as to minimize the total weight of the edges that are included in a single subset. We recall a result on the equivalence between Karush-Kuhn-Tucker points for a quadratic programming formulation and local optima for the simple flip-neighborhood. We also show that the quality of local optima with respect to a large class of neighborhoods may be arbitrarily bad and that some local optima may be hard to find.     

On coloring box graphs

We consider the chromatic number of a family of graphs we call box graphs, which arise from a box complex in n$n$-space. It is straightforward to show that any box graph in the plane has an admissible coloring with three colors, and that any box graph in n$n$-space has an admissible coloring with n+1$n+1$ colors. We show that for box graphs in n$n$-space, if the lengths of the boxes in the corresponding box complex take on no more than two values from the set {1,2,3}$\{1,2,3\}$, then the box graph is 3$3$-colorable, and for some graphs three colors are required. We also show that box graphs in 3-space which do not have cycles of length four (which we call “string complexes”) are 3$3$-colorable.     

求解图着色问题的量子蚁群算法

针对经典的图着色问题,在蚁群算法的基础上结合量子计算提出一种求解图着色问题的量子蚁群算法. 将量子比特和量子逻辑门引入到蚁群算法中,较好地避免了蚁群算法搜索易陷入局部极小的缺陷,并显著加快了算法的运算速度. 通过图着色实例的大量仿真实验,表明算法对图着色问题的求解是可行的、有效的,且具有通用性.     

Minimum sum set coloring of trees and line graphs of trees

In this paper, we study the minimum sum set coloring (MSSC) problem which consists in assigning a set of x(v) positive integers to each vertex v of a graph so that the intersection of sets assigned to adjacent vertices is empty and the sum of the assigned set of numbers to each vertex of the graph is minimum. The MSSC problem occurs in two versions: non-preemptive and preemptive. We show that the MSSC problem is strongly NP-hard both in the preemptive case on trees and in the non-preemptive case in line graphs of trees. Finally, we give exact parameterized algorithms for these two versions on trees and line graphs of trees.     

On a generalization of the Gallai-Roy-Vitaver theorem to the bandwidth coloring problem

We generalize to the bandwidth coloring problem a classical theorem, discovered independently by Gallai, Roy and Vitaver, in the context of the graph coloring problem. Two proofs are given, a simple one and a more complex one that is based on a series of equivalent mathematical programming models.     

An infinite family of subcubic graphs with unbounded packing chromatic number

Recently, Balogh et al. (2018) answered in negative the question that was posed in several earlier papers whether the packing chromatic number is bounded in the class of graphs with maximum degree 3. In this note, we present an explicit infinite family of subcubic graphs with unbounded packing chromatic number.     

Quantum annealing of the graph coloring problem

Quantum annealing extends simulated annealing by introducing artificial quantum fluctuations. The path-integral Monte Carlo version chosen is population-based and designed to be implemented on a classical computer. Its first application to the graph coloring problem is presented in this paper. It is shown by experiments that quantum annealing can outperform classical thermal simulated annealing for this particular problem. Moreover, quantum annealing proved competitive when compared with the best algorithms on most of the difficult instances from the DIMACS benchmarks. The quantum annealing algorithm has even found that the well-known benchmark graph dsjc1000.9 has a chromatic number of at most 222. This is an improvement on its best upper-bound from a large body of literature.     

List coloring of Cartesian products of graphs

A well-established generalization of graph coloring is the concept of list coloring. In this setting, each vertex v of a graph G is assigned a list L(v) of k colors and the goal is to find a proper coloring c of G with c(v)∈L(v). The smallest integer k for which such a coloring c exists for every choice of lists is called the list chromatic number of G and denoted by χ_l(G).We study list colorings of Cartesian products of graphs. We show that unlike in the case of ordinary colorings, the list chromatic number of the product of two graphs G and H is not bounded by the maximum of χ_l(G) and χ_l(H). On the other hand, we prove that χ_l(G×H)?min{χ_l(G)+col(H),col(G)+χ_l(H)}-1 and construct examples of graphs G and H for which our bound is tight.     

On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0-1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.     

Complexity of the job insertion problem in multi-stage scheduling

The job insertion problem in multi-stage scheduling is: given a schedule for n jobs and an additional job, find a feasible insertion of the additional job into the schedule that minimizes the resulting makespan. We prove that finding the optimal job insertion is NP-hard for flow shops and open shops.     

A memetic algorithm for graph coloring

Given an undirected graph G=(V,E)$G=(V\text{,}E)$ with a set V of vertices and a set E of edges, the graph coloring problem consists of partitioning all vertices into k independent sets and the number of used colors k is minimized. This paper presents a memetic algorithm (denoted by MACOL) for solving the problem of graph coloring. The proposed MACOL algorithm integrates several distinguished features such as an adaptive multi-parent crossover (AMPaX) operator and a distance-and-quality based replacement criterion for pool updating. The proposed algorithm is evaluated on the DIMACS challenge benchmarks and computational results show that the proposed MACOL algorithm achieves highly competitive results, compared with 11 state-of-the-art algorithms. The influence of some ingredients of MACOL on its performance is also analyzed.     

The graph coloring problem: A neuronal network approach

Solution of an optimization problem with linear constraints through the continuous Hopfield network (CHN) is based on an energy or Lyapunov function that decreases as the system evolves until a local minimum value is attained. This approach is extended in to optimization problems with quadratic constraints. As a particular case, the graph coloring problem (GCP) is analyzed. The mapping procedure and an appropriate parameter-setting procedure are detailed. To test the theoretical results, some computational experiments solving the GCP are shown.