On the resolution of certain discrete univariate max–min problems

Graph coloring is a classical NP-hard combinatorial optimization problem with many practical applications. A broad range of heuristic methods exist for tackling the graph coloring problem: from fast greedy algorithms to more time-consuming metaheuristics. Although the latter produce better results in terms of minimizing the number of colors, the former are widely employed due to their simplicity. These heuristic methods are centralized since they operate by using complete knowledge of the graph. However, in real-world environmets where each component only interacts with a limited number of other components, the only option is to apply decentralized methods. This paper explores a novel and simple algorithm for decentralized graph coloring that uses a fixed number of colors and iteratively reduces the edge conflicts in the graph. We experimentally demonstrate that, for most of the tested instances, the new algorithm outperforms a recent and very competitive algorithm for decentralized graph coloring in terms of coloring quality. In our experiments, the fixed number of colors used by the new algorithm is controlled in a centralized manner.     

Coloring planar graphs in parallel

We present NC algorithms for vertex and edge coloring planar graphs. The vertex coloring algorithm 5 colors any planar graph, and the edge coloring algorithm Δ edge colors planar graphs with Δ ≥ 23 (and Δ + 1 edge colors planar graphs with Δ < 23), where Δ is the maximum degree in the graph.     

Solving Graph Coloring Problems with the Douglas-Rachford Algorithm

We present the Douglas-Rachford algorithm as a successful heuristic for solving graph coloring problems. Given a set of colors, these types of problems consist in assigning a color to each node of a graph, in such a way that every pair of adjacent nodes are assigned with different colors. We formulate the graph coloring problem as an appropriate feasibility problem that can be effectively solved by the Douglas-Rachford algorithm, despite the nonconvexity arising from the combinatorial nature of the problem. Different modifications of the graph coloring problem and applications are also presented. The good performance of the method is shown in various computational experiments.     

A Branch-and-Cut algorithm for graph coloring

In this paper a Branch-and-Cut algorithm, based on a formulation previously introduced by us, is proposed for the Graph Coloring Problem. Since colors are indistinguishable in graph coloring, there may typically exist many different symmetrical colorings associated with a same number of colors. If solutions to an integer programming model of the problem exhibit that property, the Branch-and-Cut method tends to behave poorly even for small size graph coloring instances. Our model avoids, to certain extent, that bottleneck. Computational experience indicates that the results we obtain improve, in most cases, on those given by the well-known exact solution graph coloring algorithm Dsatur.     

Optimization by ant algorithms: possible roles for an individual ant

In most ant algorithms, the role of each ant is to build a solution in a constructive way, basing each decision on the greedy force and the trails. However, different roles are possible for each individual ant, ranging from a negligible help in the decision process to a refined local search heuristic. In this paper, the importance of the role assigned to each ant is discussed. Three general ant methodologies are presented. Comparative results are analyzed for the well-known graph coloring problem.     

On-line and first fit colorings of graphs

A graph coloring algorithm that immediately colors the vertices taken from a list without looking ahead or changing colors already assigned is called “on-line coloring.” The properties of on-line colorings are investigated in several classes of graphs. In many cases we find on-line colorings that use no more colors than some function of the largest clique size of the graph. We show that the first fit on-line coloring has an absolute performance ratio of two for the complement of chordal graphs. We prove an upper bound for the performance ratio of the first fit coloring on interval graphs. It is also shown that there are simple families resisting any on-line algorithm: no on-line algorithm can color all trees by a bounded number of colors.     

A new memetic algorithm based on cellular learning automata for solving the vertex coloring problem

Vertex coloring problem is a combinatorial optimization problem in graph theory in which a color is assigned to each vertex of graph such that no two adjacent vertices have the same color. In this paper a new hybrid algorithm which is obtained from combination of cellular learning automata (CLA) and memetic algorithm (MA) is proposed for solving the vertex coloring problem. CLA is an effective probabilistic learning model combining cellular automata and learning automaton (LA). Irregular CLA (ICLA) is a generalization of CLA in which the restriction of rectangular grid structure in CLA is removed. The proposed algorithm is based on the irregular open CLA (IOCLA) that is an extension of ICLA in which the evolution of CLA is influenced by both local and global environments. Similar to other IOCLA-based algorithms, in the proposed algorithm, local environment is constituted by neighboring LAs of any cell and the global environment consists of a pool of memes in which each meme corresponds to a certain local search method. Each meme is represented by a set of LAs from which the history of the corresponding local search method can be extracted. To show the superiority of the proposed algorithm over some well-known algorithms, several computer experiments have been conducted. The results show that the proposed algorithm performs better than other methods in terms of running time of algorithm and the required number of colors.     

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.     

On a generalized family of colorings

Motivated by a question in cellular telecommunication technology, we investigate a family of graph coloring problems where several colors can be assigned to each vertex and no two colors are the same within any ball of radiusR. We find bounds and coloring algorithms for different kinds of graphs including trees,n-cycles, hypercubes and lattices. We briefly examine connections to Heawood's map color theorem and state a few conjectures and open problems.Work in part performed while the author was in the Department of Mathematics, University of California, San Diego.     

Scheduling Problems and Mixed Graph Colorings

Some scheduling problems induce a mixed graph coloring, i.e., an assignment of positive integers (colors) to vertices of a mixed graph such that, if two vertices are joined by an edge, then their colors have to be different, and if two vertices are joined by an arc, then the color of the startvertex has to be not greater than the color of the endvertex. We discuss some algorithms for coloring the vertices of a mixed graph with a small number t of colors and present computational results for calculating the chromatic number, i.e., the minimal possible value of such a t . We also study the chromatic polynomial of a mixed graph which may be used for calculating the number of feasible schedules.     

Minimum sum edge colorings of multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.     

Facial unique-maximum colorings of plane graphs with restriction on big vertices

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova et al. (2018). We conclude the paper by proposing some problems.     

A simple branching scheme for vertex coloring problems

We present a branching scheme for some vertex coloring problems based on a new graph operator called extension. The extension operator is used to generalize the branching scheme proposed by Zykov for the basic problem to a broad class of coloring problems, such as graph multicoloring, where each vertex requires a multiplicity of colors, graph bandwidth coloring, where the colors assigned to adjacent vertices must differ by at least a given distance, and graph bandwidth multicoloring, which generalizes both the multicoloring and the bandwidth coloring problems. We report some computational evidence of the effectiveness of the new branching scheme.     

子立方无爪图的单射边色数至多为6

设$G$是一个图. 图$G$的一个单射边染色是指图$G$的一个边染色, 使得距离为$2$的两条边或者在同一个三角形中的两条边染不同的颜色. 图$G$的单射边色数是指图$G$的任意单射边染色所需要的最少颜色数. 关于单射边色数有一个猜想: 任意一个子立方图的单射边色数都不超过$6$. 在本文中, 我们证明了这个猜想对子立方无爪图是成立的, 并且给出图例说明上界$6$是紧的. 同时, 我们的证明隐含了求解这类图不超过$6$种颜色的单射边染色方案的一个线性时间算法.     

The harmonious coloring problem is NP-complete for interval and permutation graphs

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.     

A New Ant Colony Optimization Algorithm for the Lower Bound of Sum Coloring Problem

We consider an undirected graph G?=?(V, E), the minimum sum coloring problem (MSCP) asks to find a valid vertex coloring of G, using natural numbers (1,2,...), the aim is to minimize the total sum of colors. In this paper we are interested in the elaboration of an approximate solution for the minimum sum coloring problem (MSCP), more exactly we try to give a lower bound for MSCP by looking for a decomposition of the graph based on the metaheuristic of ant colony optimization (ACO). We test different instances to validate our approach.     

A Note on Strong Edge Coloring of Sparse Graphs

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ'_s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2 k. Yang and Zhu [J. Graph Theory, 83, 334–339(2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4 kΔ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤ 4 kΔ-2 k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most(4 k-1)Δ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤(4 k-1)Δ-2 k + 1.     

灾情巡视问题的蚁群两目标优化算法研究

分析目前灾情巡视问题求解方法存在的缺陷,归纳出灾情巡视问题两目标优化模型.针对灾情巡视问题模型特点,引入蚁群算法和多目标优化理论,提出两个灾情巡视问题的蚁群两目标优化算法:算法1将灾情巡视问题的道路网络转化为完全图,增加m-1个(m为巡视组数)虚拟巡视起点,将灾情巡视两目标优化问题转化为单旅行商两目标优化问题,然后使用蚁群算法和多目标优化理论进行迭代求解.算法2使用一只蚂蚁寻找一个子回路,m个子回路构成一个灾情巡视可行方案,采用罚函数法和多目标优化理论构建增广两目标优化评价函数,使用g组,共g×m只蚂蚁共同协作来发现灾情巡视问题的最优解.算法特点:①算法1将灾情巡视两目标优化问题转化为单旅行商两目标优化问题,可以充分利用已有蚁群算法求解单旅行商问题的研究成果;②两个算法引入蚁群算法,提高了算法效率;③两个算法克服目前灾情巡视问题的求解方法不严密性缺陷;④两目标优化算法可以为用户提供多个满足约束条件的Pareto组合解,扩大了用户选择范围,增强了算法的适用性.算法测试表明:灾情巡视问题的蚁群两目标优化算法是完全可行和有效的.     

Achromatic number of fragmentable graphs

A complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge. The achromatic number ψ(G) is the greatest number of colors in such a coloring. We say a class of graphs is fragmentable if for any positive ε, there is a constant C such that any graph in the class can be broken into pieces of size at most C by removing a proportion at most ε of the vertices. Examples include planar graphs and grids of fixed dimension. Determining the achromatic number of a graph is NP‐complete in general, even for trees, and the achromatic number is known precisely for only very restricted classes of graphs. We extend these classes very considerably, by giving, for graphs in any class which is fragmentable, triangle‐free, and of bounded degree, a necessary and sufficient condition for a sufficiently large graph to have a complete coloring with a given number of colors. For the same classes, this gives a tight lower bound for the achromatic number of sufficiently large graphs, and shows that the achromatic number can be determined in polynomial time. As examples, we give exact values of the achromatic number for several graph families. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:94–114, 2010     

A Characterization of b‐Perfect Graphs

A b‐coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b‐chromatic number of a graph G is the largest integer k such that G admits a b‐coloring with k colors. A graph is b‐perfect if the b‐chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is b‐perfect if and only if it does not contain as an induced subgraph a member of a certain list of 22 graphs. This entails the existence of a polynomial‐time recognition algorithm and of a polynomial‐time algorithm for coloring exactly the vertices of every b‐perfect graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:95–122, 2012