Vertex Coloring Problem Research Articles

Lower Bounding Techniques for DSATUR-based Branch and Bound

Given an undirected graph, the Vertex Coloring Problem (VCP) consists of assigning a color to each vertex of the graph such that two adjacent vertices do not share the same color and the total number of colors is minimized. DSATUR-based Branch-and-Bound is a well-known exact algorithm for the VCP. One of its main drawbacks is that a lower bound (equal to the size of a maximal clique) is computed once at the root of the branching scheme and it is never updated during the execution of the algorithm. In this article, we show how to update the lower bound and we compare the efficiency of several lower bounding techniques.

Vertex-coloring of fuzzy graphs: A new approach

In this paper, a new vertex-coloring problem of a fuzzy graph with crisp vertices and fuzzy edges is studied. Membership degree of a fuzzy edge is interpreted as incompatibility degree of its associated incident vertices. This interpretation can be used to define the concept of total incompatibilit y. Here, unlike the traditional graph coloring problems, two adjacent vertices can receive same colors; these type of vertices and their associated edge are named incompatible vertices and incompatible edge, respectively. In proposed coloring methodology, the total incompatibility of a vertex-coloring is defined as the sum of incompatibility degrees of all incompatible edges. Then, based on the minimum possible degree of total incompatibilities, fuzzy chromatic number of a fuzzy graph is introduced. In order to find an optimal k-coloring, with minimum degree of total incompatibly, firstly a binary programming problem is formulated. Then, a hybrid local search genetic algorithm is designed to solve the large-size problems. Practical uses of the proposed algorithm are illustrated and analyzed by different-size problems. Finally, a cell site assignment problem, as a real world application of the presented fuzzy graph vertex-coloring, is formulated and solved.

Solving the Maximum Clique and Vertex Coloring Problems on Very Large Sparse Networks

This paper explores techniques for solving the maximum clique and vertex coloring problems on very large-scale real-life networks. Because of the size of such networks and the intractability of the considered problems, previously developed exact algorithms may not be directly applicable. The proposed approaches aim to reduce the network instances to a size that is tractable for existing solvers, while preserving optimality. Two clique relaxation structures are exploited for this purpose. In addition to the known k-core structure, a newly introduced clique relaxation, k-community, is used to further reduce the instance size. Experimental results on real-life graphs (collaboration networks, P2P networks, social networks, etc.) show the proposed procedures to be effective by finding, for the first time, exact solutions for instances with over 18 million vertices.

Correction: Automatically Generated Algorithms for the Vertex Coloring Problem

In the introduction section, there was an error in the equation in the third sentence of the first paragraph. The sentence should begin: A function F: V ⟼ ℕ is defined, where ℕ is a finite set of colors ... This error is not present in the PDF. There was an error in the affiliation 2 for Gustavo Gatica. Affiliation 2 should be: Escuela de Informatica, Universidad Andres Bello, Santiago, Chile

The Elimination of Spatial-Temporal Uncertainty in Underwater Sensor Networks

Since data in underwater sensor networks (UWSNs) is transmitted by acoustic signals, the characteristics of a UWSN are different from those of a terrestrial sensor network. Specifically, due to the high propagation delay of acoustic signals in UWSNs, referred as spatial-temporal uncertainty, current terrestrial MAC schemes do not work well in UWSNs. Hence, we consider spatial-temporal uncertainty in the design of an energy-efficient TDMA-based MAC protocol for UWSNs. We first translate the TDMA-based scheduling problem in UWSNs into a special vertex-coloring problem in the context of a spatial-temporal conflict graph (ST-CG) that describes explicitly the conflict delays among transmission links. With the help of the ST-CG, we propose two novel heuristic approaches: 1) the traffic-based one-step trial approach (TOTA) to solve the coloring problem in a centralized fashion; and for scalability, 2) the distributed traffic-based one-step trial approach (DTOTA) to assign the data schedule for tree-based routing structures in a distributed manner. In addition, a mixed integer linear programming (MILP) model is derived to obtain a theoretical bound for the TDMA-based scheduling problem in UWSNs. Finally, a comprehensive performance study is presented, showing that both TOTA and DTOTA guarantee collision-free transmission. They thus outperform existing MAC schemes such as S-MAC, ECDiG, and T-Lohi in terms of network throughput and energy consumption.

Solving Vertex Coloring Problem with Cellular Ant Algorithm

Graph Coloring Problem (GCP) is one of the most studied combinatorial optimization problems. Its simplest form is called vertex coloring. It is important in theory and practice. Cellular Ant Algorithm is attempted to graph coloring problem- vertex coloring. Cellular ant algorithm is a new optimization method for solving real problems by using both the evolutionary rule of cellular, graph theory and the characteristics of ant colony optimization. Empirical study of the proposed Cellular Ant Algorithm is also carried out based on the DIMACS challenge benchmarks. The computational results show that Cellular Ant Algorithm for graph coloring problem is feasible and robust.

(k,c) – coloring via Column Generation

Abstract In this paper we study the ( k , c ) – coloring problem, a generalization of the well known Vertex Coloring Problem (VCP). We propose a new formulation and compare it computationally with another formulation from the literature. We also develop a diving heuristic that provides with good quality results at a reasonable computational effort.

Minimizing Warehouse Space with a Dedicated Storage Policy

Given the increasingly significant impact of an efficient product-location strategy on warehouses' performance from a service level and operational costs perspective, this paper presents a possible operations research-oriented solution to provide a tangible reduction of the overall required warehousing space, thereby translating the storage location assignment problem (SLAP) into a vertex colouring problem (VCP). Developing the topic of their previous work on the development of an effective multi-product slot-code optimization heuristic, the authors focused on finding a cost-effective way to solve the SLAP through a mathematical-optimization approach. The formulation validation on a real industrial case showed its high optimization potential, and benchmarking simulations displayed performances significantly close to the best theoretical case. Indeed, the optimized value results were definitively close to the SLAP lower bound calculated assuming a randomized storage policy which, distinct from the developed solution, must inevitably be supported by warehouse management system software. On the contrary, the proposed methodology relies upon a dedicated storage policy, which is easily implementable by companies of all sizes without the need for investing in expensive IT tools.

A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems

Using the semi-tensor product of matrices, this paper investigates the maximum (weight) stable set and vertex coloring problems of graphs with application to the group consensus of multi-agent systems, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for the internally stable set problem, based on which a new algorithm to find all the internally stable sets is established for any graph. Secondly, the maximum (weight) stable set problem is considered, and a necessary and sufficient condition is presented, by which an algorithm to find all the maximum (weight) stable sets is obtained. Thirdly, the vertex coloring problem is studied by using the semi-tensor product method, and two necessary and sufficient conditions are proposed for the colorability, based on which a new algorithm to find all the k-coloring schemes and minimum coloring partitions is put forward. Finally, the obtained results are applied to multi-agent systems, and a new protocol design procedure is presented for the group consensus problem. The study of illustrative examples shows that the results/algorithms presented in this paper are very effective.

Exact weighted vertex coloring via branch-and-price

We consider the Weighted Vertex Coloring Problem (WVCP), in which a positive weight is associated to each vertex of a graph. In WVCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used, where the cost of each color is given by the maximum weight of the vertices assigned to that color. This NP-hard problem arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose the first exact algorithm for the problem, which is based on column generation and branch-and-price. Computational results on a large set of instances from the literature are reported, showing excellent performance when compared with the best heuristic algorithms from the literature.

Distributed deterministic edge coloring using bounded neighborhood independence

We study the edge-coloring problem in the message-passing model of distributed computing. This is one of the most fundamental problems in this area. Currently, the best-known deterministic algorithms for (2Δ −1)-edge-coloring requires O(Δ) + log* n time (Panconesi and Rizzi in Distrib Comput 14(2):97–100, 2001), where Δ is the maximum degree of the input graph. Also, recent results of Barenboim and Elkin (2010) for vertex-coloring imply that one can get an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}\cdot \log n)}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ log n) time, for an arbitrarily small constant $${\epsilon > 0}$$ . In this paper we devise a significantly faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}) + \log* n}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ) + log* n time. This result improves the state-of-the-art running time for deterministic edge-coloring with this number of colors in almost the entire range of maximum degree Δ. Moreover, it improves it exponentially in a wide range of Δ, specifically, for 2 Ω(log*n) ≤ Δ ≤ polylog(n). In addition, for small values of Δ (up to log1 - δ n, for some fixed δ > 0) our deterministic algorithm outperforms all the existing randomized algorithms for this problem. Also, our algorithm is the first O(Δ)-edge-coloring algorithm that has running time o(Δ) + log* n, for the entire range of Δ. All previous (deterministic and randomized) O(Δ)-edge-coloring algorithms require $${\Omega(\min \{\Delta, \sqrt{\log n}\ \})}$$ time. On our way to these results we study the vertex-coloring problem on graphs with bounded neighborhood independence. This is a large family of graphs, which strictly includes line graphs of r-hypergraphs (i.e., hypergraphs in which each hyperedge contains r or less vertices) for r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to general graphs. Our main technical contribution is a subroutine that computes an O(Δ/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p 2) + log* n time, for a parameter p, 1 ≤ p ≤ Δ. In all previous efficient distributed routines for m-defective p-coloring the product m· p is super-linear in Δ. In our routine this product is linear in Δ, and this enables us to speed up the algorithm drastically.

Determining the chromatic number of triangle-free [formula omitted]-free graphs in polynomial time

Let 2P3 denote the disjoint union of two paths on three vertices. A graph G that has no subgraph isomorphic to a graph H is called H-free. The Vertex Coloring problem is the problem to determine the chromatic number of a graph. Its computational complexity for triangle-free H-free graphs has been classified for every fixed graph H on at most 6 vertices except for the case H=2P3. This remaining case is posed as an open problem by Dabrowski, Lozin, Raman and Ries. We solve their open problem by showing polynomial-time solvability.

Advice Complexity of the Online Vertex Coloring Problem

We study online algorithms with advice for the problem of coloring graphs which come as input vertex by vertex. We consider the class of all 3-colorable graphs and its sub-classes of chordal and maximal outerplanar graphs, respectively. We show that, in the case of the first two classes, for coloring optimally, essentially log2 3 advice bits per vertex (bpv) are necessary and sufficient. In the case of maximal outerplanar graphs, we show a lower bound of 1.0424 bpv and an upper bound of 1.2932 bpv. Finally, we develop algorithms for 4-coloring in these graph classes. The algorithm for 3-colorable chordal and outerplanar graphs uses 0.9865 bpv, and in case of general 3-colorable graphs, we obtain an algorithm using < 1.1583 bpv.

L-Factors and Adjacent Vertex-Distinguishing Edge-Weighting

An edge weighting problem of a graph G is an assignment of an integer weight to each edge e. Based on edge weighting problem, several types of vertex-coloring problems are put forward. A simple observation illuminates that edge weighting problem has a close relationship with special factors of graphs. In this paper, we obtain several results on the existence of factors with the pre-specified degrees, which generalizes earlier results in [2, 3]. Using these results, we investigate edge-weighting problem. In particular, we prove that every 4-colorable graph admits a vertex-coloring 4-edge-weighting.

정점 색칠 문제의 다항시간 알고리즘

본 논문은 지금까지 NP-완전인 난제로 알려진 정점 색칠 문제를 선형시간 복잡도로 해결한 알고리즘을 제안하였다. 제안된 알고리즘은 그래프 G=(V,E)의 최소 채색수 <TEX>${\chi}(G)$</TEX>=k를 결정하기 위해 사전에 k값을 알지 못한다는 가정에 기반하고 있다. 단지 주어진 그래프를 독립집합 <TEX>$\overline{C}$</TEX>와 정점 피복 집합 C로 정확히 양분하여 <TEX>$\overline{C}$</TEX>에 색을 배정하는 방법을 적용하였다. 독립집합 <TEX>$\overline{C}$</TEX>의 원소는 <TEX>${\delta}(G)$</TEX>인 정점 <TEX>${\upsilon}$</TEX>가, C의 원소는 정점 <TEX>${\upsilon}$</TEX>의 인접 정점들 u가배정된다. 축소된 그래프 C는 다시 <TEX>$\overline{C}$</TEX>와 C로 양분되며, 이 과정을 C의 간선이 없을 때까지 수행한다. 26개의 다양한 그래프를 대상으로 제안된 알고리즘을 적용한 결과 정점 <TEX>${\upsilon}$</TEX>를 선택하는 횟수는 정점의 수 n보다 작은 값을 나타내었으며, <TEX>${\chi}(G)$</TEX>=k를 찾는데 성공하였다. The Vertex Coloring Problem hasn't been solved in polynomial time, so this problem has been known as NP-complete. This paper suggests linear time algorithm for Vertex Coloring Problem (VCP). The proposed algorithm is based on assumption that we can't know a priori the minimum chromatic number <TEX>${\chi}(G)$</TEX>=k for graph G=(V,E) This algorithm divides Vertices V of graph into two parts as independent sets <TEX>$\overline{C}$</TEX> and cover set C, then assigns the color to <TEX>$\overline{C}$</TEX>. The element of independent sets <TEX>$\overline{C}$</TEX> is a vertex <TEX>${\upsilon}$</TEX> that has minimum degree <TEX>${\delta}(G)$</TEX> and the elements of cover set C are the vertices <TEX>${\upsilon}$</TEX> that is adjacent to <TEX>${\upsilon}$</TEX>. The reduced graph is divided into independent sets <TEX>$\overline{C}$</TEX> and cover set C again until no edge is in a cover set C. As a result of experiments, this algorithm finds the <TEX>${\chi}(G)$</TEX>=k perfectly for 26 Graphs that shows the number of selecting <TEX>${\upsilon}$</TEX> is less than the number of vertices n.

An exact approach for the Vertex Coloring Problem

Given an undirected graph G = ( V , E ) , the Vertex Coloring Problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. In this paper, we present an exact algorithm for the solution of VCP based on the well-known Set Covering formulation of the problem. We propose a Branch-and-Price algorithm embedding an effective heuristic from the literature and some methods for the solution of the slave problem, as well as two alternative branching schemes. Computational experiments on instances from the literature show the effectiveness of the algorithm, which is able to solve, for the first time to proven optimality, five of the benchmark instances in the literature, and reduce the optimality gap of many others.

Linear programming formulation of the vertex colouring problem

In this paper, we present a first linear programming (LP) formulation of the vertex colouring problem (VCP). The complexity orders of the number of variables and the number of constraints of the proposed LP are O (δ9·ς3) and O (δ8·ς3), respectively, where δ and ς are the number of vertices and the number of available colours in the VCP instance, respectively. Hence, the proposed model represents a reaffirmation of 'P = NP'. First, we develop a bipartite network flow (BNF) based model of the problem. Then, we use a graph-based modelling framework similar to that of Diaby to develop the proposed LP model. A numerical example is used to illustrate the approach.

A survey on vertex coloring problems

AbstractThis paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms proposed in the literature are reported. The second part of the paper is devoted to some generalizations of the problem, which are obtained by considering additional constraints [Bandwidth (Multi) Coloring Problem, Bounded Vertex Coloring Problem] or an objective function with a special structure (Weighted Vertex Coloring Problem). The extension of the models for the classical VCP to the considered problems and the best performing algorithms from the literature, as well as the corresponding computational results, are reported.

Energy-efficient power/rate control and scheduling in hybrid TDMA/CDMA wireless sensor networks

We consider a hybrid TDMA/CDMA wireless sensor network (WSN) and quantitatively investigate the energy efficiency obtained by combining adaptive power/rate control with time-domain scheduling. The energy efficiency improvement is carried out with respect to interfering-cluster scheduling, intra-cluster node scheduling, and transmission powers and times (durations) control (PTC) for individual nodes. The interfering-cluster scheduling is formulated as a vertex-coloring problem, which can be solved efficiently using existing numerical algorithms in graph theory. For the node scheduling problem, we present a heuristic algorithm, which iteratively searches for the best schedule in such a way that the energy consumption keeps decreasing after every iteration. Compared with the exponentially complicated exhaustive search algorithm, this heuristic algorithm has polynomial computing complexity and can provide optimal solutions in the most simulated cases. For the transmission power/time control, two simplified PTC schemes, namely, PTC-UT and PTC-USG, are proposed and studied based on our previous optimization work PTC-IPT. We show that PTC-UT and PTC-USG provide comparable energy efficiency to PTC-IPT at only half of its complexity. Numerical examples are used to validate our findings.

Some classical combinatorial problems on circulant and claw-free graphs: the isomorphism and coloring problems on circulant graphs and the stable set problem on claw-free graphs

This is a summary of the author’s Ph.D. thesis supervised by Sara Nicoloso and Gianpaolo Oriolo and defended on 3 April 2008 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request. This work deals with three classical combinatorial problems, namely the isomorphism, the vertex-coloring and the stable set problem, restricted to two graph classes, namely circulant and claw-free graphs. In the first part (joint work with Sara Nicoloso), we derive a necessary and sufficient condition to test isomorphism of circulant graphs, and give simple algorithms to solve the vertex-coloring problem on this class of graphs. In the second part (joint work with Gianpaolo Oriolo and Gautier Stauffer), we propose a new combinatorial algorithm for the maximum weighted stable set problem in claw-free graphs, and devise a robust algorithm for the same problem in the subclass of fuzzy circular interval graphs, which also provides recognition when the stability number is greater than three.