Graph Coloring Problem

Computing the chromatic number of a graph is an NP-hard problem. For random graphs and some other classes of graphs, estimators of the expected chromatic number have been well studied. In this paper, a new 0–1 integer programming formulation for the graph coloring problem is presented. The proposed new formulation is used to develop a method that generates graphs of known chromatic number by using the KKT optimality conditions of a related continuous nonlinear program.

Misalkan G=(V,E) adalah suatu graf. Fungsi c dari E ke {1, 2, ..., k} dikatakan pewarnaan-k pelangi pada G, jika untuk setiap pasang titik u dan v di V terdapat suatu lintasan dengan u dan v sebagai titik ujung yang setiap sisinya memperoleh warna berbeda. Bilangan terhubung pelangi graf G, dinotasikan dengan rc(G), adalah bilangan bulat positif terkecil k sehingga G mempunyai suatu pewarnaan-k pelangi. Selanjutnya,c dikatakan pewarnaan-k pelangi kuat, jika untuk setiap titik u dan v di V terdapat lintasan pelangi dengan panjangnya sama dengan jarak u dan v. Dalam hal ini, bilangan bulat positif terkecil k sehingga G mempunyai suatu pewarnaan-k pelangi kuat didefinisikan sebagai bilangan terhubung pelangi kuat yang dinotasikan dengan src(G).Pewarnaan pelangi digunakan antara lain untuk mengamankan kode rahasia yang dikirimkan.
Pada makalah ini dibahas tentang pewarnaan pelangi graf berlian. Graf berlian dengan 2n titik dinotasikan dengan Brn adalah graf yang diperoleh dari graf tangga segitiga dengan 2n-1 titik dan ditambahkan satu titik dan beberapa sisi tertentu. Kami menentukan rc(Brn) dan src(Brn) untuk n≥ 4.
Kata kunci:bilangan terhubung pelangi,bilangan terhubung pelangi kuat, graf berlian, pewarnaan pelangi.

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.

Graph Coloring Problem (GCP) bears an enormous significance to the researchers in the field of soft computing. In this paper, we demonstrate a Quantum-Inspired Evolutionary Algorithm (QEA) to solve GCP. We use two dimensional arrays of Q-bits called Q-bit individual to produce binary individual. Q-gate operation is applied as a variation operator on Q-bit individuals. In traditional evolutionary algorithm (EA) for GCP, k-coloring approach is used and the EA is run several times for decreasing value of k until lowest possible k is reached. In our QEA, we start with the number of colors equal to the theoretical upper bound of the chromatic number, which is maximum out-degree + 1, and during evolution some colors are made unused to reduce the number of color in each generation. As a result, solution is found in a single run. We test 36 datasets from DIMACS benchmark and compare the result with several recent works. For five datasets, our algorithm obtains better solution than other.

This paper discusses reformulations of the problem of coloring plane maps with four colors. We include discussion of the Eliahou–Kryuchkov conjecture, the Penrose formula, the vector cross-product formulation and the reformulations in terms of formations and factorizations due to G. Spencer-Brown.

A total coloring of a graph G = (V, E) is an assignment of colors to vertices and edges such that neither two adjacent vertices nor two incident edges get the same color, and, for each edge, the end-points and the edge itself receive different colors. Any valid total coloring induces a partition of the elements of G into total matchings, which are defined as subsets of vertices and edges that can take the same color. In this paper, we propose Integer Linear Programming models for both the Total Coloring and the Total Matching problems, and we study the strength of the corresponding Linear Programming relaxations. The total coloring is formulated as the problem of finding the minimum number of total matchings that cover all the graph elements. This covering formulation can be solved by a Column Generation algorithm, where the pricing subproblem corresponds to the Weighted Total Matching Problem. Hence, we study the Total Matching Polytope. We introduce three families of nontrivial valid inequalities: vertex-clique inequalities based on standard clique inequalities of the Stable Set Polytope, congruent-2k3 cycle inequalities based on the parity of the vertex set induced by the cycle, and even-clique inequalities induced by complete subgraphs of even order. We prove that congruent-2k3 cycle inequalities are facet-defining only when k = 4 , while the vertex-clique and even-cliques are always facet-defining. Finally, we present preliminary computational results of a Column Generation algorithm for the Total Coloring Problem and a Cutting Plane algorithm for the Total Matching Problem.

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.

We analyze synchronization phenomena in a dynamical hysteresis neural network. The dynamical hysteresis neural network accounts a phase difference in each neuron to be an information. Based on this result, we propose an application for solving 2-colorable graph coloring problems.

This research work focuses on Integer Linear Programming (ILP) formulations for the Routing and Wavelength Assignment (RWA) problem in Wavelength Division Multiplexed (WDM) networks where end-users communicate with each other by establishing all-optical, non-interfering wavelength channels which are referred as lightpaths. The RWA problem is reducible to Graph Coloring problem in polynomial time and hence found to be NP-Complete. So,

Intellectual Property Protection or IPP is an act of protecting a digital product by embedding an author's signature into the object in the form of minute errors. This signature will be the proof of ownership. However, this technique cannot be directly applied to protect intellectual properties such as solutions to hard problems which must maintain the correct functionality. This paper proposes a constraint-based watermarking technique and lays out a theoretical framework to evaluate this type of technique. Based on this framework, we analyze previously proposed two watermarking techniques for the graph coloring problem because of its theoretical importance in complexity theory and numerous applications in real life. This newly proposed algorithm is the another way to protect graph coloring solution from an intruder by coloring a graph using Modified Genetic Algorithm. We used those colored partitions for the purpose of embedding signature into the graph, so that the developer o...

this paper. We examine the data reconciliation problem within a generalized framework in which differences between multi-sets correspond to evaluations of arbitrary error functions. We limit communication between reconciling hosts to a single message and, under such conditions, the problem of data reconciliation is equivalent to a variation of the problem of graph coloring: second-order coloring or distance-2 coloring, from

In this paper, we study the problem of graph coloring and propose a novel particle swarm optimization (PSO) algorithm for it. We use the PSO evolutionary progress to improve a simple deterministic greedy algorithm. The new algorithm can achieve a result that is better than known heuristic algorithms do, as verified by an extensive simulation study.

We introduce a graph coloring challenge benchmark based on the problem of completing Latin squares. We show how the hardness of the instances can be finely controlled by varying the fraction of pre- colored squares. We compare three complete (exact) solution strategies on this benchmark: (1) a Constraint Satisfaction (CSP) based approach, (2) a hybrid Linear Programming / CSP approach,

Differential evolution was developed for reliable and versatile function optimization. It has also become interesting for other domains because of its ease to use. In this paper, we posed the question of whether differential evolution can also be used by solving of the combinatorial optimization problems, and in particular, for the graph coloring problem. Therefore, a hybrid self-adaptive differential evolution algorithm for graph coloring was proposed that is comparable with the best heuristics for graph coloring today, i.e. Tabucol of Hertz and de Werra and the hybrid evolutionary algorithm of Galinier and Hao. We have focused on the graph 3-coloring. Therefore, the evolutionary algorithm with method SAW of Eiben et al., which achieved excellent results for this kind of graphs, was also incorporated into this study. The extensive experiments show that the differential evolution could become a competitive tool for the solving of graph coloring problem in the future.

Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that {\em survey propagation} (a certain message passing algorithm) is effective in finding proper $k$-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a $k$-coloring of graphs drawn from a certain planted-solution distribution over $k$-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem. Our techniques can be applied to several...