Vertex Descent is a local search algorithm which forms the basis of a wide spectrum of tabu search, simulated annealing and hybrid evolutionary algorithms for graph colouring. These algorithms are usually treated as experimental and provide strong results on established benchmarks. As a step towards studying these heuristics analytically, an analysis of the behaviour of Vertex Descent is provided. It is shown that Vertex Descent is able to find feasible colourings for several types of instances in expected polynomial time. This includes 2-colouring of paths and 3-colouring of graphs with maximum degree 3. The same also holds for 3-colouring of a subset of 3-colourable graphs with maximum degree 4. As a consequence, Vertex Descent finds a 3-colouring in expected polynomial time for the smallest graph for which Br 'elaz's heuristic DSATUR needs 4 colours. On the other hand, Vertex Descent may fail for forests with maximum degree 3 with high probability.
Graph coloring is one of the most studied combinatorial optimization problems. This paper presents an improved extraction and expansion method (IE^2COL), initially introduced in Wu and Hao (2012) [44]. IE^2COL employs a forward independent set extraction strategy to reduce the initial graph G. From the reduced graph, IE^2COL triggers a backward coloring process which uses extracted independent sets as new color classes for intermediate subgraph coloring. The proposed method is assessed on 20 large benchmark graphs with 900 to 4000 vertices. Computational results show that it provides new upper bounds for 6 graphs and consistently matches the current best-known results for 12 other graphs. This paper presents an effective memetic approach (HEAD ) designed for coloring difficult graphs. In this algorithm a powerful tabu search is used inside a very specific population of individuals. Indeed, the main characteristic of (HEAD ) is to work with a population of only two individuals. This provides a very simple algorithm with neither selection operator nor replacement strategy. Because of its simplicity, (HEAD ) allows an easy way for managing the diversity. We focus this work on the impact of this diversity management on well-studied graphs of the DIMACS challenge benchmarks, known to be very difficult to solve. A detailed analysis is provided for three graphs on which (HEAD ) finds a legal coloring with less colors than reference algorithms: DSJC500.5 with 47 colors, DSJC1000.5 with 82 colors and flat1000_76_0 with 81 colors. The analysis performed in this work will allow to improve (HEAD ) efficiency in terms of computation time and maybe to decrease the number of needed colors for other graphs. Quantum annealing is a combinatorial optimization technique inspired by quantum mechanics. Here we show that a spin model for the k-coloring of large dense random graphs can be field tuned so that its acceptance ratio diverges during Monte Carlo quantum annealing, until a ground state is reached. We also find that simulations exhibiting such a diverging acceptance ratio are generally more effective than those tuned to the more conventional pattern of a declining and or stagnating acceptance ratio. This observation facilitates the discovery of solutions to several well-known benchmark k-coloring instances, some of which have been open for almost two decades. 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.