Simulated Annealing and Graph Colouring

Simulated Annealing is a very successful heuristic for various problems in combinatorial optimization. In this paper an application of Simulated Annealing to the 3-coloring problem is considered. In contrast to many good empirical results we will show for a certain class of graphs that the expected first hitting time of a proper coloring, given an arbitrary cooling scheme, is of exponential size. These results are complementary to those in "Nolte, Andreas and Schrader, Rainer (2000) A Note on the Finite Time Behaviour of Simulated Annealing", where the convergence of Simulated Annealing to an optimal solution in exponential time is proved

Optimization by quantum annealing for the graph colouring problem

Quantum annealing is the quantum equivalent of the well known classical simulated annealing algorithm for combinatorial optimization problems. Despite the appeal of the approach, quantum annealing algorithms competitive with the state of the art for specific problems hardly exist in the literature. Graph colouring is a difficult problem of practical significance that can be formulated as combinatorial optimization. By introducing a symmetry-breaking problem representation, and finding fast incremental techniques to calculate energy changes, a competitive graph colouring algorithm based on quantum annealing is derived. This algorithm is further enhanced by tuning simplification techniques; replica spacing techniques to increase robustness; and a messaging protocol, which enables quantum annealing to efficiently take advantage of multiprocessor environments. Additionally, observations of some patterns in the tuning for random graphs led to a more effective algorithm able to find new upper bounds for several widely-used benchmark graphs, some of which had resisted improvement in the last two decades

A Parameterisation of Algorithms for Distributed Constraint Optimisation via Potential Games

This paper introduces a parameterisation of learning algorithms for distributed constraint optimisation problems (DCOPs). This parameterisation encompasses many algorithms developed in both the computer science and game theory literatures. It is built on our insight that when formulated as noncooperative games, DCOPs form a subset of the class of potential games. This result allows us to prove convergence properties of algorithms developed in the computer science literature using game theoretic methods. Furthermore, our parameterisation can assist system designers by making the pros and cons of, and the synergies between, the various DCOP algorithm components clear

Performance Analyses of Graph Heuristics and Selected Trajectory Metaheuristics on Examination Timetable Problem

Examination timetabling problem is hard to solve due to its NP-hard nature, with a large number of constraints having to be accommodated. To deal with the problem effectually, frequently heuristics are used for constructing feasible examination timetable while meta-heuristics are applied for improving the solution quality. This paper presents the performances of graph heuristics and major trajectory metaheuristics or S-metaheuristics for addressing both capacitated and un-capacitated examination timetabling problem. For constructing the feasible solution, six graph heuristics are used. They are largest degree (LD), largest weighted degree (LWD), largest enrolment degree (LE), and three hybrid heuristic with saturation degree (SD) such as SD-LD, SD-LE, and SD-LWD. Five trajectory algorithms comprising of tabu search (TS), simulated annealing (SA), late acceptance hill climbing (LAHC), great deluge algorithm (GDA), and variable neighborhood search (VNS) are employed for improving the solution quality. Experiments have been tested on several instances of un-capacitated and capacitated benchmark datasets, which are Toronto and ITC2007 dataset respectively. Experimental results indicate that, in terms of construction of solution of datasets, hybridizing of SD produces the best initial solutions. The study also reveals that, during improvement, GDA, SA, and LAHC can produce better quality solutions compared to TS and VNS for solving both benchmark examination timetabling datasets

Compact analogue neural network: a new paradigm for neural based combinatorial optimisation

The authors present a new approach to neural based optimisation, to be termed as the compact analogue neural network (CANN), which requires substantially fewer neurons and interconnection weights as compared to the Hopfield net. They demonstrate that the graph colouring problem can be solved by using the CANN, with only O(N) neurons and O(N2) interconnections, where N is the number of nodes. In contrast, a Hopfield net would require N2 neurons and O(N4) interconnection weights. A novel scheme for realising the CANN in hardware form is discussed, in which each neuron consists of a modified phase locked loop (PLL), whose output frequency represents the colour of the relevant node in a graph. Interactions between coupled neurons cause the PLLs to equilibrate to frequencies corresponding to a valid colouring. Computer simulations and experimental results using hardware bear out the efficacy of the approach