A simple algorithm for sampling colourings of G(N,D/N) up to Gibbs Uniqueness threshold

Approximate random $k$-colouring of a graph G is a well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n,d/n), where d is a sufficiently large constant. We propose a novel efficient algorithm for approximate random k-colouring G(n,d/n) for any k>(1+\epsilon)d. To be more specific, with probability at least 1-n^{-\Omega(1)} over the input instances G(n,d/n) and for k>(1+\epsilon)d, the algorithm returns a k-colouring which is distributed within total variation distance n^{-\Omega(1)} from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a ``simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution

Graph Colouring with Input Restrictions

In this thesis, we research the computational complexity of the graph colouring problem and its variants including precolouring extension and list colouring for graph classes that can be characterised by forbidding one or more induced subgraphs. We investigate the structural properties of such graph classes and prove a number of new properties. We then consider to what extent these properties can be used for efficiently solving the three types of colouring problems on these graph classes. In some cases we obtain polynomial-time algorithms, whereas other cases turn out to be NP-complete. We determine the computational complexity of k-COLOURING, k-PRECOLOURING EXTENSION and LIST k-COLOURING on $P_k$-free graphs. In particular, we prove that k-COLOURING on $P_8$-free graphs is NP-complete, 4-PRECOLOURING EXTENSION $P_7$-free graphs is NP-complete, and LIST 4-COLOURING on $P_6$-free graphs is NP-complete. In addition, we show the existence of an integer r such that k-COLOURING is NP-complete for $P_r$-free graphs with girth 4. In contrast, we determine for any fixed girth $g\geq 4$ a lower bound $r(g)$ such that every $P_{r(g)}$-free graph with girth at least $g$ is 3-colourable. We also prove that 3-LIST COLOURING is NP-complete for complete graphs minus a matching. We present a polynomial-time algorithm for solving 4-PRECOLOURING EXTENSION on $(P_2+P_3)$-free graphs, a polynomial-time algorithm for solving LIST 3-Colouring on $(P_2+P_4)$-free graphs, and a polynomial-time algorithm for solving LIST 3-COLOURING on $sP_3$-free graphs. We prove that LIST k-COLOURING for $(K_{s,t},P_r)$-free graphs is also polynomial-time solvable. We obtain several new dichotomies by combining the above results with some known results

Frequency reassignment in cellular phone networks

In cellular communications networks, cells use beacon frequencies to ensure the smooth operation of the network, for example in handling call handovers from one cell to another. These frequencies are assigned according to a frequency plan, which is updated from time to time, in response to evolving network requirements. The migration from one frequency plan to a new one proceeds in stages, governed by the network's base station controllers. Existing methods result in periods of reduced network availability or performance during the reassgnment process. The problem posed to the Study Group was to develop a dynamic reassignment algorithm for implementing a new frequency plan so that there is little or no disruption of the network's performance during the transition. This problem was naturally formulated in terms of graph colouring and an effective algorithm was developed based on a straightforward approach of search and random colouring

Decision making with decision event graphs

We introduce a new modelling representation, the Decision Event Graph (DEG), for asymmetric multistage decision problems. The DEG explicitly encodes conditional independences and has additional significant advantages over other representations of asymmetric decision problems. The colouring of edges makes it possible to identify conditional independences on decision trees, and these coloured trees serve as a basis for the construction of the DEG. We provide an efficient backward-induction algorithm for finding optimal decision rules on DEGs, and work through an example showing the efficacy of these graphs. Simplifications of the topology of a DEG admit analogues to the sufficiency principle and barren node deletion steps used with influence diagrams