The Complexity of Change

Many combinatorial problems can be formulated as "Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?". An example of such a question is: given two k-colourings of a graph, can I transform the first k-colouring into the second one, by recolouring one vertex at a time, and always maintaining a proper k-colouring? Another example is: given two solutions of a SAT-instance, can I transform the first solution into the second one, by changing the truth value one variable at a time, and always maintaining a solution of the SAT-instance? Other examples can be found in many classical puzzles, such as the 15-Puzzle and Rubik's Cube. In this survey we shall give an overview of some older and more recent work on this type of problem. The emphasis will be on the computational complexity of the problems: how hard is it to decide if a certain transformation is possible or not?Comment: 28 pages, 6 figure

From the Ising and Potts models to the general graph homomorphism polynomial

In this note we study some of the properties of the generating polynomial for homomorphisms from a graph to at complete weighted graph on $q$ vertices. We discuss how this polynomial relates to a long list of other well known graph polynomials and the partition functions for different spin models, many of which are specialisations of the homomorphism polynomial. We also identify the smallest graphs which are not determined by their homomorphism polynomials for $q=2$ and $q=3$ and compare this with the corresponding minimal examples for the $U$-polynomial, which generalizes the well known Tutte-polynomal.Comment: V2. Extended versio

Sampling grid colourings with fewer colours

We provide an optimally mixing Markov chain for 6-colourings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colourings has strong spatial mixing. 4 and 5 are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colourings of the square grid

Sampling Colourings of the Triangular Lattice

We show that the Glauber dynamics on proper 9-colourings of the triangular lattice is rapidly mixing, which allows for efficient sampling. Consequently, there is a fully polynomial randomised approximation scheme (FPRAS) for counting proper 9-colourings of the triangular lattice. Proper colourings correspond to configurations in the zero-temperature anti-ferromagnetic Potts model. We show that the spin system consisting of proper 9-colourings of the triangular lattice has strong spatial mixing. This implies that there is a unique infinite-volume Gibbs distribution, which is an important property studied in statistical physics. Our results build on previous work by Goldberg, Martin and Paterson, who showed similar results for 10 colours on the triangular lattice. Their work was preceded by Salas and Sokal's 11-colour result. Both proofs rely on computational assistance, and so does our 9-colour proof. We have used a randomised heuristic to guide us towards rigourous results.Comment: 42 pages. Added appendix that describes implementation. Added ancillary file

Counting coloured planar maps: differential equations

We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial polynomial differential equation withrespect to the edge variable. We give explicitly a differential systemthat characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. Instatistical physics terms, we solvethe q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating functionwas proved to be algebraic for certain values of q,including q=1, 2 and 3. It isknown to be transcendental in general. In contrast, our differential system holds for an indeterminate q.For certain special cases of combinatorial interest (four colours; properq-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.Comment: 43 p