Approximation schemes for randomly sampling colorings

Graph colouring is arguably one of the most important issues in Graph Theory. However, many of the questions that arise in the area such as the chromatic number problem or counting the number of proper colorings of a graph are known to be hard. This is the reason why approximation schemes are considered. In this thesis we consider the problem of approximate sampling a proper coloring at random. Among others, approximate samplers yield approximation schemes for the problem of counting the number of colourings of a graph. These samplers are based in Markov chains, and the main requirement of these chains is to mix rapidly, namely in time polynomial in the number of vertices. Two main examples are the Glauber and the flip dynamics. In the project we study under which conditions these chains mix rapidly and hence under which conditions there exist efficient samplers

Algorithms for #BIS-hard problems on expander graphs

We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) $\Delta$-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite $\Delta$-regular tree. We also find efficient counting and sampling algorithms for proper $q$-colorings of random $\Delta$-regular bipartite graphs when $q$ is sufficiently small as a function of $\Delta$

Methods for Estimating The Diagonal of Matrix Functions

Many applications such as path integral evaluation in Lattice Quantum Chromodynamics (LQCD), variance estimation of least square solutions and spline ts, and centrality measures in network analysis, require computing the diagonal of a function of a matrix, Diag(f(A)) where A is sparse matrix, and f is some function. Unfortunately, when A is large, this can be computationally prohibitive. Because of this, many applications resort to Monte Carlo methods. However, Monte Carlo methods tend to converge slowly. One method for dealing with this shortcoming is probing. Probing assumes that nodes that have a large distance between them in the graph of A, have only a small weight connection in f(A). to determine the distances between nodes, probing forms Ak. Coloring the graph of this matrix will group nodes that have a high distance between them together, and thus a small connection in f(A). This enables the construction of certain vectors, called probing vectors, that can capture the diagonals of f(A). One drawback of probing is in many cases it is too expensive to compute and store A^k for the k that adequately determines which nodes have a strong connection in f(A). Additionally, it is unlikely that the set of probing vectors required for A^k is a subset of the probing vectors needed for Ak+1. This means that if more accuracy in the estimation is required, all previously computed work must be discarded. In the case where the underlying problem arises from a discretization of a partial dierential equation (PDE) onto a lattice, we can make use of our knowledge of the geometry of the lattice to quickly create hierarchical colorings for the graph of A^k. A hierarchical coloring is one in which colors for A^{k+1} are created by splitting groups of nodes sharing a color in A^k. The hierarchical property ensures that the probing vectors used to estimate Diag(f(A)) are nested subsets, so if the results are inaccurate the estimate can be improved without discarding the previous work. If we do not have knowledge of the intrinsic geometry of the matrix, we propose two new classes of methods that improve on the results of probing. One method seeks to determine structural properties of the matrix f(A) by obtaining random samples of the columns of f(A). The other method leverages ideas arising from similar problems in graph partitioning, and makes use of the eigenvectors of f(A) to form effective hierarchical colorings. Our methods have thus far seen successful use in computational physics, where they have been applied to compute observables arising in LQCD. We hope that the renements presented in this work will enable interesting applications in many other elds