Linear Programming Bounds for Randomly Sampling Colorings

Here we study the problem of sampling random proper colorings of a bounded degree graph. Let $k$ be the number of colors and let $d$ be the maximum degree. In 1999, Vigoda showed that the Glauber dynamics is rapidly mixing for any $k > \frac{11}{6} d$. It turns out that there is a natural barrier at $\frac{11}{6}$, below which there is no one-step coupling that is contractive, even for the flip dynamics. We use linear programming and duality arguments to guide our construction of a better coupling. We fully characterize the obstructions to going beyond $\frac{11}{6}$. These examples turn out to be quite brittle, and even starting from one, they are likely to break apart before the flip dynamics changes the distance between two neighboring colorings. We use this intuition to design a variable length coupling that shows that the Glauber dynamics is rapidly mixing for any $k\ge \left(\frac{11}{6} - \epsilon_0\right)d$ where $\epsilon_0 \geq 9.4 \cdot 10^{-5}$. This is the first improvement to Vigoda's analysis that holds for general graphs.Comment: 30 pages, 3 figures; fixed some typo

Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph

Shortest Reconfiguration of Colorings Under Kempe Changes

International audienc

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