We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any function of a polylog number of polynomial-weight halfspaces under any distribution. As special cases of these results we obtain algorithms for learning intersections and thresholds of halfspaces. Our uniform distribution learning algorithms involve a novel non-geometric approach to learning halfspaces; we use Fourier techniques together with a careful analysis of the noise sensitivity of functions of halfspaces. Our algorithms for learning under any distribution use techniques from real approximation theory to construct low degree polynomial threshold functions.

/pdf/learning-intersections-and-thresholds-of-halfspaces-3o168bklfp.pdf

Learning intersections and thresholds of halfspaces

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.

Rainbow Generalizations of Ramsey Theory: A Survey

The main focus of this thesis is on using the divide and conquer technique toeciently solve graph problems that are in general intractable. We work inthe eld of parameterized algorithms, using width parameters of graphs thatindicate the complexity inherent in the structure of the input graph. We usethe notion of branch decompositions of a set function introduced by Robert-son and Seymour to de ne three new graph parameters, boolean-width, max-imum matching-width (MM-width) and maximum induced matching-width(MIM-width). We compare these new graph width parameters to existinggraph parameters by de ning partial orders of width parameters. We focuson tree-width, branch-width, clique-width, module-width and rank-width,and include a Hasse diagram of these orders containing 32 graph parameters.We use the size of a maximum matching in a bipartite graph as a setfunction to de ne MM-width and show that MM-width never di ers by morethan a multiplicative factor 3 from tree-width. The main reason for introduc-ing MM-width is that it simpli es the comparison between tree-width andparameters de ned via branch decomposition of a set function.We use the logarithm of the number of maximal independent sets in a bi-partite graph as set function to de ne boolean-width. We show that boolean-width of a graph class is bounded if and only if rank-width is bounded, andshow that the boolean-width of a graph can be as low as the logarithm of therank-width of the graph. Given a decomposition of boolean-width k, we de-sign FPT algorithms parameterized by k, for a large class of graph problems,whose runtime has a single exponential dependency in the boolean-width,i.e. O

/pdf/new-width-parameters-of-graphs-579q8o70bm.pdf

New Width Parameters of Graphs

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 211–216, 2004

/pdf/edge-colorings-of-complete-graphs-without-tricolored-49vtxd9ghh.pdf

Edge colorings of complete graphs without tricolored triangles

We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the existence of special monochromatic spanning trees in such colorings. We also determine the size of largest monochromatic stars guaranteed to occur. 2004 Wiley Periodicals, Inc. J Graph Theory 46: 211–216, 2004

/pdf/edge-colorings-of-complete-graphs-without-tricolored-4gohb8fiay.pdf

Edge colorings of complete graphs without tricolored triangles: EDGE COLORING WITHOUT TRICOLORED TRIANGLES

https://pagesperso.g-scop.grenoble-inp.fr/~levequeb/Publications/Meyniel-TCS.pdf

Coloring vertices of a graph or finding a Meyniel obstruction

http://www.iasi.cnr.it/jack/material/(15)_The_Traveling_Preacher.pdf

The travelling preacher, projection, and a lower bound for the stability number of a graph

Brambles and independent packings in chordal graphs

A PPA parity theorem about trees in a bipartite graph

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The diamond is the graph obtained from $K_4$ by removing an edge. In this paper we show that every ($P_6$, diamond)-free graph $G$ satisfies $\chi(G)\le \omega(G)+3$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the famous 27-vertex Schlafli graph. Our result unifies previously known results on the existence of linear $\chi$-binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect ($P_6$, diamond)-free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well-known characterization of 3-colourable $(P_6,K_3)$-free graphs.

Kathie Cameron

Papers

Coloring vertices of a graph or finding a Meyniel obstruction

The travelling preacher, projection, and a lower bound for the stability number of a graph

Brambles and independent packings in chordal graphs

A PPA parity theorem about trees in a bipartite graph

An Optimal $\chi$-Bound for ($P_6$, diamond)-Free Graphs