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.

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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

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

Finding a maximum induced matching in weakly chordal graphs

Induced matchings in intersection graphs

The graphs with maximum induced matching and maximum matching the same size

Lambda composition

Packing a maximum number of disjoint triangles into a given graph G is NP-hard, even for most classes of structured graphs. In contrast, we show that packing a maximum number of independent (that is, disjoint and nonadjacent) triangles is polynomial-time solvable for many classes of structured graphs, including weakly chordal graphs, asteroidal triple-free graphs, polygon-circle graphs, and interval-filament graphs. These classes contain other well-known classes such as chordal graphs, cocomparability graphs, circle graphs, circular-arc graphs, and outerplanar graphs. Our results apply more generally to independent packings by members of any family of connected graphs.

Kathie Cameron

Papers

Finding a maximum induced matching in weakly chordal graphs

Induced matchings in intersection graphs

The graphs with maximum induced matching and maximum matching the same size

Lambda composition

Independent packings in structured graphs