We present a fairly general method for finding deterministic constructions obeying what we call k-restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n,k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2/sup k/ configurations appear) and families of perfect hash functions. The near-optimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixed-subgraph finding algorithms, and of near optimal /spl Sigma/II/spl Sigma/ threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a local-coloring protocol, and for exhaustive testing of circuits.

/pdf/splitters-and-near-optimal-derandomization-3z9xxkmysh.pdf

Splitters and near-optimal derandomization

The problem of error-free transmission capacity of a noisy channel was posed by Shannon in 1956 and remains unsolved, Nevertheless, partial results for this and similar channel and source coding problems have had a considerable impact on information theory, computer science, and mathematics. We review the techniques, results, information measures, and challenges encountered in this ongoing quest.

Zero-error information theory

We propose a randomized divide-and-conquer technique that leads to improved randomized and deterministic algorithms for NP-hard path, matching, and packing problems. For the parameterized max-path problem, our randomized algorithm runs in time $O(4^{k}k^{2.7}m)$ and polynomial space (where m is the number of edges in the input graph), improving the previous best randomized algorithm for the problem that runs in time $O(5.44^{k}km)$ and exponential space. Our randomized algorithms for the parameterized max r-d matching and max r-set packing problems run in time $4^{(r-1)k}n^{O(1)}$ and polynomial space, improving the previous best algorithms for the problems that run in time $10.88^{rk}n^{O(1)}$ and exponential space. Moreover, our randomized algorithms can be derandomized to result in significantly improved deterministic algorithms for the problems, and they can be extended to solve other matching and packing problems.

/pdf/randomized-divide-and-conquer-improved-path-matching-and-3559s7icrq.pdf

Randomized Divide-and-Conquer: Improved Path, Matching, and Packing Algorithms

i Acknowledgements First of all, I would like to thank my supervisor Pinar Heggernes for her help and guidance, both with my reasearch and with all other aspects of being a graduate student. This thesis would not have existed without her. Thomassen and Yngve Villanger. A special thanks goes to Saket Saurabh, with whom I have shared many sleepless nights and cups of coffee preparing this work. Thanks to my family, especially to my mother and father. In more than one way, i am here largely because of you. I would also like to thank my friends for filling my life with things other than discrete mathematics. You mean a lot to me, hugs to all of you and kisses to one in particular. Thanks also to the Algorithms group at the University of Bergen, you are a great lot. An extra thanks goes to my table-tennis opponents, even though this thesis exists despite of your efforts to keep me away from it. Finally, thanks to Pasha for being furry and having whiskers.

/pdf/new-methods-in-parameterized-algorithms-and-complexity-28bfbxb0kb.pdf

New Methods in Parameterized Algorithms and Complexity

An out-tree T is an oriented tree with exactly one vertex of in-degree zero and a vertex x of T is called internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a subgraph isomorphic to a given out-tree with k vertices. Both algorithms run in O *(5.704 k ) time. We apply the deterministic algorithm to obtain an algorithm of runtime O *(c k ), where c is a constant, for deciding whether an input digraph contains a spanning out-tree with at least k internal vertices. This answers in affirmative a question of Gutin, Razgon and Kim (Proc. AAIM'08).

pdf/algorithm-for-finding-k-vertex-out-trees-and-its-application-4hb4lqswty.pdf

Algorithm for Finding k-Vertex Out-trees and Its Application to k-Internal Out-branching Problem

A simple proof is given of the best-known upper bound on the cardinality of a set of vectors of length t over an alphabet of size b, with the property that, for every subset of k vectors, there is a coordinate in which they all differ This question is motivated by the study of perfect hash functions

Perfect Hashing and Probability

The average size of an independent set in graphs with a given chromatic number

Recursive bounds for perfect hashing

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Perfect Hashing and Probability

The average size of an independent set in graphs with a given chromatic number

The average size of an independent set in graphs with a given chromatic number

Recursive bounds for perfect hashing