Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computing witnesses for the Boolean product of two matrices. That is, if A and B are two n by n matrices, and C = AB is their Boolean product, the algorithm finds for every entry Cij = 1 a witness: an index k so that Aik = Bkj = 1. Its running time exceeds that of computing the product of two n by n matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a given n-subset of {1,..., m}.

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 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 \Sigma\Pi\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.

Let C be a code of length n over an alphabet of q letters. For a pair of integers 2 t < u, C is (t; u)-hashing if for any two subsets T ; U C, satisfying T U , jT j = t, jU j = u, there is a coordinate 1 i n such that for any x 2 T , y 2 U x, x and y dier in the i-th coordinate. This de nition, generalizing the standard notion of a t-hashing family, is motivated by an application in designing the so-called parent identifying codes, used in digital ngerprinting. In this paper we provide lower and upper bounds on the best possible rate of (t; u)-hashing families for xed t; u and growing n. We also describe an explicit construction of (t; u)-hashing families. The obtained lower bound on the rate of (t; u)-hashing families is applied to get a new lower bound on the rate of t-parent identifying codes.

We sharpen a result of Hansel on separating set systems. We also extend a theorem of Spencer on completely separating systems by proving an analogue of Hansel’s result. 1

Abstract. We present a randomized subexponential time, polynomial space parameterized algorithm for the k-Weighted Feedback Arc Set in Tournaments (k-FAST) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kind of universal hash functions, that we coin universal coloring families. For integers m, k and r, a family F of functions from [m] to [r] is called a universal (m, k, r)-coloring family if for any graph G on the set of vertices [m] with at most k edges, there exists an f ∈ F which is a proper vertex coloring of G. Our algorithm is the first non-trivial subexponential time parameterized algorithm outside the framework of bidimensionality. 1

Let C be a code of length n over an alphabet of q letters. An n-word y is called a descendant of a set of t codewords x 1 ; : : : ; x t if y i 2 fx 1 i ; : : : ; x t i g for all i = 1; : : : ; n: A code is said to have the t-identifying parent property if for any n-word that is a descendant of at most t parents it is possible to identify at least one of them. We study a generalization of hashing, (t; u)-hashing, which ensures identication, and provide tight estimates of the rates.