Thomas Hofmeister

Contrast-optimal k out of n secret sharing schemes in visual cryptography

Visual cryptography and (k, n)-visual secret sharing schemes were introduced by Naor and Shamir in [NaSh1]. A sender wishing to transmit a secret message distributes n transparencies among n recipients, where the transparencies contain seemingly random pictures. A (k, n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k - 1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message.

A Probabilistic 3-SAT Algorithm Further Improved

In [Sch99], Schoning proposed a simple yet efficient randomized algorithm for solving the k-SAT problem. In the case of 3-SAT, the algorithm has an expected running time of poly(n) ? (4/3)n = O(1.3334n) when given a formula F on n variables. This was the up to now best running time known for an algorithm solving 3-SAT. Here, we describe an algorithm which improves upon this time bound by combining an improved version of the above randomized algorithm with other randomized algorithms. Our new expected time bound for 3-SAT is O(1.3302n).

Contrast-Optimal k out of n Secret Sharing Schemes in Visual Cryptography

Visual cryptography and (k, n)-visual secret sharing schemes were introduced by Naor and Shamir in [NaSh1]. A sender wishing to transmit a secret message distributes n transparencies among n recipients, where the transparencies contain seemingly random pictures. A (k, n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k - 1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message.

Some notes on threshold circuits, and multiplication in depth 4

Abstract It is known that two n -bit numbers can be added by polynomial-size ∧ - ∨-circuits of depth 3 and multiplied by threshold circuits of depth 10. Two tricks for the reduction of depth in threshold circuits are formalized. Further, threshold circuits for the addition of m numbers of length n , m ⩽ n , in depth 3 (with O( n 2 ) gates and O( nm 3 + n 3 m ) wires) and for the multiplication of two n -bit numbers in depth 4 (O( n 2 ) gates and O( n 4 ) wires) are presented.

A Note on the Simulation of Exponential Threshold Weights

It is known (see [GHR],[GK]) that for F(x) = w0 + w1x1 + ...+ w n x n , a threshold gate G(x) = sgn(F(x)) which may have arbitrarily large integer weights w i can be computed (“simulated”) in threshold circuits of depth 2 with polynomial size. In this paper, we modify the method from [GK] to obtain an improvement in two respects: The approach described here is simpler and the size of the simulating circuit is smaller.

Approximating Maximum Independent Sets in Uniform Hypergraphs

We consider the problems of approximating the independence number and the chromatic number of k-uniform hypergraphs on n vertices. For fixed k≥2, we describe for both problems polynomial time approximation algorithms with approximation ratios O(n/(log(k-1) n)2). This extends results of Boppana and Halldorsson [5] who showed for the case of graphs that an approximation ratio of O(n/(log n)2) can be achieved in polynomial time. On the other hand, assuming NP ⊋ ZPP, there are no polynomial time algorithms for the independence number and the chromatic number of k-uniform hypergraphs with approximation ratio of n 1-ɛ for any fixed e > 0.

A Comparison of Approximation Algorithms for the MaxCut-Problem

In this paper we compare, from a practical point of view, approximation algorithms for the problem MaxCut. For this problem, we are given an undirected graph G = (V;E) with vertex set V and edge set E, and we are looking for a partition V = V1 [ V2 with V1 \ V2 = ; of the vertex set which maximizes the number of edges e 2 E which have one endpoint in V1 and the other in V2. The investigated algorithms include semide nite programming, a random strategy, genetic algorithms, two combinatorial algorithms and a divide{and{conquer strategy.

A combinatorial design approach to MAXCUT

The k-MAXCUT problem for undirected graphs G=(V, E) consists of finding a partition such that the number of edges with endpoints in two different sets Vi is maximized. We offer a new approach to this problem by showing that the combinatorial notion of block designs can be used to algorithmically obtain partitions which achieve lower bounds for which until now only existence proofs were known.

A geometric framework for solving subsequence problems in computational biology efficiently

In this paper, we introduce the notion of a constrained Minkowski sumwhich for two (finite) point-sets P,Q⊆ R2 and a set of k inequalities Ax≥ b is defined as the point-set (P ⊕ Q)Ax≥ b= x = p+q | ∈ P, q ∈ Q, , Ax ≥ b. We show that typical subsequenceproblems from computational biology can be solved by computing a setcontaining the vertices of the convex hull of an appropriatelyconstrained Minkowski sum. We provide an algorithm for computing such a setwith running time O(N log N), where N=|P|+|Q| if k is fixed. For the special case (P⊕ Q)x1≥ β, where P and Q consistof points with integer x1-coordinates whose absolute values arebounded by O(N), we even achieve a linear running time O(N). Wethereby obtain a linear running time for many subsequence problemsfrom the literature and improve upon the best known running times forsome of them.The main advantage of the presented approach is that it provides a generalframework within which a broad variety of subsequence problems canbe modeled and solved.This includes objective functions and constraintswhich are even more complexthan the ones considered before.

An Application of Codes to Attribute-Efficient Learning

We design asymptotically optimal query strategies for the class of parity functions which contain at most k essential variables. The number of questions asked is at most twice the number asked by an optimal strategy. The strategy presented is even non-adaptive. For fixed k, the number of questions is optimal up to additive constants. Our results improve upon results by Uehara, Tsuchida and Wegener [6].