Henning Thomas

Playing Mastermind With Many Colors

We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n1 − ε, ε > 0 a constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case k = n, our results imply that Codebreaker can find the secret code with O(nlog log n) guesses. This bound is valid also when only black answer pegs are used. It improves the O(nlog n) bound first proven by Chvátal. We also show that if both black and white answer pegs are used, then the O(nlog log n) bound holds for up to n2log log n colors. These bounds are almost tight, as the known lower bound of Ω(n) shows. Unlike for k ≤ n1 − ε, simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal nonadaptive strategy (deterministic or randomized) needs Θ(nlog n) guesses.

Approximation Algorithms for 3D Orthogonal Knapsack

We study non-overlapping axis-parallel packings of 3D boxes with profits into a dedicated bigger box where rotation is either forbidden or permitted, and we wish to maximize the total profit. Since this optimization problem is NP-hard, we focus on approximation algorithms. We obtain fast and simple algorithms for the non-rotational scenario with approximation ratios 9 + ϵ and 8 + ϵ, as well as an algorithm with approximation ratio 7 + ϵ that uses more sophisticated techniques; these are the smallest approximation ratios known for this problem. Furthermore, we show how the used techniques can be adapted to the case where rotation by 90° either around the z-axis or around all axes is permitted, where we obtain algorithms with approximation ratios 6 + ϵ and 5 + ϵ, respectively. Finally our methods yield a 3D generalization of a packability criterion and a strip packing algorithm with absolute approximation ratio 29/4, improving the previously best known result of 45/4.

Playing mastermind with many colors

We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n1−e, e > 0 constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case k = n, our results imply that Codebreaker can find the secret code with O(n log log n) guesses. This bound is valid also when only black answer-pegs are used. It improves the O(n log n) bound first proven by Chvatal (Combinatorica 3 (1983), 325--329). We also show that if both black and white answer-pegs are used, then the O(n log log n) bound holds for up to n2 log log n colors. These bounds are almost tight as the known lower bound of Ω(n) shows. Unlike for k ≤ n1−e, simply guessing at random until the secret code is determined is not sufficient. In fact, we show that any non-adaptive strategy needs an expected number of Ω(n log n) guesses.

Internal DLA: Efficient Simulation of a Physical Growth Model

The internal diffusion limited aggregation (IDLA) process places n particles on the two dimensional integer grid. The first particle is placed on the origin; every subsequent particle starts at the origin and performs an unbiased random walk until it reaches an unoccupied position.

A randomized version of Ramsey's theorem

The standard randomization of Ramseys theorem asks for a fixed graph F and a fixed number r of colors: for what densities p = p(n) can we asymptotically almost surely color the edges of the random graph G(n, p) with r colors without creating a monochromatic copy of F. This question was solved in full generality by Rodl and Rucinski [Combinatorics, Paul Erdős is eighty, vol. 1, 1993, 317–346; J Am Math Soc 8(1995), 917–942]. In this paper we consider a different randomization that was recently suggested by Allen et al. [Random Struct Algorithms, in press]. Let \documentclass{article} \usepackage{amsmath,amsfonts} \pagestyle{empty} \begin{document}

Coloring the edges of a random graph without a monochromatic giant component

{{\mathcal R}_F(n,q)}

A Randomized Version of Ramseyʼs Theorem

\end{document} **image** be a random subset of all copies of F on a vertex set Vn of size n, in which every copy is present independently with probability q. For which functions q = q(n) can we color the edges of the complete graph on Vn with r colors such that no monochromatic copy of F is contained in \documentclass{article} \usepackage{amsmath,amsfonts} \pagestyle{empty} \begin{document}

Explosive Percolation in Erdős-Rényi-like Random Graph Processes

{{\mathcal R}_F(n,q)}

Small subgraphs in random graphs and the power of multiple choices

\end{document} **image** ? We answer this question for strictly 2-balanced graphs F. Moreover, we combine bsts an r-edge-coloring of G(n, p)oth randomizations and prove a threshold result for the property that there exi such that no monochromatic copy of F is contained in \documentclass{article} \usepackage{amsmath,amsfonts} \pagestyle{empty} \begin{document}

Balanced online Ramsey games in random graphs

{{\mathcal R}_F(n,q)}