Improved Approximation Algorithm for the Number of Queries Necessary to Identify a Permutation

In the past three decades, deductive games have become interesting from the algorithmic point of view. Deductive games are two players zero sum games of imperfect information. The first player, called "codemaker", chooses a secret code and the second player, called "codebreaker", tries to break the secret code by making as few guesses as possible, exploiting information that is given by the codemaker after each guess. A well known deductive game is the famous Mastermind game. In this paper, we consider the so called Black-Peg variant of Mastermind, where the only information concerning a guess is the number of positions in which the guess coincides with the secret code. More precisely, we deal with a special version of the Black-Peg game with n holes and k >= n colors where no repetition of colors is allowed. We present a strategy that identifies the secret code in O(n log n) queries. Our algorithm improves the previous result of Ker-I Ko and Shia-Chung Teng (1985) by almost a factor of 2 for the case k = n. To our knowledge there is no previous work dealing with the case k > n. Keywords: Mastermind; combinatorial problems; permutations; algorithm

The Exact Query Complexity of Yes-No Permutation Mastermind

Mastermind is famous two-player game. The first player (codemaker) chooses a secret code which the second player (codebreaker) is supposed to crack within a minimum number of code guesses (queries). Therefore, the codemaker’s duty is to help the codebreaker by providing a well-defined error measure between the secret code and the guessed code after each query. We consider a variant, called Yes-No AB-Mastermind, where both secret code and queries must be repetition-free and the provided information by the codemaker only indicates if a query contains any correct position at all. For this Mastermind version with n positions and k ≥ n colors and ` := k + 1 − n, we prove a lower bound of ∑ k j=` log 2 j and an upper bound of n log 2 n + k on the number of queries necessary to break the secret code. For the important case k = n, where both secret code and queries represent permutations, our results imply an exact asymptotic complexity of Θ (n log n) queries

Algorithm Engineering für einige komplexe PraxisproblemeExakte Verfahren, Heuristiken und hybride evolutionäre Algorithmen

This work deals with the design of exact algorithms and heuristics for complex optimization problems that origin from three practical applications and one classical combinatorial task. We obtain exact algorithms by modeling our problems in terms of mathematical optimization problems and applying suitable software tools to solve this models. Because of the complexity of our problems, exact algorithms cannot solve large instances within adequate time. Therefore, on the one hand, we derive efficient heuristics by adapting algorithms for similar mathematical problems in a suitable manner. On the other hand, evolutionary algorithms have shown to be successfully applicable to many hard mathematical problems. For that reason, we experimentally determine appropriate EA frameworks, hybridizing the evolutionary operators with our problem specific heuristics.Diese Arbeit befasst sich mit dem Entwurf von exakten und heuristischen Lösungsverfahren für schwere Optimierungsprobleme, die aus drei praktischen Anwendungen und einer klassischen kombinatorischen Fragestellung stammen. Exakte Verfahren bekommen wir durch die Modellierung als mathematische Optimierungsprobleme und die Anwendung geeigneter Software für deren Lösung. Da die Komplexität der behandelten Probleme das exakte Lösen großer Instanzen in adäquater Zeit verbietet, werden zum Einen effiziente Heuristiken durch geeignete Erweiterungen bekannter Verfahren für verwandte mathematische Probleme der vorliegenden Praxisaufgaben gewonnen. Zum Anderen haben sich evolutionäre Algorithmen sehr erfolgreich bei der Bewältigung vieler mathematisch schwerer Optimierungsprobleme gezeigt. Daher werden experimentell geeignete EA-Rahmenwerke für die gegebenen Problemstellungen ermittelt und mit den zuvor gewonnenen Heuristiken hybridisiert

On the Query Complexity of Black-Peg AB-Mastermind

Mastermind is a two players zero sum game of imperfect information. Starting with Erd˝os and Rényi (1963), its combinatorics have been studied to date by several authors, e.g., Knuth (1977), Chvátal (1983), Goodrich (2009). The first player, called “codemaker”, chooses a secret code and the second player, called “codebreaker”, tries to break the secret code by making as few guesses as possible, exploiting information that is given by the codemaker after each guess. For variants that allow color repetition, Doerr et al. (2016) showed optimal results. In this paper, we consider the so called Black-Peg variant of Mastermind, where the only information concerning a guess is the number of positions in which the guess coincides with the secret code. More precisely, we deal with a special version of the Black-Peg game with n holes and k ≥ n colors where no repetition of colors is allowed. We present upper and lower bounds on the number of guesses necessary to break the secret code. For the case k = n, the secret code can be algorithmically identified within less than (n − 3)dlog 2 ne + 5 2 n − 1 queries. This result improves the result of Ker-I Ko and Shia-Chung Teng (1985) by almost a factor of 2. For the case k > n, we prove an upper bound of (n − 2)dlog 2 ne + k + 1. Furthermore, we prove a new lower bound of n for the case k = n, which improves the recent n − log log(n) bound of Berger et al. (2016). We then generalize this lower bound to k queries for the case k ≥ n

Parameter Optimization and Validation of a Marine Biogeochemical Model using a Hybrid Algorithm

Sensitivity computations, parameter identification and optimization for an 1-D marine biogeochemical model of $NPZD$ type are presented. For the optimization a hybrid algorithm combining quantum-evolutionary and local gradient-based search methods is used. It turns out to be an efficient and flexible tool for optimization and can be easily adopted for other simulation models. For the model under investigation attainable data could be exactly identified. For realistic measure ment data we argue that a certain parameter set leading to a non-optimal fit cannot be improved. Moreover we show that data uncertainty leads to a significant parameter spread. Thus we conclude that the $NPZD$ model needs to be modified or extended, maybe including a modification of external forcings and/or initial conditions

One size fits all? Calibrating an ocean biogeochemistry model for different circulations

Global biogeochemical ocean models are often tuned to match the observed distributions and fluxes of inorganic and organic quantities. This tuning is typically carried out “by hand”. However, this rather subjective approach might not yield the best fit to observations, is closely linked to the circulation employed and is thus influenced by its specific features and even its faults. We here investigate the effect of model tuning, via objective optimisation, of one biogeochemical model of intermediate complexity when simulated in five different offline circulations. For each circulation, three of six model parameters have been adjusted to characteristic features of the respective circulation. The values of these three parameters – namely, the oxygen utilisation of remineralisation, the particle flux parameter and potential nitrogen fixation rate – correlate significantly with deep mixing and ideal age of North Atlantic Deep Water (NADW) and the outcrop area of Antarctic Intermediate Waters (AAIW) and Subantarctic Mode Water (SAMW) in the Southern Ocean. The clear relationship between these parameters and circulation characteristics, which can be easily diagnosed from global models, can provide guidance when tuning global biogeochemistry within any new circulation model. The results from 20 global cross-validation experiments show that parameter sets optimised for a specific circulation can be transferred between similar circulations without losing too much of the model's fit to observed quantities. When compared to model intercomparisons of subjectively tuned, global coupled biogeochemistry–circulation models, each with different circulation and/or biogeochemistry, our results show a much lower range of oxygen inventory, oxygen minimum zone (OMZ) volume and global biogeochemical fluxes. Export production depends to a large extent on the circulation applied, while deep particle flux is mostly determined by the particle flux parameter. Oxygen inventory, OMZ volume, primary production and fixed-nitrogen turnover depend more or less equally on both factors, with OMZ volume showing the highest sensitivity, and residual variability. These results show a beneficial effect of optimisation, even when a biogeochemical model is first optimised in a relatively coarse circulation and then transferred to a different finer-resolution circulation model

Multiobjective Calibration of a Global Biogeochemical Ocean Model Against Nutrients, Oxygen, and Oxygen Minimum Zones

Global biogeochemical ocean models rely on many parameters, which govern the interaction between individual components, and their response to the physical environment. They are often assessed/calibrated against quasi-synoptic data sets of dissolved inorganic tracers. However, a good fit to one observation might not necessarily imply a good match to another. We investigate whether two different metrics—the root-mean-square error to nutrients and oxygen and a metric measuring the overlap between simulated and observed oxygen minimum zones (OMZs)—help to constrain a global biogeochemical model in different aspects of performance. Three global model optimizations are carried out. Two single-objective optimizations target the root-mean-square metric and a sum of both metrics, respectively. We then present and explore multiobjective optimization, which results in a set of compromise solutions. Our results suggest that optimal parameters for denitrification and nitrogen fixation differ when applying different metrics. Optimization against observed OMZs leads to parameters that enhance fixed nitrogen cycling; this causes too low nitrate concentrations and a too high global pelagic denitrification rate. Optimization against nutrient and oxygen concentrations leads to different parameters and a lower global fixed nitrogen turnover; this results in a worse fit to OMZs. Multiobjective optimization resolves this antagonistic effect and provides an ensemble of parameter sets, which help to address different research questions. We finally discuss how systematic model calibration can help to improve models used for projecting climate change and its effect on fisheries and climate gas emissions

Algorithm Engineering for some Complex Practise ProblemsExact Algorithms, Heuristics and Hybrid Evolutionary Algorithms

This work deals with the design of exact algorithms and heuristics for complex optimization problems that origin from three practical applications and one classical combinatorial task. We obtain exact algorithms by modeling our problems in terms of mathematical optimization problems and applying suitable software tools to solve this models. Because of the complexity of our problems, exact algorithms cannot solve large instances within adequate time. Therefore, on the one hand, we derive efficient heuristics by adapting algorithms for similar mathematical problems in a suitable manner. On the other hand, evolutionary algorithms have shown to be successfully applicable to many hard mathematical problems. For that reason, we experimentally determine appropriate EA frameworks, hybridizing the evolutionary operators with our problem specific heuristics