We analyze the classic board game of Mastermind with n holes and a constant number of colors. The classic result of Chvátal (Combinatorica 3:325–329, 1983 ) states that the codebreaker can find the secret code with Θ ( n /log n ) questions. We show that this bound remains valid if the codebreaker may only store a constant number of guesses and answers. In addition to an intrinsic interest in this question, our result also disproves a conjecture of Droste, Jansen, and Wegener (Theory Comput. Syst. 39:525–544, 2006 ) on the memory-restricted black-box complexity of the OneMax function class.

Randomized search heuristics such as evolutionary algorithms, simulated annealing, and ant colony optimization are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime analysis results have appeared in the last 20 years, a powerful complexity theory for such algorithms is yet to be developed. We enrich the existing notions of black-box complexity by the additional restriction that not the actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the black-box algorithm. Many randomized search heuristics belong to this class of algorithms. We show that the new ranking-based model can give more realistic complexity estimates. The class of all binary-value functions has a black-box complexity of O (log n ) in the previous black-box models, but has a ranking-based complexity of Θ ( n ). On the other hand, for the class of all OneMax functions, we present a ranking-based black-box algorithm that has a runtime of Θ ( n /log n ), which shows that the OneMax problem does not become harder with the additional ranking-basedness restriction.

We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n 1 − 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 ( 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 Chvátal. 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 n 2 log log n colors. These bounds are almost tight, as the known lower bound of Ω( n ) shows. Unlike for k ≤ n 1 − 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( n log n ) guesses.

In a recent work, Doerr and Fouz [\emph{Asymptotically Optimal Randomized Rumor Spreading}, in ArXiv] present a new quasi-random PUSH algorithm for the rumor spreading problem (also known as gossip spreading or message propagation problem). Their \emph{hybrid protocol} outperforms all known PUSH protocols. In this work, we add to the hybrid protocol a direction-reversing element. We show that this \emph{direction-reversing quasi-random rumor spreading protocol with random restarts} yields a constant factor improvement over the hybrid model, if we allow the same dose of randomness. Put differently, our protocol achieves the same broadcasting time as the hybrid model by employing only (roughly) half the number of random choices.

Black-box complexity is a complexity theoretic measure for how difficult a problem is to be optimized by a general purpose optimization algorithm. It is thus one of the few means trying to understand which problems are tractable for genetic algorithms and other randomized search heuristics.Most previous work on black-box complexity is on artificial test functions. In this paper, we move a step forward and give a detailed analysis for the two combinatorial problems minimum spanning tree and single-source shortest paths. Besides giving interesting bounds for their black-box complexities, our work reveals that the choice of how to model the optimization problem is non-trivial here. This in particular comes true where the search space does not consist of bit strings and where a reasonable definition of unbiasedness has to be agreed on.

We analyze the network congestion game with atomic players, asymmetric strategies, and the maximum latency among all players as social cost. This important social cost function is much less understood than the average latency. We show that the price of anarchy is at most two, when the network is a ring and the link latencies are linear. Our bound is tight. This is the first sharp bound for the maximum latency objective.

We introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. Our multiplicative version of the classical drift theorem allows easier analyses in the often encountered situation that the optimization progress is roughly proportional to the current distance to the optimum. To display the strength of this tool, we regard the classical problem of how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time O ( n log n ), where n is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most (1+ o (1))1.39e n ln n , again using multiplicative drift analysis. We also prove a corresponding lower bound of (1− o (1))e n ln n which actually holds for all functions with a unique global optimum. We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours in graphs.

Extending previous analyses on function classes like linear functions, we analyze how the simple (1+1) evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotonic. These functions have the property that whenever only 0-bits are changed to 1, then the objective value strictly increases. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant c in the mutation probability p(n) = c/n can make a decisive difference. We show that if c < 1, then the (1+1) EA finds the optimum of every such function in Θ(n log n) iterations. For c = 1, we can still prove an upper bound of O(n(3/2)). However, for c ≥ 16, we present a strictly monotonic function such that the (1+1) EA with overwhelming probability needs 2(Ω(n)) iterations to find the optimum. This is the first time that we observe that a constant factor change of the mutation probability changes the runtime by more than a constant factor.

We show that for all 1 < k d log n the k-ary unbiased black-box complexity of the n-dimensional OneMax function class is O(n/k). This indicates that the power of higher arity operators is much stronger than what the previous O(n/log k) bound by Doerr et al. (Faster black-box algorithms through higher arity operators, Proc. of FOGA 2011, pp. 163--172, ACM, 2011) suggests. The key to this result is an encoding strategy, which might be of independent interest. We show that, using k-ary unbiased variation operators only, we may simulate an unrestricted memory of size O(2k) bits.

We show that the black-box complexity with memory restriction one of the n-dimensional OneMax function class is at most 2n. This disproves the @Q(nlogn) conjecture of Droste, Jansen, and Wegener (Theory Computing Systems 39 (2006) 525-544).

We study the LeadingOnes game, a Mastermind-type guessing game first regarded as a test case in the complexity theory of randomized search heuristics. The first player, Carole, secretly chooses a string z ∈ {0, 1} and a permutation π of [n]. The goal of the second player, Paul, is to identify the secret (z, π) with a small number of queries. A query is a string x ∈ {0, 1}, and the score of x is fz,π(x) := max{i ∈ [0..n] | ∀j ≤ i : zπ(j) = xπ(j)} , the length of the longest common prefix of x and z with respect to π. We are interested in the number of queries needed by Paul to identify the secret. By using a relatively straightforward strategy, Paul can identify the secret with O(n log n) queries and recently only a modest improvement of this to O(n log n/ log log n) was available [DW12]. In this paper, we completely resolve the problem by offering the following results. We show that when limited to deterministic strategies, O(n log n) queries is the best possible. On the other hand, by using randomization Paul can find the secret code with an expected number of O(n log log n) queries, which we prove is optimal by matching it with a lower bound of the same asymptotic magnitude. Finally, we prove that a number of problems that are naturally related to our problem (such as deciding whether a sequence of queries and scores is consistent) can be solved in polynomial time.

Drift analysis has become a powerful tool to prove bounds on the runtime of randomized search heuristics. It allows, for example, fairly simple proofs for the classical problem how the (1+1) Evolutionary Algorithm (EA) optimizes an arbitrary pseudo-Boolean linear function. The key idea of drift analysis is to measure the progress via another pseudo-Boolean function (called drift function) and use deeper results from probability theory to derive from this a good bound for the runtime of the EA. Surprisingly, all these results manage to use the same drift function for all linear objective functions. In this work, we show that such universal drift functions only exist if the mutation probability is close to the standard value of $1/n$.

We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal., 34 (1997), pp. 2028-2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a nonuniform sampling strategy, which is more suited for higher-dimensional inputs and additionally takes into account the topological characteristics of given point sets, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into critical test boxes. These critical test boxes provably yield higher discrepancy values and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimensions the new method behaves clearly better than all previously known methods.

We show that the unrestricted black-box complexity of the n-dimensional XOR- and permutation-invariant LeadingOnes function class is O(n log(n) / loglogn). This shows that the recent natural looking O(nlogn) bound is not tight. The black-box optimization algorithm leading to this bound can be implemented in a way that only 3-ary unbiased variation operators are used. Hence our bound is also valid for the unbiased black-box complexity recently introduced by Lehre and Witt. The bound also remains valid if we impose the additional restriction that the black-box algorithm does not have access to the objective values but only to their relative order (ranking-based black-box complexity).

We study fault tolerant rumor spreading algorithms in the complete graph topology. Our focus is on algorithms that use minimum communication both in a global and local sense: they establish the minimum possible number of inter-processor connections in total, and in each round each processor is involved in at most one connection. The challenge is in designing such algorithms that have an asymptotically optimal, that is, logarithmic, time complexity even in the presence of failed nodes. We rst show, using a coupling argument, that if nodes are crashed not adversarially, but independently at random with constant probability less than one, then already the basic GP algorithm of Gasieniec and Pelc (Parallel Computing 22:903{912, 1996) with high probability has an asymptotically optimal O(logn) time complexity. This improves signicantly over the worst-case guarantee of f +O(logn) given there for f crashed nodes. We then show that by adding randomization to the algorithm, these time and communication complexities can be maintained also against adversarial failures. This is easily achieved by running the GP-algorithm with randomly permuted node labels , at the price, however, of increasing the communication overhead to an average (log

Randomized search heuristics are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime bounds exist, a powerful complexity theory for such algorithms is yet to be developed. We contribute to this goal in several aspects. In a first step, we analyze existing black-box complexity models. Our results indicate that these models are not restrictive enough. This remains true if we restrict the memory of the algorithms under consideration. These results motivate us to enrich the existing notions of black-box complexity by the additional restriction that not actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the algorithms. Many heuristics belong to this class of algorithms. We show that our ranking-based model gives more realistic complexity estimates for some problems, while for others the low complexities of the previous models still hold. Surprisingly, our results have an interesting game-theoretic aspect as well. We show that analyzing the black-box complexity of the OneMaxn function class—a class often regarded to analyze how heuristics progress in easy parts of the search space—is the same as analyzing optimal winning strategies for the generalized Mastermind game with 2 colors and length-n codewords. This connection was seemingly overlooked so far in the search heuristics community.

Randomisierte Suchheuristiken sind problemunabhangige Algorithmen, die sowohl im wissenschaftlichen als auch im industriellen Kontext zur Optimierung von schwierigen Problemen genutzt werden. Sie sind einfach zu implementieren, lassen sich vielseitig einsetzen und liefern uberraschend haufig bereits in kurzer Zeit sehr gute Ergebnisse. Daher sind randomisierte Suchheuristiken weit verbreitet. Ein groses Problem in Anwendung von randomisierten Suchheuristiken ist jedoch die Tatsache, dass sich schwer vorhersagen lasst, ob sich das zu optimierende Problem gut durch eine geeignete Heuristik losen lasst oder ob andere problemspezifische Verfahren deutliche besser geeignet sind. Mit meiner Dissertation leisten wir einen Beitrag zur Entwicklung einer Komplexitatstheorie fur randomisierte Suchheuristiken. Unser langfristiges Ziel ist die Charakterisierung von Problemklassen in solche, die sich schnell und zuverlassig durch Suchheuristiken optimieren lassen und solche, fur die grundsatzlich andere Methoden besser geeignet sind.

We extend the work of Lehre and Witt (GECCO 2010) on the unbiased black-box model by considering higher arity variation operators. In particular, we show that already for binary operators the black-box complexity of \leadingones drops from $\Theta(n^2)$ for unary operators to $O(n \log n)$. For \onemax, the $\Omega(n \log n)$ unary black-box complexity drops to O(n) in the binary case. For $k$-ary operators, $k \leq n$, the \onemax-complexity further decreases to $O(n/\log k)$.

Randomized search heuristics are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. A big step forward would be a useful complexity theory for such algorithms. We enrich the two existing black-box complexity notions due to Wegener and other authors by the restrictions that not actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the algorithm. Many randomized search heuristics belong to this class of algorithms. We show that the new ranking-based model gives more realistic complexity estimates for some problems, while for others the low complexities of the previous models still hold.