arXiv subjects

Carola Winzen

Publications and source records attributed to Carola Winzen.

12 recordsLinked to original sources

arXiv2012

Black-Box Complexity: Breaking the $O(n \log n)$ Barrier of LeadingOnes

We show that the unrestricted black-box complexity of the $n$-dimensional XOR- and permutation-invariant LeadingOnes function class is $O(n \log (n) / \log \log n)$. This shows that the recent natural looking $O(n\log n)$ 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 (GECCO 2010). 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).

cs.DS↗

arXiv2012

The Price of Anarchy for Selfish Ring Routing is Two

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.

cs.GT↗

arXiv2012

Ranking-Based Black-Box Complexity

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 gives more realistic complexity estimates for some problems. For example, 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 $Θ(n)$. 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.

cs.NE↗

arXiv2012

Reducing the Arity in Unbiased Black-Box Complexity

We show that for all $1<k \leq \log n$ 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 $O(2^k)$ bits.

cs.NE↗

arXiv2011

Playing Mastermind With Constant-Size Memory

We analyze the classic board game of Mastermind with $n$ holes and a constant number of colors. A result of Chvátal (Combinatorica 3 (1983), 325-329) 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 of Computing Systems 39 (2006), 525-544) on the memory-restricted black-box complexity of the OneMax function class.

cs.DS↗

arXiv2011

Black-Box Complexities of Combinatorial Problems

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.

cs.NE↗

arXiv2011

A Randomized Algorithm Based on Threshold Accepting to Approximate the Star Discrepancy

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), 2028--2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a non-uniform sampling strategy which is more suited for higher-dimensional inputs, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into \emph{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 dimension the new method behaves clearly better than all previously known methods.

cs.DS↗

arXiv2011

Direction-Reversing Quasi-Random Rumor Spreading with Restarts

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.

cs.DS↗

arXiv2011

Multiplicative Drift Analysis

In this work, we introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. We give a multiplicative version of the classical drift theorem. This allows easier analyses in those settings where 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 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 ${(1+o(1)) 1.39 \euler n\ln (n)}$, 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.

cs.NE↗

arXiv2010

Faster Black-Box Algorithms Through Higher Arity Operators

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 $Θ(n^2)$ for unary operators to $O(n \log n)$. For \onemax, the $Ω(n \log n)$ 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 $O(n/\log k)$.

cs.NE↗

arXiv2010

Non-Existence of Linear Universal Drift Functions

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 $1/n$.

cs.NE↗

arXiv2010

Optimizing Monotone Functions Can Be Difficult

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 monotone. 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) evolutionary algorithm 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 $c > 33$, we present a strictly monotone function such that the (1+1) evolutionary algorithm with overwhelming probability does not find the optimum within $2^{Ω(n)}$ iterations. This is the first time that we observe that a constant factor change of the mutation probability changes the run-time by more than constant factors.

cs.NE↗