Carola Winzen

Multiplicative drift analysis

Drift analysis is one of the strongest tools in the analysis of evolutionary algorithms. Its main weakness is that it is often very hard to find a good drift function. In this paper, we make progress in this direction. We prove 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 objective value. Our drift theorem immediately gives natural proofs for the best known run-time bounds for the (1+1) Evolutionary Algorithm computing minimum spanning trees and shortest paths, since here we may simply take the objective function as drift function. As a more challenging example, we give a relatively simple proof for the fact that any linear function is optimized in time O(n log n). In the multiplicative setting, a simple linear function can be used as drift function (without taking any logarithms). However, we also show that, both in the classical and the multiplicative setting, drift functions yielding good results for all linear functions exist only if the mutation probability is at most c/n for a small constant c.

Drift analysis and linear functions revisited

We regard the classical problem how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. We show that any such function is optimized in time (1 + o(1)) 1.39en ln (n), where n is the length of the bit string. We also prove a lower bound of (1 −o(1))en ln(n), which in fact holds for all functions with a unique global optimum. This shows that for linear functions, even though the optimization behavior might differ, the resulting runtimes are very similar. Our experimental results suggest that the true optimization times are even closer than what the theoretical guarantees promise.

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(<i>n</i>²) for unary operators to <i>O</i>(<i>n</i> log <i>n</i>). For OneMax, the Ω(<i>n</i> log <i>n</i>) unary black-box complexity drops to <i>O</i>(<i>n</i>) in the binary case. For <i>k</i>-ary operators, <i>k</i> ≤ <i>n</i>, the OneMax-complexity further decreases to <i>O</i>(<i>n</i> / log <i>k</i>).

Finding optimal volume subintervals with k points and calculating the star discrepancy are NP-hard problems

The well-known star discrepancy is a common measure for the uniformity of point distributions. It is used, e.g., in multivariate integration, pseudo random number generation, experimental design, statistics, or computer graphics. We study here the complexity of calculating the star discrepancy of point sets in the d-dimensional unit cube and show that this is an NP-hard problem. To establish this complexity result, we first prove NP-hardness of the following related problems in computational geometry: Given n points in the d-dimensional unit cube, find a subinterval of minimum or maximum volume that contains k of the n points. Our results for the complexity of the subinterval problems settle a conjecture of E. Thiemard [E. Thiemard, Optimal volume subintervals with k points and star discrepancy via integer programming, Math. Meth. Oper. Res. 54 (2001) 21-45].

Playing Mastermind with Constant-size Memory

We analyze the classic board game of Mastermind with n holes and a constant number of colors. The classic result of Chvatal (Combinatorica 3 (1983), 325-329) states that the codebreaker can find the secret code with Theta(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.

Mutation rate matters even when optimizing monotonic functions

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 iterations. For c=1, we can still prove an upper bound of O(n3/2). However, for , we present a strictly monotonic function such that the (1+1) EA with overwhelming probability needs 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.

Towards a complexity theory of randomized search heuristics: ranking-based black-box complexity

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.

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

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.

Playing Mastermind with Constant-Size Memory

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/logn) 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.

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) 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(n3/2). However, for c > 33, we present a strictly monotone function such that the (1+1) EA with overwhelming probability does not find the optimum within 2Ω(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.