We conduct a rigorous analysis of the (1+1) evolutionary algorithm for the single source shortest path problem proposed by Scharnow, Tinnefeld, and Wegener (The analyses of evolutionary algorithms on sorting and shortest paths problems, 2004, Journal of Mathematical Modelling and Algorithms, 3(4):349-366). We prove that with high probability, the optimization time is O(n2 max{ℓ, log(n)}), where ℓ is the smallest integer such that any vertex can be reached from the source via a shortest path having at most ℓ edges. This bound is tight. For all values of n and ℓ we provide a graph with edge weights such that, with high probability, the optimization time is of order Ω(n2 max{ℓ, log(n)}). To obtain such sharp bounds, we develop a new technique that overcomes the coupon collector behavior of previously used arguments. Also, we exhibit a simple Chernoff type inequality for sums of independent geometrically distributed random variables, and one for sequences of random variables that are not independent, but show a desired behavior independent of the outcomes of the previous random variables. We are optimistic that these tools find further applications in the analysis of evolutionary algorithms.

In this paper, we examine how adding objectives to a given optimization problem affects the computational effort required to generate the set of Pareto-optimal solutions. Experimental studies show that additional objectives may change the running time behavior of an algorithm drastically. Often it is assumed that more objectives make a problem harder as the number of different tradeoffs may increase with the problem dimension. We show that additional objectives, however, may be both beneficial and obstructive depending on the chosen objective. Our results are obtained by rigorous running time analyses that show the different effects of adding objectives to a well-known plateau function. Additional experiments show that the theoretically shown behavior can be observed for problems with more than one objective.

We show that the natural evolutionary algorithm for the all-pairs shortest path problem is significantly faster with a crossover operator than without. This is the first theoretical analysis proving the usefulness of crossover for a non-artificial problem.

Evolutionary algorithms operating on bit strings usually employ a global mutation where each bit is flipped independently with some mutation probability. Most often the mutation probability is set fixed in a way that on average exactly one bit is flipped in a mutation. A seemingly very similar concept is a local one realized by an operator that flips exactly one bit chosen uniformly at random.

Rigorous runtime analyses of evolutionary algorithms (EAs) mainly investigate algorithms that use elitist selection methods. Two algorithms commonly studied are Randomized Local Search (RLS) and the (1+1) EA and it is well known that both optimize any linear pseudo-Boolean function on n bits within an expected number of O(n log n) fitness evaluations. In this paper, we analyze variants of these algorithms that use fitness proportional selection.

We conduct a rigorous analysis of the (1 + 1) evolutionary algorithm for the single source shortest path problem proposed by Scharnow, Tinnefeld and Wegener (Journal of Mathematical Modelling and Algorithms, 2004). We prove a tight bound of Theta(n2max{log(n),lscr}) on the optimization time, where lscr is the maximum number of edges of a shortest path with minimum number of edges from the source to another vertex. Using various tools from probability theory we show that these bounds not only hold in expectation, but also with high probability. We are optimistic that these tools can also be used to analyze the run-time of evolutionary algorithms for other problems.

In this paper, we examine how adding objectives to a given optimization problem affects the computation effort required to generate the set of Pareto-optimal solutions. Experimental studies show that additional objectives may change the runtime behavior of an algorithm drastically. Often it is assumed that more objectives make a problem harder as the number of different trade-offs may increase with the problem dimension. We show that additional objectives, however, may be both beneficial and obstructive depending on the chosen objective. Our results are obtained by rigorous runtime analyses that show the different effects of adding objectives to a well-known plateau-function.

Wireless sensor networks typically consist of small, very simple network nodes without any positioning device like GPS. After an initialization phase, the nodes know with whom they can talk directly, but have no idea about their relative geographic locations. We examine how much geometry information is nevertheless hidden in the communication graph of the network: Assuming that the connectivity is determined by the well-known unit-disk graph model, we prove that using an extremely simple linear-time algorithm one can identify nodes on the boundaries of holes of the network. That is, there is enough geometry information hidden in the connectivity structure to identify topological features—in our example the holes in the network. While the theoretical analysis turns out to be quite conservative,an actual implementation shows that the algorithm works well under less stringent conditions.