A tight analysis of the (1 + 1)-EA for the single source shortest path problem

Abstract

Abstract-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 Θ(n 2 max{log(n), }) on the optimization time, where 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.

Computing single source shortest paths using single-objective fitness

“…Note that, contrary to the multiobjective setting, this length L may decrease. We shall adopt the proof of [4] for the multi-objective setting to deal with this issue. In particular, we denote by Lt the maximum length ℓ such that the path (v0, .…”

“…In addition to n denoting the number of vertices of the input graph, let ℓ denote the maximum number of edges of a shortest path. Then [4] shows that the optimization time of the multi-objective EA proposed in [17] is O(n 2 max{log(n), ℓ}) with high probability. The methods of [17] would only yield an optimization time of O(n 2 ℓ log(n)) in expectation.…”

“…We should note that the results in [17] have recently been improved in [4]. In addition to n denoting the number of vertices of the input graph, let ℓ denote the maximum number of edges of a shortest path.…”

Runtime analysis of the (1+1) evolutionary algorithm on strings over finite alphabets

“…In contrast to linear pseudo-Boolean functions, the optimization behavior of the (1+1) EA on combinatorial problems like the minimum spanning tree problems [15,17,18] or the shortest path tree problem [2,3,4,19] are still poorly understood with respect to structural insights.…”

Exploring the Runtime of an Evolutionary Algorithm for the Multi-Objective Shortest Path Problem

“…The paper by Scharnow et al (2004) analyzes the runtime of a simple EA for the single-source variant. Doerr, Happ, et al (2007) refine this analysis by classifying the input instances according to where is the smallest integer such that any vertex can be reached from the source via a shortest path with at most edges. The all-pairs variant, presented in Doerr et al (2008), gives a natural example where the use of an appropriate recombination operator can improve the runtime of an EA.…”