Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem☆
Introduction
Evolutionary algorithms (EAs) are randomized search heuristics which have found many applications in generating good solutions for real-world problems as well as for NP-hard combinatorial optimization problems. Especially in the case of multi-objective optimization problems they have been successfully applied. In this case the population of a multi-objective evolutionary algorithm (MOEA) is used to compute or approximate the Pareto front. In contrast to their wide applications there are only a few theoretical results on the runtime behavior of MOEAs. Such analyses are necessary to get a better understanding how these heuristics work and which parts of the Pareto front can be computed quickly.
In the last years, a lot of progress has been made in the analysis of evolutionary algorithms with respect to the expected runtime and the probability to find an optimum after a fixed number of steps (see Droste et al. (2002) for an overview). The first analyses consider EAs for simple single-objective problems. Principally, the optimization of special pseudo boolean functions has been investigated to show the behavior of EAs in different situations and to develop new methods for analyzing the runtime of EAs.
The analysis of evolutionary algorithms for combinatorial optimization problems has been started for some well-known problems. There are results on sorting and shortest path problems (Scharnow et al., 2002), on maximum matchings (Giel and Wegener, 2003), on Eulerian cycles (Neumann, 2004), and on minimum spanning trees (Neumann and Wegener, 2004). All these mentioned problems can be solved in polynomial time by deterministic algorithms. An analysis of evolutionary algorithms on problems belonging to the complexity class P is helpful to understand how evolutionary algorithms work on NP-hard problems as well as to develop better algorithms for such problems. Recently, Witt (2005) has considered the NP-hard partition problem and analyzed it with respect to its approximability in expected polynomial time by simple EAs. In addition he has presented an average case analysis for two input distributions.
There are only a few results on the expected runtime of multi-objective evolutionary algorithms. Laumanns et al., 2004a, Laumanns et al., 2004b have analyzed two multi-objective local search algorithms (SEMO and FEMO) for a problem with conflicting objectives. Giel (2003) has investigated a simple MOEA that searches globally (Global SEMO). He has presented bounds on the expected runtime and success probability for simple pseudo boolean functions. Neumann and Wegener (2005) have examined whether a multi-objective model can help to construct faster evolutionary algorithms for single-objective optimization problems. They have shown that minimum spanning trees can be computed faster by using a multi-objective EA instead of a corresponding single-objective one.
In this paper we analyze MOEAs on a NP-hard multi-objective combinatorial optimization problem. Laumanns et al., 2004a, Laumanns et al., 2004b have considered a special instance of the multi-objective knapsack problem. An analysis of MOEAs on a combinatorial problem, not only a class of instances, is still missing. This is indeed the first paper analyzing a MOEA with respect to the expected runtime on a NP-hard combinatorial optimization problem.
We consider the multi-objective minimum spanning tree problem. Many successful evolutionary algorithms have been proposed for this problem (see e.g Zhou and Gen, 1999, Knowles and Corne, 2001). Neumann and Wegener (2004) have shown that randomized search heuristics are able to compute minimum spanning trees in expected polynomial time. Their analysis is based on the investigation of the expected multiplicative weight decrease (with respect to the difference of the weight of the current graph and the weight of a minimum spanning tree) and serves as a starting point for our analysis.
We analyze Global SEMO until it has produced a population including for each extremal point of the Pareto front a corresponding solution. The extremal points of the Pareto front are of particular interest as we show that they constitute a 2-approximation of the Pareto front.
After having given a motivation for our work we introduce in Section 2 the basic concepts of Pareto optimality and our model of the multi-objective minimum spanning tree problem. In Section 3 we show that the extremal points constitute a 2-approximation of the Pareto front, and present the multi-objective evolutionary algorithm which we will consider in Section 4. We analyze this algorithm in Section 5 with respect to the expected time until it has produced a population that includes for each extremal point of the Pareto front a corresponding spanning tree and finish with some conclusions.
Section snippets
Multi-objective minimum spanning trees
In the scenario of multi-objective optimization k, often conflicting, objectives have to be optimized at the same time. The aim is to find solutions such that an improvement of one objective can only be achieved at the expense of another one. These concepts lead to the following definition of Pareto optimality. Definition 1 Let the search space S and the problem to minimize a vector-valued function f = (f1, … , fk), f : S → O, be given. A search point x dominates a search point x′ (x ≺f x′) if fi(x) ⩽ fi(x′) holds
The extremal points of the convex hull
Let F be the Pareto front of a given instance. If we consider the bi-objective problem conv(F) is a piecewise linear function (see Fig. 1). Note that for each spanning tree T on the convex hull there is a λ ∈ [0, 1] such that T is a minimum spanning tree with respect to the single weight function λw1 + (1 − λ)w2 (see e.g. Knowles and Corne, 2001). We will use this in Section 5 to transform an arbitrary spanning tree S into a desired Pareto optimal spanning tree T on conv(F) using Theorem 1.
Let q1 and q
The algorithm
We consider the algorithm called Global SEMO (Simple Evolutionary Multi-objective Optimizer) which has already been discussed by Giel (2003) applying it to pseudo boolean functions. This algorithm can be seen as a generalization of the (1 + 1) EA, which has been considered in a number of theoretical analyses, to the multi-objective case. Recently, Neumann and Wegener (2005) have shown that the minimum spanning tree problem can be solved more easily by Global SEMO than by the (1 + 1) EA. Global SEMO
Analysis of Global SEMO
Lemma 4 Global SEMO working on the fitness function f or f′ constructs a population consisting of connected graphs in expected time O(m log n). Proof Due to the fitness functions no steps increasing the number of connected components are accepted. At any time the current population P consists of solutions having the same number of connected components as otherwise the solution s with the smallest number of connected components would dominate all solutions with a larger number of connected components in P. The
Conclusions
The multi-objective minimum spanning tree problem is one of the best-known multi-objective combinatorial optimization problems. For the first time evolutionary algorithms have been analyzed with respect to the expected time until they produce solutions of the Pareto front. In the case of a strongly convex Pareto front, we have achieved a pseudopolynomial bound on the expected time until the population includes for each Pareto optimal objective vector a corresponding spanning tree. For an
Acknowledgements
Thanks to Ingo Wegener for helpful discussions during the preparation of this work and to the reviewers for useful suggestions.
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