Fixed Parameter Evolutionary Algorithms and Maximum Leaf Spanning Trees: A Matter of Mutation

Abstract

Abstract. Evolutionary algorithms have been shown to be very successful for a wide range of NP-hard combinatorial optimization problems. We investigate the NP-hard problem of computing a spanning tree that has a maximal number of leaves by evolutionary algorithms in the context of fixed parameter tractability (FPT) where the maximum number of leaves is the parameter under consideration. Our results show that simple evolutionary algorithms working with an edge-set encoding are confronted with local optima whose size of the inferior neighborhood grows with the value of an optimal solution. Investigating two common mutation operators, we show that an operator related to spanning tree problems leads to an FPT running time in contrast to a general mutation operator that does not have this property.

Fixed-parameter evolutionary algorithms for the Euclidean Traveling Salesperson problem

“…. , q k ) on the inner points; 2 repeat 3 Construct π from π by applying s + 1 random basic operations where s is chosen according to P ois(1); 4 if Dyn(π ) ≤ Dyn(π) then π ← π 5 until termination condition;…”

“…This approach allows one to take an additional parameter measuring the hardness of an instance into account. Results have been obtained for vertex cover [3], minimum leaf spanning trees [4], makespan scheduling [5], and the Euclidean Traveling Salesperson problem [1]. The parameterized complexity analysis of bio-inspired computing seems to be very promising for bridging theory and practice in the field of bio-inspired computing.…”

On the analysis of the (1+1) evolutionary algorithm for the maximum leaf spanning tree problem

“…As for the MLST problem, Kratsch et al [11] presented a parameterized analysis of EAs on this problem, where the performance of EAs with two different mutation operators has been investigated, that is, standard bit-mutation and more problem-specific operator. However, the approximation performances of EAs on the MLST problem are still unknown.…”

“…Various methods have been proposed for solving this NP-hard problem, such as exact algorithm [7], and fixed parameter tractable algorithms [4,5,11]. Besides, some variations of this problem for cubic graphs [9,12] and bipartite graphs [13] have also been investigated.…”

A Parameterized Runtime Analysis of Simple Evolutionary Algorithms for Makespan Scheduling

“…It facilitates the understanding of which features in an instance of a given problem makes the problem hard to solve. Parameterized runtime results have been obtained in the context of evolutionary computation for the vertex cover problem [9], the computation of maximum leaf spanning trees [8], the MAX-2-SAT problem [12], and the Euclidean TSP [13].…”