Efficient Forest Data Structure for Evolutionary Algorithms Applied to Network Design

The design of a network is a solution to several engineering and science problems. Several network design problems are known to be NP-hard, and population-based metaheuristics like evolutionary algorithms (EAs) have been largely investigated for such problems. Such optimization methods simultaneously generate a large number of potential solutions to investigate the search space in breadth and, consequently, to avoid local optima. Obtaining a potential solution usually involves the construction and maintenance of several spanning trees, or more generally, spanning forests. To efficiently explore the search space, special data structures have been developed to provide operations that manipulate a set of spanning trees (population). For a tree with n nodes, the most efficient data structures available in the literature require time O(n) to generate a new spanning tree that modifies an existing one and to store the new solution. We propose a new data structure, called node-depth-degree representation (NDDR), and we demonstrate that using this encoding, generating a new spanning forest requires average time O(root n). Experiments with an EA based on NDDR applied to large-scale instances of the degree-constrained minimum spanning tree problem have shown that the implementation adds small constants and lower order terms to the theoretical bound.

Genetic variability and demographic history of Heliothis virescens (Lepidoptera: Noctuidae) populations from Brazil inferred by mtDNA sequences

Intra-and inter-population genetic variability and the demographic history of Heliothis virescens (F.) populations were evaluated by using mtDNA markers (coxI, coxII and nad6) with samples from the major cotton-and soybean-producing regions in Brazil in the growing seasons 2007/08, 2008/09 and 2009/10. AMOVA indicated low and non-significant genetic structure, regardless of geographical scale, growing season or crop, with most of genetic variation occurring within populations. Clustering analyzes also indicated low genetic differentiation. The haplotype network obtained with combined datasets resulted in 35 haplotypes, with 28 exclusive occurrences, four of them sampled only from soybean fields. The minimum spanning network showed star-shaped structures typical of populations that underwent a recent demographic expansion. The recent expansion was supported by other demographic analyzes, such as the Bayesian skyline plot, the unimodal distribution of paired differences among mitochondrial sequences, and negative and significant values of neutrality tests for the Tajima's D and Fu's F-S parameters. In addition, high values of haplotype diversity ((H) over cap) and low values of nucleotide diversity (pi), combined with a high number of low frequency haplotypes and values of theta(pi)<theta(W), suggested a recent demographic expansion of H. virescens populations in Brazil. This demographic event could be responsible for the low genetic structure currently found; however, haplotypes present uniquely at the same geographic regions and from one specific host plant suggest an initial differentiation among H. virescens populations within Brazil.

Maximum Series-Parallel Subgraph

Consider the NP-hard problem of, given a simple graph G, to find a series-parallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a 1/2-approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has n-1 edges and any series-parallel graph on n vertices has at most 2n-3 edges. We present a 7/12 -approximation for this problem and results showing the limits of our approach.