An Iterated Local Search Approach for Finding Provably Good Solutions for Very Large TSP Instances

Abstract. Meta-heuristics usually lack any kind of performance guar-antee and therefore one cannot be certain whether the resulting solutions are (near) optimum solutions or not without relying on additional algo-rithms for providing lower bounds (in case of minimization). In this paper, we present a highly effective hybrid evolutionary local search algorithm based on the iterated Lin-Kernighan heuristic combined with a lower bound heuristic utilizing 1-trees. Since both upper and lower bounds are improved over time, the gap between the two bounds is minimized by means of effective heuristics. In experiments, we show that the proposed approach is capable of finding short tours with a gap of 0.8 % or less for TSP instances up to 10 million cities. Hence, to the best of our knowledge, we present the first evolutionary algorithm and meta-heuristic in general that delivers provably good solutions and is highly scalable with the problem size. We show that our approach outperforms all existing heuristics for very large TSP instances.

Polynomial-time separation of simple comb inequalities

The comb inequalities are a well-known class of facet-inducing inequalities for the Traveling Salesman Problem, defined in terms of certain vertex sets called the handle and the teeth. We say that a comb inequality is simple if the following holds for each tooth: either the intersection of the tooth with the handle has cardinality one, or the part of the tooth outside the handle has cardinality one, or both. The simple comb inequalities generalize the classical 2-matching inequalities of Edmonds, and also the so-called Chv´atal comb inequalities. In 1982, Padberg and Rao gave a polynomial-time algorithm for separating the 2-matching inequalities – i.e., for testing if a given fractional solution to an LP relaxation violates a 2-matching inequality. We extend this significantly by giving a polynomial-time algorithm for separating the simple comb inequalities. The key is a result due to Caprara and Fischetti

Growth performance and management of a mixed rainforest tree plantation

Monoculture plantations of Pinus, Eucalyptus and Acacia have been established oil rainforest lands throughout the world. However, this type of reforestation generally supplies low quality timber and contributes to landscape simplification. Alternatives to exotic monoculture plantations are now beginning to gain momentum with farmers and landholders attempting to establish a variety of rainforest trees in small plantations. When compared to the well studied commercial species, knowledge concerning the growth and management of many of these rainforest species is in its infancy. To help expand this limited knowledge base an experimental plantation of 16 rainforest tree species in a randomised design was established near Mt. Mee, in south-eastern Queensland, Australia. Changes in growth, form (based on stem straightness, branch size and branchiness), crown diameters and leaf area of each species were examined over 5 years. Patterns of height growth were also measured monthly for 31 months. Species in this trial could be separated into three groups based on their overall growth after 5 years and their growth patterns. Early successional status, low timber density, high maximum photosynthetic rates and large total leaf areas were generally correlated to rapid height growth. Several species (including Araucaria cunninghamii, Elaeocarpus grandis, Flindersia brayleyana, Grevillea robusta and Khaya nyasica) had above average form and growth, while all species in the trial had considerable potential to have increased productivity through tree selection. As canopy closure occurred at the site between years four and five, growth increments declined. To reduce stand competition a number of different thinning techniques could be employed. However, simple geometric or productivity based thinnings appear to be inappropriate management techniques for this mixed species stand as they would either remove many of the best performing trees or nearly half the species in the trial. Alternatively, a form based thinning would maintain the site's diversity, increase the average form of the plantation and provide some productivity benefits

Combining Metaheuristics and Exact Algorithms in Combinatorial Optimization: A Survey and Classification

International audienceIn this survey we discuss different state-of-the-art approaches of combining exact algorithms and metaheuristics to solve combinatorial optimization problems. Some of these hybrids mainly aim at providing optimal solutions in shorter time, while others primarily focus on getting better heuristic solutions. The two main categories in which we divide the approaches are collaborative versus integrative combinations. We further classify the different techniques in a hierarchical way. Altogether, the surveyed work on combinations of exact algorithms and metaheuristics documents the usefulness and strong potential of this research direction