Recognizing Treelike k-Dissimilarities

A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called "4-point condition". However, in case k > 2 Pachter and Speyer recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k >= 3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2k-element subset of X arises from some tree, and that 2k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.Comment: 18 pages, 4 figure

Understanding Phase Transitions with Local Optima Networks: Number Partitioning as a Case Study

Phase transitions play an important role in understanding search difficulty in combinatorial optimisation. However, previous attempts have not revealed a clear link between fitness landscape properties and the phase transition. We explore whether the global landscape structure of the number partitioning problem changes with the phase transition. Using the local optima network model, we analyse a number of instances before, during, and after the phase transition. We compute relevant network and neutrality metrics; and importantly, identify and visualise the funnel structure with an approach (monotonic sequences) inspired by theoretical chemistry. While most metrics remain oblivious to the phase transition, our results reveal that the funnel structure clearly changes. Easy instances feature a single or a small number of dominant funnels leading to global optima; hard instances have a large number of suboptimal funnels attracting the search. Our study brings new insights and tools to the study of phase transitions in combinatorial optimisation

Transoceanic Dispersal and Subsequent Diversification on Separate Continents Shaped Diversity of the Xanthoparmelia pulla Group (Ascomycota)

In traditional morphology-based concepts many species of lichenized fungi have world-wide distributions. Molecular data have revolutionized the species delimitation in lichens and have demonstrated that we underestimated the diversity of these organisms. The aim of this study is to explore the phylogeography and the evolutionary patterns of the Xanthoparmelia pulla group, a widespread group of one of largest genera of macrolichens. We used a dated phylogeny based on nuITS and nuLSU rDNA sequences and performed an ancestral range reconstruction to understand the processes and explain their current distribution, dating the divergence of the major lineages in the group. An inferred age of radiation of parmelioid lichens and the age of a Parmelia fossil were used as the calibration points for the phylogeny. The results show that many species of the X. pulla group as currently delimited are polyphyletic and five major lineages correlate with their geographical distribution and the biosynthetic pathways of secondary metabolites. South Africa is the area where the X. pulla group radiated during the Miocene times, and currently is the region with the highest genetic, morphological and chemical diversity. From this center of radiation the different lineages migrated by long-distance dispersal to others areas, where secondary radiations developed. The ancestral range reconstruction also detected that a secondary lineage migrated from Australia to South America via long-distance dispersal and subsequent continental radiation

Solving the Multidimensional Knapsack Problem Using an Evolutionary Algorithm Hybridized with Branch and Bound

Abstract. A hybridization of an evolutionary algorithm (EA) with the branch and bound method (B&B) is presented in this paper. Both tech-niques cooperate by exchanging information, namely lower bounds in the case of the EA, and partial promising solutions in the case of the B&B. The multidimensional knapsack problem has been chosen as a bench-mark. To be precise, the algorithms have been tested on large problems instances from the OR-library. As it will be shown, the hybrid approach can provide high quality results, better than those obtained by the EA and the B&B on their own.