Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

An unrecognized ancient lineage of green plants persists in deep marine waters

We provide molecular phylogenetic evidence that the obscure genera Palmophyllum Kütz. and Verdigellas D. L. Ballant. et J. N. Norris form a distinct and early diverging lineage of green algae. These palmelloid seaweeds generally persist in deep waters, where grazing pressure and competition for space are reduced. Their distinctness warrants recognition as a new order, the Palmophyllales. Although phylogenetic analyses of both the 18S rRNA gene and two chloroplast genes (atpB and rbcL) are in agreement with a deep-branching Palmophyllales, the genes are in conflict about its exact phylogenetic placement. Analysis of the nuclear ribosomal DNA allies the Palmophyllales with the prasinophyte genera Prasinococcus and Prasinoderma (Prasinococcales), while the plastid gene phylogeny placed Palmophyllum and Verdigellas as sister clade to all other Chlorophyta