2009/06/28 by Raymond Hemmecke, Matthias Köppe, Jon Lee +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Optimization Algorithms Research #Complexity and Algorithms in Graphs #Cover (algebra) #Focus (optics) #Function (biology) #Integer (computer science) #Integer programming #Nonlinear system #Polynomial #Simple (philosophy) #math.OC #msc:90C10 #msc:90C11 #msc:90C30
paper · pdf · doi:10.1007/978-3-540-68279-0_15
57 pages. To appear in: M. Jünger, 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 3540682740
arxiv created 2009/06/28 · openalex publication_date 2009/11/06 · openalex created_date 2016/06/24 · arxiv updated 2017/01/03 · openalex updated_date 2026/08/05
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.