Combinatorial Optimization: Theory and Algorithms

This comprehensive textbook on combinatorial optimization puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. It has arisen as the basis of several courses on combinatorial optimization and more special topics at graduate level. Since the complete book contains enough material for at least four semesters (4 hours a week), one usually selects material in a suitable way. The book contains complete but concise proofs, also for many deep results, some of which did not appear in a book before. Many very recent topics are covered as well, and many references are provided. Thus this book represents the state of the art of combinatorial optimization. This third edition contains a new chapter on facility location problems, an area which has been extremely active in the past few years. Furthermore there are several new sections and further material on various topics. New exercises and updates in the bibliography were added. From the reviews of the 2nd edition: "This book on combinatorial optimization is a beautiful example of the ideal textbook." __Operations Resarch Letters 33 (2005), p.216-217__ "The second edition (with corrections and many updates) of this very recommendable book documents the relevant knowledge on combinatorial optimization and records those problems and algorithms that define this discipline today. To read this is very stimulating for all the researchers, practitioners, and students interested in combinatorial optimization." __OR News 19 (2003), p.42__ This Book Provides The First Rigorous Derivation Of Mesoscopic And Macroscopic Equations From A Deterministic System Of Microscopic Equations. The Microscopic Equations Are Cast In The Form Of A Deterministic (newtonian) System Of Coupled Nonlinear Oscillators For N Large Particles And Infinitely Many Small Particles. The Mesoscopic Equations Are Stochastic Ordinary Differential Equations (sodes) And Stochastic Partial Differential Equatuions (spdes), And The Macroscopic Limit Is Described By A Parabolic Partial Differential Equation. A Detailed Analysis Of The Sodes And (quasi-linear) Spdes Is Presented. Semi-linear (parabolic) Spdes Are Represented As First Order Stochastic Transport Equations Driven By Stratonovich Differentials. The Time Evolution Of Correlated Brownian Motions Is Shown To Be Consistent With The Depletion Phenomena Experimentally Observed In Colloids. A Covariance Analysis Of The Random Processes And Random Fields As Well As A Review Section Of Various Approaches To Spdes Are Also Provided. An Extensive Appendix Makes The Book Accessible To Both Scientists And Graduate Students Who May Not Be Specialized In Stochastic Analysis. Probabilists, Mathematical And Theoretical Physicists As Well As Mathematical Biologists And Their Graduate Students Will Find This Book Useful. Peter Kotelenez Is A Professor Of Mathematics At Case Western Reserve University In Cleveland, Ohio. From Microscopic Dynamics To Mesoscopic Kinematics -- Heuristics: Microscopic Model And Space—time Scales -- Deterministic Dynamics In A Lattice Model And A Mesoscopic (stochastic) Limit -- Proof Of The Mesoscopic Limit Theorem -- Mesoscopic A: Stochastic Ordinary Differential Equations -- Stochastic Ordinary Differential Equations: Existence, Uniqueness, And Flows Properties -- Qualitative Behavior Of Correlated Brownian Motions -- Proof Of The Flow Property -- Comments On Sodes: A Comparison With Other Approaches -- Mesoscopic B: Stochastic Partial Differential Equations -- Stochastic Partial Differential Equations: Finite Mass And Extensions -- Stochastic Partial Differential Equations: Infinite Mass -- Stochastic Partial Differential Equations:homogeneous And Isotropic Solutions -- Proof Of Smoothness, Integrability, And Itô’s Formula -- Proof Of Uniqueness -- Comments On Other Approaches To Spdes -- Macroscopic: Deterministic Partial Differential Equations -- Partial Differential Equations As A Macroscopic Limit -- General Appendix. Peter Kotelenez. Includes Bibliographical References (p. 445-458) And Index. The present volume analyzes mathematical models of time-dependent physical p- nomena on three levels: microscopic, mesoscopic, and macroscopic. We provide a rigorous derivation of each level from the preceding level and the resulting me- scopic equations are analyzed in detail. Following Haken (1983, Sect. 1. 11. 6) we deal, “at the microscopic level, with individual atoms or molecules, described by their positions, velocities, and mutual interactions. At the mesoscopic level, we describe the liquid by means of ensembles of many atoms or molecules. The - tension of such an ensemble is assumed large compared to interatomic distances but small compared to the evolving macroscopic pattern.... At the macroscopic level we wish to study the corresponding spatial patterns. ” Typically, at the mac- scopic level, the systems under consideration are treated as spatially continuous systems such as?uids or a continuous distribution of some chemical reactants, etc. Incontrast,onthemicroscopiclevel,Newtonianmechanicsgovernstheequationsof 1 motion of the individual atoms or molecules. These equations are cast in the form 2 of systems of deterministic coupled nonlinear oscillators. The mesoscopic level is probabilistic in nature and many models may be faithfully described by stochastic 3 ordinary and stochastic partial differential equations (SODEs and SPDEs), where the latter are de?ned on a continuum. The macroscopic level is described by ti- dependent partial differential equations (PDE's) and its generalization and simpl- cations. In our mathematical framework we talk of particles instead of atoms and mo- cules. The transition from the microscopic description to a mesoscopic (i. e.