基本説明
"..covers the dominant theoretical approaches to the approximate solution of hard combinatorial optimization and enumeration problems..."- Richard Karp, University of California at Berkeley.
Full Description
Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872-1970) Most natural optimization problems, including those arising in important application areas, are NP-hard. Therefore, under the widely believed con jecture that P -=/= NP, their exact solution is prohibitively time consuming. Charting the landscape of approximability of these problems, via polynomial time algorithms, therefore becomes a compelling subject of scientific inquiry in computer science and mathematics. This book presents the theory of ap proximation algorithms as it stands today. It is reasonable to expect the picture to change with time. This book is divided into three parts. In Part I we cover combinato rial algorithms for a number of important problems, using a wide variety of algorithm design techniques. The latter may give Part I a non-cohesive appearance. However, this is to be expected - nature is very rich, and we cannot expect a few tricks to help solve the diverse collection of NP-hard problems. Indeed, in this part, we have purposely refrained from tightly cat egorizing algorithmic techniques so as not to trivialize matters. Instead, we have attempted to capture, as accurately as possible, the individual character of each problem, and point out connections between problems and algorithms for solving them.
Contents
1 Introduction.- I. Combinatorial Algorithms.- 2 Set Cover.- 3 Steiner Tree and TSP.- 4 Multiway Cut and k-Cut.- 5 k-Center.- 6 Feedback Vertex Set.- 7 Shortest Superstring.- 8 Knapsack.- 9 Bin Packing.- 10 Minimum Makespan Scheduling.- 11 Euclidean TSP.- II. LP-Based Algorithms.- 12 Introduction to LP-Duality.- 13 Set Cover via Dual Fitting.- 14 Rounding Applied to Set Cover.- 15 Set Cover via the Primal—Dual Schema.- 16 Maximum Satisfiability.- 17 Scheduling on Unrelated Parallel Machines.- 18 Multicut and Integer Multicommodity Flow in Trees.- 19 Multiway Cut.- 20 Multicut in General Graphs.- 21 Sparsest Cut.- 22 Steiner Forest.- 23 Steiner Network.- 24 Facility Location.- 25 k-Median.- 26 Semidefinite Programming.- III. Other Topics.- 27 Shortest Vector.- 28 Counting Problems.- 29 Hardness of Approximation.- 30 Open Problems.- A An Overview of Complexity Theory for the Algorithm Designer.- A.3.1 Approximation factor preserving reductions.- A.4 Randomized complexity classes.- A.5 Self-reducibility.- A.6 Notes.- B Basic Facts from Probability Theory.- B.1 Expectation and moments.- B.2 Deviations from the mean.- B.3 Basic distributions.- B.4 Notes.- References.- Problem Index.
