Springer Science & Business Media, 14.03.2013 - 380 Seiten
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.
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Steiner Tree and TSP
Feedback Vertex Set
Multicut and Integer Multicommodity Flow in Trees
Multicut in General Graphs
A An Overview of Complexity Theory
B Basic Facts from Probability Theory
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achieves an approximation approximation algorithm approximation factor approximation guarantee assume bipartite graph bound on OPT capacity Chapter clauses Clearly complementary slackness conditions compute connected components Consider the following constraints corresponding cut in G cycle defined denote distance labels dual program elements endpoints Exercise extreme point solution facility factor algorithm feasible solution feedback vertex set find a minimum given graph G greedy algorithm Hence Hint instance integer program integrality gap iteration k-center least Lemma length linear program lower bound LP-relaxation maximize metric minimum weight multicommodity flow multicover multiway cut problem node nonnegative NP-complete NP-hard O(log objective function value obtain optimal solution output packing partition path picked polynomial polynomial time algorithm primal primal-dual schema probability Proof random relaxation s-t cut satisfied semidefinite programming set cover problem shortest superstring Show string subset tight example tour triangle inequality truth assignment undirected graph vertex cover problem