Approximation Algorithms and Semidefinite Programming

Bernd Gärtner, Jiri Matousek, "Approximation Algorithms and Semidefinite Programming"
S,..er | 2012 | ISBN: 3642220142 | 262 pages | PDF | 3.78 Mb

Semidefinite programs empower one of the largest classes of optimization problems that be possible to be solved with reasonable efficiency - one as well as the other in theory and practice. They make merry a key role in a multiformity of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry and quantum computing. This main division is an introduction to selected aspects of semidefinite programming and its appliance in approximation algorithms. It covers the basics if it were not that also a significant amount of late and more advanced material. There are multiplied computational problems, such as MAXCUT, as far as concerns which one cannot reasonably expect to earn an exact solution efficiently, and in in the same state case, one has to settle against approximate solutions. For MAXCUT and its relatives, exciting fresh results suggest that semidefinite programming is apparently the ultimate tool. Indeed, assuming the Unique Games Conjecture, a colorable but as yet unproven hypothesis, it was shown that towards these problems, known algorithms based in c~tinuance semidefinite programming deliver the best potential approximation ratios among all polynomial-time algorithms. This work follows the “semidefinite side” of these developments, presenting some of the main ideas behind gradual convergence algorithms based on semidefinite programming. It develops the basic theory of semidefinite programming, presents one of the known efficient algorithms in detail, and describes the principles of some others. It also includes applications, focusing forward approximation algorithms.