Randomized Rounding for Semidefinite Programs-Variations on the MAX CUT Example
نویسنده
چکیده
MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. Goemans and Williamson gave an algorithm that approximates MAX CUT within a ratio of 0.87856. Their algorithm first uses a semidefinite programming relaxation of MAX CUT that embeds the vertices of the graph on the surface of an n dimensional sphere, and then cuts the sphere
منابع مشابه
Csc5160: Combinatorial Optimization and Approximation Algorithms Topic: Semidefinite Programming 22.1 Semidefinite Programming Problem
In this lecture, we provide another class of relaxations, called Semidefinite Programming Relaxation. These serve as relaxations for several NP-hard problems, in particular, for problems that can be expressed as strict quadratic programs. The relaxed problems, together with techniques like randomized rounding, give good approximation algorithms to hard combinatorial problems. We will illustrate...
متن کاملRank-Two Relaxation Heuristics for MAX-CUT and Other Binary Quadratic Programs
The Goemans-Williamson randomized algorithm guarantees a high-quality approximation to the Max-Cut problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized ...
متن کاملApproximating generalizations of Max Cut
This thesis is written for the Swedish degree Licentiate of Science, teknisk licentiat. It is a university degree, intermediate between that of master and that of doctor. We study the approximability of different generalizations of the Max Cut problem. First, we show that Max Set Splitting and Max Not-All-Equal Sat are both approximable within 1.380 in probabilistic polynomial time. The algorit...
متن کاملLecture 17 ( Nov 3 , 2011 ) : Approximation via rounding SDP : Max - Cut
The next technique we learn is designing approximation algorithms using rounding semidefinite programs. This was first introduced to obtain improved approximation algorithms for the problem of Max-Cut. A trivial 1 2-approximation is to obtain a random partition of the vertices; i.e. place every vertex v ∈ V into set S with probability 1/2. We get E(weight of cut) = 1 2 e∈E w(e) ≥ 1 2 opt and th...
متن کاملA sub-constant improvement in approximating the positive semidefinite Grothendieck problem
Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we make a sub-constant improvement in the approximation ratio of one such problem. Precisely, we describe a polynomial-time algorithm for the positive semidefinit...
متن کامل