# The RPR2 rounding technique for semidefinite programs

@article{Feige2006TheRR,
title={The RPR2 rounding technique for semidefinite programs},
author={Uriel Feige and Michael Langberg},
journal={J. Algorithms},
year={2006},
volume={60},
pages={1-23}
}
• Published 8 July 2001
• Computer Science
• J. Algorithms
62 Citations

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## References

SHOWING 1-10 OF 19 REFERENCES
Constructing worst case instances for semidefinite programming based approximation algorithms
• Computer Science
SODA '01
• 2001
It is shown, for the first time, that the local analyses of the Goemans and Williamson MAX CUT algorithm, as well as its extension by Zwick, are tight for every possible relative size of the maximum cut.
Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems
Using outward rotations, an approximation algorithm is obtained for MAX CUT that performs better than the algorithm of Goemans and Williamson and an improved approximation algorithm for MAX NAE{3}-SAT is obtained.
Convex quadratic and semidefinite programming relaxations in scheduling
A provably good convex quadratic programming relaxation of strongly polynomial size is proposed for this problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs.
Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
• Computer Science
JACM
• 1995
This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.
On the optimality of the random hyperplane rounding technique for MAX CUT
• Computer Science, Mathematics
Random Struct. Algorithms
• 2002
There are graphs and optimal embeddings for which the best hyperplane approximates opt within a ratio no better than α, even in the presence of additional valid constraints, which strengthens a result of Karloff that applied only to the expected number of edges cut by a random hyperplane.
Improved approximation of Max-Cut on graphs of bounded degree
• Computer Science, Mathematics
J. Algorithms
• 2000
New 3/4-Approximation Algorithms for the Maximum Satisfiability Problem
• Computer Science, Mathematics
SIAM J. Discret. Math.
• 1994
It is shown that although standard randomized rounding does not give a good approximate result, the best solution of the two given by randomized rounding and a well-known algorithm of Johnson is always within $\frac{3}{4}$ of the optimal solution.
Improved approximation algorithms for MAXk-CUT and MAX BISECTION
• Computer Science
Algorithmica
• 2006
Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph intok blocks so as to maximize the
A 7/8-approximation algorithm for MAX 3SAT?
• Computer Science
Proceedings 38th Annual Symposium on Foundations of Computer Science
• 1997
A randomized approximation algorithm which takes an instance of MAX 3SAT as input that is optimal if the instance-a collection of clauses each of length at most three-is satisfiable, and a method of obtaining direct semidefinite relaxations of any constraint satisfaction problem of the form MAX CSP(F), where F is a finite family of Boolean functions.
Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT
• Computer Science, Mathematics
Proceedings Third Israel Symposium on the Theory of Computing and Systems
• 1995
The approach combines the Feige-Lovasz (STOC92) semidefinite programming relaxation of one-round two-prover proof systems, together with rounding techniques for the solutions of semideFinite programs, as introduced by Goemans and Williamson (SToc94).