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}
}
View via Publisher
doi.org
65 Citations
Highly Influential Citations
11
Background Citations
13
Methods Citations
17
Results Citations
4

Figures from this paper

65 Citations

A Complex Semidefinite Programming Rounding Approximation Algorithm for the Balanced Max-3-Uncut Problem
TLDR
A complex discrete linear program is proposed for the balanced Max-3-Uncut problem using the complex semidefinite programming rounding technique and a 0.3456 -approximation algorithm is presented by further integrating a greedy swapping process after the rounding step.
Approximation Algorithms and Semidefinite Programming
TLDR
The basic theory of semidefinite programming is developed, one of the known efficient algorithms is presented in detail, and the principles of some others are described, focusing on approximation algorithms.
Approximation Algorithms for Max 3-Section Using Complex Semidefinite Programming Relaxation
TLDR
An approximation algorithm for the Max 3-section problem which divides a weighted graph into 3 parts of equal size so as to maximize the weight of the edges connecting different parts that can hit the 2/3 bound is presented.
Approximation bounds for quadratic maximization and max-cut problems with semidefinite programming relaxation
TLDR
It is shown that the actual worst-case performance ratio of the SDP relaxation approach (with the triangle inequalities added) is at least α + δd if every weight is strictly positive, whereδd > 0 is a constant depending on the problem dimension and data.
Improved Approximation Algorithms for Maximum Graph Partitioning Problems
TLDR
It is proved that a sub-optimal choice of these parameters can be obtained by the solution of a linear program which leads to the desired improvement of the approximation factors.
Rounding Two and Three Dimensional Solutions of the SDP Relaxation of MAX CUT
TLDR
An improved rounding procedure for SDP solutions that lie in ℝ3 with a performance ratio of about 0.8818 is presented, which resolves an open problem posed by Feige and Schechtman [STOC'01].
Improved approximation algorithms for the max-bisection and the disjoint 2-catalog segmentation problems
TLDR
This work considers the max-bisection problem and the disjoint 2-catalog segmentation problem, two well-known NP-hard combinatorial optimization problems and applies the semidefinite programming relaxation and the RPR2 technique to obtain a performance curve as a function of the ratio of the optimal SDP value over the total weight.
Towards Sharp Inapproximability For Any 2-CSP
  • Per Austrin
  • Computer Science
    48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
  • 2007
TLDR
It is shown how to reduce the search for a good inapproximability result to a certain numeric minimization problem, and conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly.
Conic Linear Programming
TLDR
Although CLP has long been known to be convex optimization problems, no efficient solution algorithm was known until about two decades ago, when it was discovered that interior-point algorithms for LP can be adapted to solve certain CLPs with both theoretical and practical efficiency.
The Approximability of Learning and Constraint Satisfaction Problems
  • Yi Wu
  • Computer Science, Mathematics
  • 2010
TLDR
It is NP-hard to give a better than 5/8-approximation problem for satisfiable 3-CSP, and the following two results on agnostic learning monomials and low degree polynomial threshold functions are proved.
...
...

References

SHOWING 1-10 OF 19 REFERENCES
Constructing worst case instances for semidefinite programming based approximation algorithms
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
Improved approximation algorithms for MAXk-CUT and MAX BISECTION
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?
  • H. Karloff, U. Zwick
  • Computer Science
    Proceedings 38th Annual Symposium on Foundations of Computer Science
  • 1997
TLDR
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
  • U. Feige, M. Goemans
  • Computer Science, Mathematics
    Proceedings Third Israel Symposium on the Theory of Computing and Systems
  • 1995
TLDR
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).
Randomized rounding: A technique for provably good algorithms and algorithmic proofs
TLDR
A randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0–1 problem is given and can be extended to provide bounds on the disparity between the rational and 0-1 optima for a given problem instance.
...
...