# Approximation Algorithms for MAX 4-SAT and Rounding Procedures for Semidefinite Programs

@inproceedings{Halperin1999ApproximationAF,
title={Approximation Algorithms for MAX 4-SAT and Rounding Procedures for Semidefinite Programs},
author={Eran Halperin and Uri Zwick},
booktitle={J. Algorithms},
year={1999}
}
• Published in J. Algorithms 9 June 1999
• Computer Science
Karloff and Zwick obtained recently an optimal 7/8-approximation algorithm for MAX 3-SAT. In an attempt to see whether similar methods can be used to obtain a 7/8-approximation algorithm for MAX SAT, we consider the most natural generalization of MAX 3-SAT, namely MAX 4-SAT. We present a semidefinite programming relaxation of MAX 4-SAT and a new family of rounding procedures that try to cope well with clauses of various sizes. We study the potential, and the limitations, of the relaxation and…
55 Citations

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## References

SHOWING 1-10 OF 46 REFERENCES

### 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.

### 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 MAX SAT

• Computer Science
SODA '00
• 2000
This paper considers approximation algorithms for MAX SAT proposed by Goemans and Williamson and presents a sharpened analysis of their performance guarantees, and shows that these algorithms, combined with recent approximation algorithm for MAX 2SAT, MAX 3S AT, and MAX SAT due to Feige and Goeman, Karloff and Zwick, andZwick, respectively, lead to an improved approximation algorithm.

### 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.

### Approximation Algorithms for the Maximum Satisfiability Problem

• Computer Science
Nord. J. Comput.
• 1996
This paper presents approximation algorithms for MAX SAT, including a 0.76544-approximation algorithm, based on semidefinite programming and the 0.75-Approximation algorithms of Yannakakis and Goemans-Williamson.

### Gadgets, approximation, and linear programming

• Computer Science
Proceedings of 37th Conference on Foundations of Computer Science
• 1996
The authors present a linear-programming based method for finding "gadgets", i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in

### Free bits, PCPs and non-approximability-towards tight results

• Computer Science, Mathematics
Proceedings of IEEE 36th Annual Foundations of Computer Science
• 1995
A proof system for NP is presented using logarithmic randomness and two amortized free bits, so that Max clique is hard within N/sup 1/3/ and chromatic number within N-Sup 1/5/, and a comprehensive study of PCP and FPCP parameters is initiated.

### 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).

### Approximation algorithms for MAX SAT: Yannakakis vs. Goemans-Williamson

• Takao Asano
• Computer Science
Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems
• 1997
This paper considers approximation algorithms for MAX SAT proposed by Yannnkakis and Goemans-Williamson and presents an approximation algorithm which is an improvement of Yannakakis' algorithm.

### Efficient probabilistically checkable proofs and applications to approximations

• Computer Science
STOC '93
• 1993
This work constructs multi-prover proof systems for NP which use only a constant number of provers to simultaneously achieve low error, low randomness and low answer size, and shows that approximating minimum set cover within any constant is NP-complete.