Abstract
Hypergraph 2-colorability, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph 2-colorability, set splitting and MAX-CUT (which is a special case of hypergraph 2-colorability) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of \( \frac{1} {2} + \in \) . This is an improvement over the ratio of \( \frac{1} {2} \) attributed to another local improvement heuristic for MAX-CUT [6]. In fact we provide a bound of \( \frac{k} {{k + 1}} \) for this problem, where k ≥ 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques [3] appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both [4] yield a bound of \( \frac{1} {2} \) for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of \( \frac{1} {2} + \in \) for this problem.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M. Garey and D. Johnson. Computers and intractability: a guide to the theory of NP-Completeness. W. H. Freeman, 1979.
M. R. Garey, D. S. Johnson, and L. J. Stockmeyer. Some simplified np-complete graph problems. Theoretical Computer Science, 1:237–267, 1976.
M. Goemans and D. P. Williamson. 0.878 approximation algorithms for MAX-CUT and max-2sat. In Proceedings of the 26th. annual ACM symposium on theory of computing, pages 422–431, 1994.
M. Goemans and D. P. Williamson. New 3/4 approximation algorithms for MAX-CUT. SIAM J. Disc. Math., 7:656–666, 1994.
D. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and Systems Science, 9:256–278, 1974.
C. Papadimitriou. Computational Complexity. Addison Wesley, 1994.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gaur, D.R., Krishnamurti, R. (1999). Simple Approximation Algorithms for MAXNAESP and Hypergraph 2-colarability . In: Algorithms and Computation. ISAAC 1999. Lecture Notes in Computer Science, vol 1741. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46632-0_6
Download citation
DOI: https://doi.org/10.1007/3-540-46632-0_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66916-6
Online ISBN: 978-3-540-46632-1
eBook Packages: Springer Book Archive