Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T06:52:12.587Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Hardness of Approximating Some NP-optimization Problems Relatedto Minimum Linear Ordering Problem

Published online by Cambridge University Press:  15 April 2002

Sounaka Mishra  and
Kripasindhu Sikdar
Sounaka Mishra
Affiliation:
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Calcutta 700 035, India; e-mail: res9513@isical.ac.in
Kripasindhu Sikdar
Affiliation:
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Calcutta 700 035, India; e-mail: sikdar@isical.ac.in
Get access

Abstract

We study hardness of approximating several minimaximal and maximinimal NP-optimizationproblems related to the minimum linear ordering problem (MINLOP). MINLOP is to find a minimum weight acyclic tournament in a given arc-weighted completedigraph. MINLOP is APX-hard but its unweighted version is polynomial timesolvable. We prove that MIN-MAX-SUBDAG problem, which is a generalization ofMINLOP and requires to find a minimum cardinality maximal acyclic subdigraphof a given digraph, is, however, APX-hard. Using results of Håstad concerning hardness of approximating independence number of a graph we then prove similar results concerning MAX-MIN-VC (respectively, MAX-MIN-FVS) which requires to find a maximum cardinality minimal vertex cover in a given graph (respectively,a maximum cardinality minimal feedback vertex set in a given digraph). We alsoprove APX-hardness of these and several related problems on various degree bounded graphs and digraphs.

Keywords

NP-optimization problemsMinimaximal and maximinimalNP-opt- imization problemsApproximation algorithmsHardness of approximationAPX-hardnessAP-reductionL-reductionS-reduction.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 3 , May 2001 , pp. 287 - 309
Copyright
© EDP Sciences, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

P. Alimonti and V. Kann, Hardness of approximating problems on cubic graphs, in Proc. 3rd Italian Conf. on Algorithms and Complexity. Springer-Verlag, Lecture Notes in Comput. Sci. 1203 (1997) 288-298.
Ausiello, G., Crescenzi, P. and Protasi, M., Fundamental Study: Approximate solution of NP optimization problems. Theoret. Comput. Sci. 150 (1995) 1-55. CrossRef
G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela and M. Protasi, Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer-Verlag, Berlin Heidelberg (1999).
V. Bafna, P. Berman and T. Fujito, Constant ratio approximations of feedback vertex sets in weighted undirected graphs, in 6th Annual International Symposium on Algorithms and Computation (1995).
Brandstädt, A., Chepoi, V.D. and Dragan, F.F., The algorithmic use of hypertree structure and maximum neighborhood orderings. Discrete Appl. Math. 82 (1998) 43-77. CrossRef
Brandstädt, A. and Kratsch, D., On domination problems for permutation and other graphs. Theoret. Comput. Sci. 54 (1987) 181-198. CrossRef
Chanas, S. and Kobylanski, P., A new heuristic algorithm solving the linear ordering problem. Comput. Optim. Appl. 6 (1996) 191-205. CrossRef
Chang, M.S., Efficient algorithms for the domination problems on interval and circular-arc graphs. SIAM J. Comput. 27 (1998) 1671-1694. CrossRef
A. Chaudhary and S. Vishwanathan, Approximation algorithms for achromatic number, in Proc. 8th Ann. ACM-SIAM Symp. on Discrete Algorithms. ACM-SIAM (1997) 558-563.
Cheston, G.A., Fricke, G., Hedetniemi, S.T. and Jacobs, D.P., On the computational complexity of upper fractional domination. Discrete Appl. Math. 27 (1990) 195-207. CrossRef
P. Crescenzi, V. Kann, R. Silvestri and L. Trevisan, Structures in approximation classes, in 1st. Annu. Int. Conf. on Computing and Combinatorics. Springer-Verlag, Lecture Notes in Comput. Sci. 959 (1995) 539-548.
U. Feige and J. Kilian, Zero knowledge and the chromatic number. Proc. Comp. Complexity (1996).
Fraber, M., Independent domination in chordal graphs. Oper. Res. Lett. 1 (1982) 134-138. CrossRef
Fraber, M. and Keil, J.M., Domination in permutation graphs. J. Algorithms 6 (1985) 309-321. CrossRef
T. Fujito, Personal communication (1999).
Grötschel, M., Jünger, M. and Reinelt, G., A cutting plane algorithm for linear ordering problem. Oper. Res. 32 (1984) 1195-1220. CrossRef
Grötschel, M., Jünger, M. and Reinelt, G., On the acyclic subgraph polytope. Math. Programming 33 (1985) 28-42. CrossRef
J. Håstad, Clique is hard to approximate within n 1-ε, in Proc. 37th IEEE Sympo. on Foundation of Comput. Sci. (1996) 627-636.
F. Harary, Graph Theory. Addition-Wesley, Reading, MA (1969).
Harary, F., Maximum versus minimum invariants for graphs. J. Graph Theory 7 (1983) 275-284.
Harary, F. and Hedetniemi, S., The achromatic number of a graph. J. Combin. Theory 8 (1970) 154-161. CrossRef
Halldórsson, M.M., Approximating the minimum maximal independence number. Inform. Process. Lett. 46 (1993) 169-172. CrossRef
Irving, R.W., On approximating the minimum independent dominating set. Inform. Process. Lett. 37 (1991) 197-200. CrossRef
V. Kann, On the Approximability of NP-complete Optimization Problems, Ph.D. Thesis. Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm (1992).
Kann, V., Polynomially bounded minimization problems that are hard to approximate. Nordic J. Comput. 1 (1994) 317-331.
S. Khanna, R. Motwani, M. Sudan and U. Vazirani, On syntactic versus computational views of approximability, in Proc. 35th Ann. IEEE Symp. on Foundations of Computer Science (1994) 819-836.
Lund, C. and Yannakakis, M., On the hardness of approximating minimization problems. J. ACM 41 (1994) 960-981. CrossRef
Papadimitriou, C.H. and Yannakakis, M., Optimization, Approximation, and Complexity Classes. J. Comput. System Sci. 43 (1991) 425-440. CrossRef
Peters, K., Laskar, R. and Hedetniemi, S.T., Maximinimal/Minimaximal connectivity in graphs. Ars Combinatoria 21 (1986) 59-70.
D.F. Manlove, Minimaximal and maximinimal optimization problems: A partial order-based approach, Ph.D. Thesis. University of Glasgow (1998).
S. Mishra and K. Sikdar, On approximation solutions of linear ordering and related NP-Optimization problems on graphs (Extended Abstract), Electronic Notes in Discrete Mathematics, Vol. 8, edited by H. Broersma, U. Faigle, J. Hurink and S. Pickl. Elsevier Science Publishers (2001), Full version submitted for publication.
Mishra, S. and Sikdar, K., On the hardness of approximating some NP-optimization problems related to minimum linear ordering problem (extended abstract), edited by J. van Leeuwen et al., IFIP TCS 2000. Lecture Notes in Comput. Sci. 1872 (2000) 186-199. CrossRef
Ueno, S., Kajtani, Y. and Gotoh, S., On the nonseparating independent set problem and feedback set problem for graphs with no vertex exceeding three. Discrete Math. 72 (1988) 355-360. CrossRef