Recognizing when heuristics can approximate minimum vertex covers is complete for parallel access to NP

Hemaspaandra, Edith; Rothe, Jörg; Spakowski, Holger

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 1, pp. 75-91.

For both the edge deletion heuristic and the maximum-degree greedy heuristic, we study the problem of recognizing those graphs for which that heuristic can approximate the size of a minimum vertex cover within a constant factor of $r$, where $r$ is a fixed rational number. Our main results are that these problems are complete for the class of problems solvable via parallel access to $\text{NP}$. To achieve these main results, we also show that the restriction of the vertex cover problem to those graphs for which either of these heuristics can find an optimal solution remains NP-hard.

[1] H. Bodlaender, D. Thilikos and K. Yamazaki, It is hard to know when greedy is good for finding independent sets. Inform. Process. Lett. 61 (1997) 101-106.

[2] V. Chvátal, A greedy heuristic for the set-covering problem. Math. Oper. Res. 4 (1979) 233-235. | Zbl

[3] T. Eiter and G. Gottlob, The complexity class ${\Theta }_2^p$: Recent results and applications, in Proc. of the 11th Conference on Fundamentals of Computation Theory. Lect. Notes Comput. Sci. 1279 (1997) 1-18.

[4] U. Feige, A threshold of $lnn$ for approximating set cover. J. ACM 45 (1998) 634-652. | Zbl

[5] M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979). | MR | Zbl

[6] R. Graham, D. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley (1989). | MR | Zbl

[7] L. Hemachandra, The strong exponential hierarchy collapses. J. Comput. Syst. Sci. 39 (1989) 299-322. | Zbl

[8] E. Hemaspaandra and L. Hemaspaandra, Computational politics: Electoral systems, in Proc. of the 25th International Symposium on Mathematical Foundations of Computer Science. Lect. Notes Comput. Sci. 1893 (2000) 64-83. | Zbl

[9] E. Hemaspaandra, L. Hemaspaandra and J. Rothe, Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP. J. ACM 44 (1997) 806-825. | Zbl

[10] E. Hemaspaandra, L. Hemaspaandra and J. Rothe, Raising NP lower bounds to parallel NP lower bounds. SIGACT News 28 (1997) 2-13.

[11] E. Hemaspaandra and J. Rothe, Recognizing when greed can approximate maximum independent sets is complete for parallel access to NP. Inform. Proc. Lett. 65 (1998) 151-156.

[12] E. Hemaspaandra, H. Spakowski and J. Vogel, The complexity of Kemeny elections. Theor. Comput. Sci. Accepted subject to minor revision. | MR | Zbl

[13] E. Hemaspaandra and G. Wechsung, The minimization problem for boolean formulas. SIAM J. Comput. 31 (2002) 1948-1958. | Zbl

[14] D. Johnson, Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9 (1974) 256-278. | Zbl

[15] J. Kadin, ${\mathrm{P}}^{\mathrm{N}\mathrm{P}[log\mathrm{n}]}$ and sparse Turing-complete sets for NP. J. Comput. Syst. Sci. 39 (1989) 282-298. | Zbl

[16] M. Krentel, The complexity of optimization problems. J. Comput. Syst. Sci. 36 (1988) 490-509. | Zbl

[17] J. Köbler, U. Schöning and K. Wagner, The difference and truth-table hierarchies for NP. RAIRO-Inf. Theor. Appl. 21 (1987) 419-435. | Numdam | Zbl

[18] L. Lovász, On the ratio of optimal integral and fractional covers. Discrete Math. 13 (1975) 383-390. | Zbl

[19] C. Lund and M. Yannakakis, On the hardness of approximating minimization problems. J. ACM 41 (1994) 960-981. | Zbl

[20] C. Papadimitriou, Computational Complexity. Addison-Wesley (1994). | MR | Zbl

[21] C. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall (1982). | MR | Zbl

[22] C. Papadimitriou and S. Zachos, Two remarks on the power of counting, in Proc. of the 6th GI Conference on Theoretical Computer Science. Lect. Notes Comput. Sci. 145 (1983) 269-276. | Zbl

[23] T. Riege and J. Rothe, Complexity of the exact domatic number problem and of the exact conveyor flow shop problem. Theory of Computing Systems (2004). On-line publication DOI 10.1007/s00224-004-1209-8. Paper publication to appear. | MR | Zbl

[24] J. Rothe, H. Spakowski and J. Vogel, Exact complexity of the winner problem for Young elections. Theory Comput. Syst. 36 (2003) 375-386. | Zbl

[25] H. Spakowski and J. Vogel, ${\Theta }_2^p$-completeness: A classical approach for new results, in Proc. of the 20th Conference on Foundations of Software Technology and Theoretical Computer Science. Lect. Notes Comput. Sci. 1974 (2000) 348-360. | Zbl

[26] K. Wagner, More complicated questions about maxima and minima, and some closures of NP. Theor. Comput. Sci. 51 (1987) 53-80. | Zbl

[27] K. Wagner, Bounded query classes. SIAM J. Comput. 19 (1990) 833-846. | Zbl