Abstract
We refine the complexity analysis of approximation problems by relating it to a new parameter calledgap location. Many of the results obtained so far for approximations yield satisfactory analysis with respect to this refined parameter, but some known results (e.g.,max-k-colorability, max 3-dimensional matching andmax not-all-equal 3sat) fall short of doing so. As a second contribution, our work fills the gap in these cases by presenting new reductions.
Next, we present definitions and hardness results of new approximation versions of some NP-complete optimization problems. The problems we treat arevertex cover (for which we define a different optimization problem from the one treated in Papadimitriou & Yannakakis 1991),k-edge coloring, andset splitting.
Similar content being viewed by others
References
M. Ajtai, Recursive Construction for 3-Regular Expanders. InProc. 28th Ann. Symp. Found. Comput. Sci., 1987, 295–304.
N. Alon, R. A. Duke, H. Lefmann, V. Rödl, and R. Yuster, The Algorithmic Aspects of the Regularity Lemma. InProc. 33th Ann. Symp. Found. Comput. Sci., 1992, 473–482.
S. Arora and S. Safra, Probabilistic Checking of Proofs: A New Characterization of NP. InProc. 33th Ann. Symp. Found. Comput. Sci., 1992, 1–13.
S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof Verification and Intractability of Approximation Problems. InProc. 33th Ann. Symp. Found. Comput. Sci., 1992, 14–23.
M. Bellare and E. Petrank, private communication, 1992.
C. Berge,Graph and Hypergraphs. North-Holland, Amsterdam, 1973.
M. Bern andP. Plassmann, The Steiner problem with edge lengths 1 and 2.Inform. Process. Lett. 32 (1989), 171–176.
A. Blum, T. Jiang, M. Li, J. Tromp, and M. Yannakakis, Linear Approximation of Shortest Superstrings. InProc. 31th Ann. Symp. Found. Comput. Sci., 1990, 554–562.
E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, and M. Yannakakis, The Complexity of Multiway Cuts. InProc. Twenty-forth Ann. ACM Symp. Theor. Comput., 1992, 241–251.
U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy, Approximating clique is almost NP-complete. InProc. 32th Ann. Symp. Found. Comput. Sci., 1991, 2–12.
O. Gabber andZ. Galil, Explicit Construction of linear sized superconcentrators.J. Comput. System Sci. 22 (1981), 407–420.
M. R. Garey andD. S. Johnson, The complexity of near-optimal graph coloring.J. Assoc. Comput. Mach. 23 (1976), 43–49.
M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
M. R. Garey, D. S. Johnson, andL. J. Stockmeyer, Some Simplified NP-Complete Graph Problems.Theoret. Comput. Sci. 1 (1976), 237–267.
J. Hastad, S. Phillips, and S. Safra, A well Characterized Approximation Problem. InProceedings of the 2nd Israel Symposium on Theory of Computing and Systems, 1993, 261–265.
I. Hoyler, The NP-Competeness of Edge Coloring.SIAM J. Comput. 10 (1981), 718–720.
V. Kann Maximum Bounded 3-Dimensional Matching is MAX SNP-Complete.Inform. Process. Lett. 37 (1991), 27–35.
R. M. Karp, Reducibility among combinatorial problems. InComplexity of Computer Computations, ed.Raymond E. Miller and James W. Thatcher, Plenum Press, 1972, 85–103.
D. Leven andZ. Galil, NP-completness of finding the chromatic index of regular graphs.J. Algorithms 4 (1983), 35–44.
L. Lovász, Coverings and Colorings of Hypergraphs. InProc. 4-th Southern Conference on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica Publishing, 1973, 3–12.
C. Lund and M. Yannakakis, On the Hardness of Approximating Minimization Problems. InProc. Twenty-fifth Ann. ACM Symp. Theor. Comput., 1993, 286–293.
G. A. Margulis, Explicit Constructions of Concentrators.Comm. ACM 9 (1973), 71–80. (English translation inProblems Inform. Transmission, 1975, 325–332.).
R. Motwani, Lecture Notes on Approximation Algorithms. Technical report, Dept. of Computer Science, Stanford University, 1992.
C. H. Papadimitriou andM. Yannakakis, Optimization, Approximation, and Complexity Classes.J. Comput. System Sci. 43 (1991), 425–440.
C. H. Papadimitriou and M. Yannakakis, The Traveling Salesman Problem with Distances One and Two.Mathematics of Operations Research (to appear).
E. Petrank, The Hardness of Approximation: Gap Location. InProceedings of the 2nd Israel Symposium on Theory of Computing and Systems, 1993, 275–284.
S. Sahni andT. Gonzalez, P-complete approximation problems.J. Assoc. Comput. Mach. 23 (1976), 555–565.
L. J. Stockmeyer, Planar 3-Colorability is NP-Complete.SIGACT News 5(3) (1973), 19–25.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Petrank, E. The hardness of approximation: Gap location. Comput Complexity 4, 133–157 (1994). https://doi.org/10.1007/BF01202286
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01202286