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Volumn 45, Issue 3, 1998, Pages 501-555

Proof Verification and the Hardness of Approximation Problems

Author keywords

Algorithms; NP completeness; Optimization; Proof verification; Randomness; Theory

Indexed keywords

ALGORITHMS; OPTIMIZATION; RANDOM PROCESSES;

EID: 0032058198     PISSN: 00045411     EISSN: None     Source Type: Journal    
DOI: 10.1145/278298.278306     Document Type: Article
Times cited : (1260)

References (106)
  • 2
  • 3
    • 0031119485 scopus 로고    scopus 로고
    • The hardness of approximate optima in lattices, codes, and systems of linear equations
    • ARORA, S., BABAI, L., STERN, J., AND SWEEDYK, Z. 1997. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54, 2 (Apr.), 317-331.
    • (1997) J. Comput. Syst. Sci. , vol.54 , Issue.2 APR. , pp. 317-331
    • Arora, S.1    Babai, L.2    Stern, J.3    Sweedyk, Z.4
  • 4
    • 0002019995 scopus 로고    scopus 로고
    • Hardness of approximations
    • D. Hochbaum, ed. PWS Publishing, Boston, Mass.
    • ARORA, S., AND LUND, C. 1996. Hardness of approximations. In Approximation Algorithms for NP-Hard Problems, D. Hochbaum, ed. PWS Publishing, Boston, Mass., pp. 339-446.
    • (1996) Approximation Algorithms for NP-Hard Problems , pp. 339-446
    • Arora, S.1    Lund, C.2
  • 6
    • 0031651077 scopus 로고    scopus 로고
    • Probabilistic checking of proofs: A new characterization of NP
    • ARORA, S., AND SAFRA, S. 1998. Probabilistic checking of proofs: A new characterization of NP. J. ACM 45, 1 (Jan.), 70-122.
    • (1998) J. ACM , vol.45 , Issue.1 JAN. , pp. 70-122
    • Arora, S.1    Safra, S.2
  • 8
    • 84995344698 scopus 로고
    • Structure preserving reductions among convex optimization problems
    • AUSIELLO, G., D'ATRI, A., AND PROTASI, M. 1980a. Structure preserving reductions among convex optimization problems. J. Comput. Syst. Sci. 21, 136-153.
    • (1980) J. Comput. Syst. Sci. , vol.21 , pp. 136-153
    • Ausiello, G.1    D'Atri, A.2    Protasi, M.3
  • 9
    • 0002641025 scopus 로고
    • Towards a unified approach for the classification of NP-complete optimization problems
    • AUSIELLO, G., MARCHETTI-SPACCAMELA, A., AND PROTASI, M. 1980b. Towards a unified approach for the classification of NP-complete optimization problems. Theoret. Comput. Sci. 12, 83-96.
    • (1980) Theoret. Comput. Sci. , vol.12 , pp. 83-96
    • Ausiello, G.1    Marchetti-Spaccamela, A.2    Protasi, M.3
  • 11
    • 21144464095 scopus 로고
    • Transparent (holographic) proofs
    • Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computer Science. Springer-Verlag, New York
    • BABAI, L. 1993. Transparent (holographic) proofs. In Proceedings of the 10th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 665. Springer-Verlag, New York, pp. 525-534.
    • (1993) Lecture Notes in Computer Science , vol.665 , pp. 525-534
    • Babai, L.1
  • 12
    • 0002234652 scopus 로고
    • Arithmetization: A new methods in structural complexity theory
    • BABAI, L., AND FORTNOW, L. 1991. Arithmetization: a new methods in structural complexity theory. Computat. Complex. 1, 41-66.
    • (1991) Computat. Complex. , vol.1 , pp. 41-66
    • Babai, L.1    Fortnow, L.2
  • 14
    • 0002771903 scopus 로고
    • Non-deterministic exponential time has two-prover interactive protocols
    • BABAI, L., FORTNOW, L., AND LUND, C. 1991b. Non-deterministic exponential time has two-prover interactive protocols. Comptat. Complex. 1, 3-40.
    • (1991) Comptat. Complex. , vol.1 , pp. 3-40
    • Babai, L.1    Fortnow, L.2    Lund, C.3
  • 16
    • 0023995534 scopus 로고
    • Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes
    • BABAI, L., AND MORAN, S. 1988. Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. J. Comput. Syst. Sci. 36, 254-276.
    • (1988) J. Comput. Syst. Sci. , vol.36 , pp. 254-276
    • Babai, L.1    Moran, S.2
  • 17
    • 33745511846 scopus 로고
    • Hiding instances in multioracle queries
    • Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science. Springer-Verlag, New York
    • BEAVER, D., AND FEIGENBAUM, J. 1990. Hiding instances in multioracle queries. In Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 415. Springer-Verlag, New York.
    • (1990) Lecture Notes in Computer Science , vol.415
    • Beaver, D.1    Feigenbaum, J.2
  • 18
    • 0013117132 scopus 로고
    • Interactive proofs and approximation: Reductions trom two provers in one round
    • IEEE Computer Society Press, Los Alamitos, Calif.
    • BELLARE, M. 1993. Interactive proofs and approximation: Reductions trom two provers in one round. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems. IEEE Computer Society Press, Los Alamitos, Calif., pp. 266-274.
    • (1993) Proceedings of the 2nd Israel Symposium on Theory and Computing Systems , pp. 266-274
    • Bellare, M.1
  • 20
    • 0029519801 scopus 로고    scopus 로고
    • Free bits, PCPs and non-approximability - Towards tight results
    • to appear
    • BELLARE, M., GOLDREICH, O., AND SUDAN, M. 1995/1998. Free bits, PCPs and non-approximability - towards tight results. SIAM J. Comput., to appear. (Preliminary version in Proceedings of the 36th Annual Symposium on the Foundations of Computer Science. IEEE, New York, 1995, pp. 422-431. Full version available as TR95-024 of ECCC, the Electronic Colloquium on Computational Complexity, http://www.eccc.uni-trier.de/eccc/.)
    • (1995) SIAM J. Comput.
    • Bellare, M.1    Goldreich, O.2    Sudan, M.3
  • 21
    • 0029519801 scopus 로고    scopus 로고
    • IEEE, New York
    • BELLARE, M., GOLDREICH, O., AND SUDAN, M. 1995/1998. Free bits, PCPs and non-approximability - towards tight results. SIAM J. Comput., to appear. (Preliminary version in Proceedings of the 36th Annual Symposium on the Foundations of Computer Science. IEEE, New York, 1995, pp. 422-431. Full version available as TR95-024 of ECCC, the Electronic Colloquium on Computational Complexity, http://www.eccc.uni-trier.de/eccc/.)
    • (1995) Proceedings of the 36th Annual Symposium on the Foundations of Computer Science , pp. 422-431
  • 22
    • 0029519801 scopus 로고    scopus 로고
    • TR95-024 of ECCC
    • BELLARE, M., GOLDREICH, O., AND SUDAN, M. 1995/1998. Free bits, PCPs and non-approximability - towards tight results. SIAM J. Comput., to appear. (Preliminary version in Proceedings of the 36th Annual Symposium on the Foundations of Computer Science. IEEE, New York, 1995, pp. 422-431. Full version available as TR95-024 of ECCC, the Electronic Colloquium on Computational Complexity, http://www.eccc.uni-trier.de/eccc/.)
    • The Electronic Colloquium on Computational Complexity
  • 23
    • 0027308413 scopus 로고
    • Efficient probabilistically checkable proofs and applications to approximation
    • San Diego, Calif., May 16-18. ACM, New York, 1993
    • BELLARE, M., GOLDWASSER, S., LUND, C., AND RUSSELL, A. 1993/1994. Efficient probabilistically checkable proofs and applications to approximation. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, 1993, pp. 294-304. (See also errata sheet in Proceedings of the 26th Annual ACM Symposium on the Theory of Computing. ACM, New York, 1994, p. 820.)
    • (1993) Proceedings of the 25th Annual ACM Symposium on the Theory of Computing , pp. 294-304
    • Bellare, M.1    Goldwasser, S.2    Lund, C.3    Russell, A.4
  • 24
    • 0027308413 scopus 로고
    • ACM, New York
    • BELLARE, M., GOLDWASSER, S., LUND, C., AND RUSSELL, A. 1993/1994. Efficient probabilistically checkable proofs and applications to approximation. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, 1993, pp. 294-304. (See also errata sheet in Proceedings of the 26th Annual ACM Symposium on the Theory of Computing. ACM, New York, 1994, p. 820.)
    • (1994) Proceedings of the 26th Annual ACM Symposium on the Theory of Computing , pp. 820
  • 25
    • 0000011818 scopus 로고
    • The complexity of approximating a nonlinear program
    • BELLARE, M., AND ROGAWAY, P. 1993/1995. The complexity of approximating a nonlinear program. J. Math. Prog. B 69, 3 (Sept. 1995), 429-441. Also in Complexity of Numerical Optimization, P. M. Pardalos, ed. World Scientific, 1993.
    • (1993) J. Math. Prog. B , vol.69 , Issue.3 SEPT. 1995 , pp. 429-441
    • Bellare, M.1    Rogaway, P.2
  • 26
    • 58149362273 scopus 로고
    • World Scientific
    • BELLARE, M., AND ROGAWAY, P. 1993/1995. The complexity of approximating a nonlinear program. J. Math. Prog. B 69, 3 (Sept. 1995), 429-441. Also in Complexity of Numerical Optimization, P. M. Pardalos, ed. World Scientific, 1993.
    • (1993) Complexity of Numerical Optimization
    • Pardalos, P.M.1
  • 29
    • 0026624588 scopus 로고
    • On the complexity of approximating the independent set problem
    • BERMAN, P., AND SCHNITGER, G. 1992. On the complexity of approximating the independent set problem. Inf. Comput. 96, 77-94.
    • (1992) Inf. Comput. , vol.96 , pp. 77-94
    • Berman, P.1    Schnitger, G.2
  • 30
    • 0024735782 scopus 로고
    • The Steiner problem with edge lengths 1 and 2
    • BERN, M., AND PLASSMANN, P. 1989. The Steiner problem with edge lengths 1 and 2. Inf. Proc. Lett. 32, 171-176.
    • (1989) Inf. Proc. Lett. , vol.32 , pp. 171-176
    • Bern, M.1    Plassmann, P.2
  • 31
    • 1842712522 scopus 로고
    • Program checking
    • Proceedings of FST&TCS. Springer-Verlag, New York
    • BLUM, M. 1991. Program checking. In Proceedings of FST&TCS. Lecture Notes in Computer Science, vol. 560. Springer-Verlag, New York, pp. 1-9.
    • (1991) Lecture Notes in Computer Science , vol.560 , pp. 1-9
    • Blum, M.1
  • 32
    • 0028462698 scopus 로고
    • Linear approximation of shortest superstrings
    • BLUM, A., JIANG, T., LI, M., TROMP, J., AND YANNAKAKIS, M. 1994. Linear approximation of shortest superstrings. J. ACM 41, 4 (July), 630-647.
    • (1994) J. ACM , vol.41 , Issue.4 JULY , pp. 630-647
    • Blum, A.1    Jiang, T.2    Li, M.3    Tromp, J.4    Yannakakis, M.5
  • 34
    • 0000619172 scopus 로고
    • Self-testing/correcting with applications to numerical problems
    • BLUM, M., LUBY, M., AND RUBINFELD, R. 1993. Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci. 47, 3 (Dec.), 549-595.
    • (1993) J. Comput. Syst. Sci. , vol.47 , Issue.3 DEC. , pp. 549-595
    • Blum, M.1    Luby, M.2    Rubinfeld, R.3
  • 35
    • 0028370954 scopus 로고
    • PSPACE is provable by two provers in one round
    • CAI, J., CONDON, A., AND LIPTON, R. 1994. PSPACE is provable by two provers in one round. J. Comput. Syst. Sci. 48, 1 (Feb.), 183-193.
    • (1994) J. Comput. Syst. Sci. , vol.48 , Issue.1 FEB. , pp. 183-193
    • Cai, J.1    Condon, A.2    Lipton, R.3
  • 37
    • 1542686050 scopus 로고
    • The complexity of the max word problem, or the power of one-way interactive proof systems
    • Proceedings of the 8th Annual Symposium on Theoretical Aspects of Computer Science. Springer-Verlag, New York
    • CONDON, A. 1991. The complexity of the max word problem, or the power of one-way interactive proof systems. In Proceedings of the 8th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 480. Springer-Verlag, New York.
    • (1991) Lecture Notes in Computer Science , vol.480
    • Condon, A.1
  • 38
    • 0027206131 scopus 로고
    • Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions
    • San Diego, Calif., May 16-18. ACM, New York
    • CONDON, A., FEIGENBAUM, J., LUND, C., AND SHOR, P. 1993. Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, pp. 305-314.
    • (1993) Proceedings of the 25th Annual ACM Symposium on the Theory of Computing , pp. 305-314
    • Condon, A.1    Feigenbaum, J.2    Lund, C.3    Shor, P.4
  • 39
    • 0000575605 scopus 로고    scopus 로고
    • Random debaters and the hardness of approximating stochastic functions
    • CONDON, A., FEIGENBAUM, J., LUND, C., AND SHOR, P. 1996. Random debaters and the hardness of approximating stochastic functions. SIAM J. Comput. 26, 2 (Apr.), 369-400.
    • (1996) SIAM J. Comput. , vol.26 , Issue.2 APR. , pp. 369-400
    • Condon, A.1    Feigenbaum, J.2    Lund, C.3    Shor, P.4
  • 40
    • 85059770931 scopus 로고
    • The complexity of theorem-proving procedures
    • Shaker Heights, Ohio, May 3-5. ACM, New York
    • COOK, S. A. 1971. The complexity of theorem-proving procedures. In Proceedings of the 3rd Annual ACM Symposium on Theory of Computing (Shaker Heights, Ohio, May 3-5). ACM, New York, pp. 151-158.
    • (1971) Proceedings of the 3rd Annual ACM Symposium on Theory of Computing , pp. 151-158
    • Cook, S.A.1
  • 41
    • 0004148969 scopus 로고
    • Tech. Rep. SI/RR-95/02, Dipartimento di Scienze dell'Informazione, Uinversita di Roma "La Sapienza". The list is updated continuously. (The latest version is available by anonymous ftp from nada.kth.se. as Theory/Viggo-Kann/compendium.ps.Z.)
    • CRESCENZI, P., AND KANN, V. 1995. A compendium of NP optimization problems. Tech. Rep. SI/RR-95/02, Dipartimento di Scienze dell'Informazione, Uinversita di Roma "La Sapienza". The list is updated continuously. (The latest version is available by anonymous ftp from nada.kth.se. as Theory/Viggo-Kann/compendium.ps.Z.)
    • (1995) A Compendium of NP Optimization Problems
    • Crescenzi, P.1    Kann, V.2
  • 43
    • 51249181640 scopus 로고
    • Bin packing can be solved within 1 + ∈ in linear time
    • DE LA VEGA, W., AND LUEKER, G. 1981. Bin packing can be solved within 1 + ∈ in linear time. Combinatorica 1, 349-355.
    • (1981) Combinatorica , vol.1 , pp. 349-355
    • De La Vega, W.1    Lueker, G.2
  • 44
    • 0002188798 scopus 로고
    • Generalized first-order spectra and polynomial-time recognizable sets
    • Richard Karp, ed. American Mathematical Society, Providence, R.I.
    • FAGIN, R. 1974. Generalized first-order spectra and polynomial-time recognizable sets. In Complexity of Computation, Richard Karp, ed. American Mathematical Society, Providence, R.I.
    • (1974) Complexity of Computation
    • Fagin, R.1
  • 46
    • 0030100766 scopus 로고    scopus 로고
    • Interactive proofs and the hardness of approximating cliques
    • FEIGE, U., GOLDWASSER, S., LOVASZ, L., SAFRA, S., AND SZEGEDY, M. 1996. Interactive proofs and the hardness of approximating cliques. J. ACM 43, 2 (Mar.), 268-292.
    • (1996) J. ACM , vol.43 , Issue.2 MAR. , pp. 268-292
    • Feige, U.1    Goldwasser, S.2    Lovasz, L.3    Safra, S.4    Szegedy, M.5
  • 49
    • 0026991175 scopus 로고
    • Two-prover one-round proof systems: Their power and their problems
    • Victoria, B.C., Canada, May 4-6. ACM, New York
    • FEIGE, U., AND LOVÁSZ, L. 1992. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th Annual Symposium on the Theory of Computing (Victoria, B.C., Canada, May 4-6). ACM, New York, pp. 733-744.
    • (1992) Proceedings of the 24th Annual Symposium on the Theory of Computing , pp. 733-744
    • Feige, U.1    Lovász, L.2
  • 50
    • 0028544648 scopus 로고
    • On the power of multi-prover interactive protocols
    • FORTNOW, L., ROMPEL, J., AND SIPSER, M. 1994. On the power of multi-prover interactive protocols. Theoret. Comput. Sci. 134, 2 (Nov.), 545-557.
    • (1994) Theoret. Comput. Sci. , vol.134 , Issue.2 NOV. , pp. 545-557
    • Fortnow, L.1    Rompel, J.2    Sipser, M.3
  • 51
    • 85034453168 scopus 로고
    • Fast probabilistic algorithms
    • Proceedings of Symposium on Mathematical Foundations of Computer Science. Springer-Verlag, New York
    • FRIEVALDS, R. 1979. Fast probabilistic algorithms. In Proceedings of Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 74. Springer-Verlag, New York, pp. 57-69.
    • (1979) Lecture Notes in Computer Science , vol.74 , pp. 57-69
    • Frievalds, R.1
  • 55
    • 0016871461 scopus 로고
    • The complexity of near-optimal graph coloring
    • GAREY, M., AND JOHNSON, D. 1976. The complexity of near-optimal graph coloring. J. ACM 23, 1 (Jan.), 43-49.
    • (1976) J. ACM , vol.23 , Issue.1 JAN. , pp. 43-49
    • Garey, M.1    Johnson, D.2
  • 56
    • 84976822151 scopus 로고
    • "Strong" NP-completeness results: Motivation, examples and implications
    • GAREY, M., AND JOHNSON, D. 1978. "Strong" NP-completeness results: Motivation, examples and implications. J. ACM 25, 499-508.
    • (1978) J. ACM , vol.25 , pp. 499-508
    • Garey, M.1    Johnson, D.2
  • 59
    • 0000377057 scopus 로고
    • Highly resilient correctors for polynomials
    • GEMMELL, P., AND SUDAN, M. 1992. Highly resilient correctors for polynomials. Inf. Proc. Lett. 43, 4 (Sept.), 169-174.
    • (1992) Inf. Proc. Lett. , vol.43 , Issue.4 SEPT. , pp. 169-174
    • Gemmell, P.1    Sudan, M.2
  • 61
    • 84893574327 scopus 로고
    • Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
    • GOEMANS, M., AND WILLIAMSON, D. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 6 (Nov.), 1115-1145.
    • (1995) J. ACM , vol.42 , Issue.6 NOV. , pp. 1115-1145
    • Goemans, M.1    Williamson, D.2
  • 62
    • 0343851477 scopus 로고    scopus 로고
    • A taxonomy of proof systems
    • L. A. Hemaspaandra and A. Selman, eds. Springer-Verlag, New York
    • GOLDREICH, O. 1997. A taxonomy of proof systems. In Complexity Theory Retrospective II, L. A. Hemaspaandra and A. Selman, eds. Springer-Verlag, New York.
    • (1997) Complexity Theory Retrospective II
    • Goldreich, O.1
  • 63
    • 0024611659 scopus 로고
    • The knowledge complexity of interactive proof-systems
    • GOLDWASSER, S., MICALI, S., AND RACKOFF, C. 1989. The knowledge complexity of interactive proof-systems. SIAM J. Comput. 18, 1 (Feb.), 186-208.
    • (1989) SIAM J. Comput. , vol.18 , Issue.1 FEB. , pp. 186-208
    • Goldwasser, S.1    Micali, S.2    Rackoff, C.3
  • 64
    • 84944813080 scopus 로고
    • Bounds for certain multiprocessing anomalies
    • GRAHAM, R. 1966. Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563-1581.
    • (1966) Bell Syst. Tech. J. , vol.45 , pp. 1563-1581
    • Graham, R.1
  • 68
    • 0016560084 scopus 로고
    • Fast approximation algorithms for the knapsack and sum of subset problems
    • IBARRA, O., AND KIM, C. 1975. Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM 22, 463-468.
    • (1975) J. ACM , vol.22 , pp. 463-468
    • Ibarra, O.1    Kim, C.2
  • 70
    • 0016349356 scopus 로고
    • Approximation algorithms for combinatorial problems
    • JOHNSON, D. 1974. Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256-278.
    • (1974) J. Comput. Syst. Sci. , vol.9 , pp. 256-278
    • Johnson, D.1
  • 71
    • 0025855578 scopus 로고
    • Maximum bounded 3-dimensional matching is MAX SNP-complete
    • KANN, V. 1991. Maximum bounded 3-dimensional matching is MAX SNP-complete. Inf. Proc. Lett. 37, 27-35.
    • (1991) Inf. Proc. Lett. , vol.37 , pp. 27-35
    • Kann, V.1
  • 72
    • 0001867234 scopus 로고    scopus 로고
    • On approximating the longest path in a graph
    • KARGER, D., MOTWANI, R., AND RAMKUMAR, G. 1997. On approximating the longest path in a graph. Algorithmica 18, 1 (May), 82-98.
    • (1997) Algorithmica , vol.18 , Issue.1 MAY , pp. 82-98
    • Karger, D.1    Motwani, R.2    Ramkumar, G.3
  • 74
    • 0003037529 scopus 로고
    • Reducibility among combinatorial problems
    • Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, ed. Plenum Press, New York
    • KARP, R. 1972. Reducibility among combinatorial problems. In Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, ed. Advances in Computing Research. Plenum Press, New York, pp. 85-103.
    • (1972) Advances in Computing Research , pp. 85-103
    • Karp, R.1
  • 77
    • 1842662523 scopus 로고    scopus 로고
    • to appear
    • KHANNA, S., MOTWANI, R., SUDAN, M., AND VAZIRANI, U. 1994/1998. On syntactic versus computational views of approximability. In Proceedings of the 35th Annual Symposium on the Foundations of Computer Science. IEEE, New York, 1994, pp. 819-830. (Also, SIAM J. Comput., to appear.)
    • SIAM J. Comput.
  • 78
    • 0023569164 scopus 로고
    • The decision problem for the probabilities of higher-order properties
    • New York, N.Y., May 25-27. ACM, New York
    • KOLAITIS, P., AND VARDI, M. 1987. The decision problem for the probabilities of higher-order properties. In Proceedings of the 19th Annual Symposium on the Theory of Computing (New York, N.Y., May 25-27). ACM, New York, pp. 425-435.
    • (1987) Proceedings of the 19th Annual Symposium on the Theory of Computing , pp. 425-435
    • Kolaitis, P.1    Vardi, M.2
  • 79
    • 0031122054 scopus 로고    scopus 로고
    • Fully parallelized multi-prover protocols for NEXP-tie
    • LAPIDOT, D., AND SHAMIR, A. 1997. Fully parallelized multi-prover protocols for NEXP-tie. J. Comput. Syst. Sci. 54, 2 (Apr.), 215-220.
    • (1997) J. Comput. Syst. Sci. , vol.54 , Issue.2 APR. , pp. 215-220
    • Lapidot, D.1    Shamir, A.2
  • 80
    • 0000202647 scopus 로고
    • Universal'nyie perebornyie zadachi
    • Universal Search Problems: in Russian. A corrected English translation appears in an appendix to Trakhtenbrot
    • LEVIN, L. 1973. Universal'nyie perebornyie zadachi (Universal Search Problems: in Russian). Problemy Peredachi Informatsii 9, 3, 265-266. (A corrected English translation appears in an appendix to Trakhtenbrot [1984].)
    • (1973) Problemy Peredachi Informatsii , vol.9 , Issue.3 , pp. 265-266
    • Levin, L.1
  • 82
    • 0026930922 scopus 로고
    • Algebraic methods for interactive proof systems
    • LUND, C., FORTNOW, L., KARLOFF, H., AND NISAN, N. 1992. Algebraic methods for interactive proof systems. J. ACM 39, 859-868.
    • (1992) J. ACM , vol.39 , pp. 859-868
    • Lund, C.1    Fortnow, L.2    Karloff, H.3    Nisan, N.4
  • 83
    • 85028836239 scopus 로고
    • The approximation of maximum subgraph problems
    • Proceedings of ICALP 93, Springer-Verlag, New York
    • LUND, C., AND YANNAKAKIS, M. 1993. The approximation of maximum subgraph problems. In Proceedings of ICALP 93, Lecture Notes in Computer Science, vol. 700. Springer-Verlag, New York.
    • (1993) Lecture Notes in Computer Science , vol.700
    • Lund, C.1    Yannakakis, M.2
  • 84
    • 0028514351 scopus 로고
    • On the hardness of approximating minimization problems
    • LUND, C., AND YANNAKAKIS, M. 1994. On the hardness of approximating minimization problems. J. ACM 41, 5 (Sept.), 960-981.
    • (1994) J. ACM , vol.41 , Issue.5 SEPT. , pp. 960-981
    • Lund, C.1    Yannakakis, M.2
  • 85
    • 1542480559 scopus 로고    scopus 로고
    • Guillotine subdivisions approximate polygonal subdivision: Part II - A simple PTAS for geometric k-MST, TSP, and related problems
    • to appear
    • MITCHELL, J. 1998. Guillotine subdivisions approximate polygonal subdivision: Part II - A simple PTAS for geometric k-MST, TSP, and related problems. SIAM J. Comput., to appear.
    • (1998) SIAM J. Comput.
    • Mitchell, J.1
  • 86
    • 1842712516 scopus 로고
    • Tech. Rep. Department of Computer Science, Stanford Univ., Stanford, Calif.
    • Springer-Verlag, New York
    • MOTWANI, R. 1992. Tech. Rep. Department of Computer Science, Stanford Univ., Stanford, Calif. In Lecture Notes on Approximation Algorithms, Springer-Verlag, New York.
    • (1992) Lecture Notes on Approximation Algorithms
    • Motwani, R.1
  • 87
    • 0026366408 scopus 로고
    • Optimization, approximation and complexity classes
    • PAPADIMITRIOU, C., AND YANNAKAKIS, M. 1991. Optimization, approximation and complexity classes. J. Comput. Syst. Sci. 43, 425-440.
    • (1991) J. Comput. Syst. Sci. , vol.43 , pp. 425-440
    • Papadimitriou, C.1    Yannakakis, M.2
  • 88
    • 0001805669 scopus 로고
    • The traveling salesman problem with distances one and two
    • PAPADIMITRIOU, C., AND YANNAKAKIS, M. 1993. The traveling salesman problem with distances one and two. Math. Oper. Res. 18, 1-11.
    • (1993) Math. Oper. Res. , vol.18 , pp. 1-11
    • Papadimitriou, C.1    Yannakakis, M.2
  • 89
    • 0006494976 scopus 로고
    • Non-deterministic polynomial optimization problems and their approximation
    • PAZ, A., AND MORAN, S. 1981. Non-deterministic polynomial optimization problems and their approximation. Theoret. Comput. Sci. 15, 251-277.
    • (1981) Theoret. Comput. Sci. , vol.15 , pp. 251-277
    • Paz, A.1    Moran, S.2
  • 93
    • 0030714232 scopus 로고    scopus 로고
    • A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP
    • El Paso, Tex., May 4-6. ACM, New York
    • RAZ, R., AND SAFRA, S. 1997. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (El Paso, Tex., May 4-6). ACM, New York, pp. 475-484.
    • (1997) Proceedings of the 28th Annual ACM Symposium on the Theory of Computing , pp. 475-484
    • Raz, R.1    Safra, S.2
  • 95
    • 0030128622 scopus 로고    scopus 로고
    • Robust characterizations of polynomials with applications to program testing
    • RUBINFELD, R., AND SUDAN, M. 1996. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25, 2 (Apr.), 252-271.
    • (1996) SIAM J. Comput. , vol.25 , Issue.2 APR. , pp. 252-271
    • Rubinfeld, R.1    Sudan, M.2
  • 96
    • 0016437461 scopus 로고
    • Approximate algorithms for the 0/1 knapsack problem
    • SAHNI, S. 1975. Approximate algorithms for the 0/1 knapsack problem. J. ACM 22, 115-124.
    • (1975) J. ACM , vol.22 , pp. 115-124
    • Sahni, S.1
  • 97
    • 0016973706 scopus 로고
    • P-complete approximation problems
    • SAHNI, S., AND GONZALEZ, T. 1976. P-complete approximation problems. J. ACM 23, 555-565.
    • (1976) J. ACM , vol.23 , pp. 555-565
    • Sahni, S.1    Gonzalez, T.2
  • 98
    • 84976779342 scopus 로고
    • Probabilistic algorithms for verification of polynomial identities
    • SCHWARTZ, J. 1980. Probabilistic algorithms for verification of polynomial identities. J. ACM 27, 701-717.
    • (1980) J. ACM , vol.27 , pp. 701-717
    • Schwartz, J.1
  • 99
    • 0026930543 scopus 로고
    • IP = PSPACE
    • SHAMIR, A. 1992. IP = PSPACE. J. ACM 39, 4 (Oct.), 869-877.
    • (1992) J. ACM , vol.39 , Issue.4 OCT. , pp. 869-877
    • Shamir, A.1
  • 100
    • 0003881012 scopus 로고
    • Ph.D. dissertation. Univ. California, Berkeley, Berkeley, Calif., 1992
    • SUDAN, M. 1992/1996. Efficient checking of polynomials and proofs and the hardness of approximation problems. Ph.D. dissertation. Univ. California, Berkeley, Berkeley, Calif., 1992. Also appears as ACM Distinguished Theses. In Lecture Notes in Computer Science, vol. 1001. Springer-Verlag, New York, 1996.
    • (1992) Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
    • Sudan, M.1
  • 101
    • 1842813373 scopus 로고    scopus 로고
    • ACM Distinguished Theses
    • Springer-Verlag, New York
    • SUDAN, M. 1992/1996. Efficient checking of polynomials and proofs and the hardness of approximation problems. Ph.D. dissertation. Univ. California, Berkeley, Berkeley, Calif., 1992. Also appears as ACM Distinguished Theses. In Lecture Notes in Computer Science, vol. 1001. Springer-Verlag, New York, 1996.
    • (1996) Lecture Notes in Computer Science , vol.1001
  • 103
    • 78951485210 scopus 로고
    • A survey of Russian approaches to Perebor (brute-force search) algorithms
    • TRAKHTENBROT, B. 1984. A survey of Russian approaches to Perebor (brute-force search) algorithms. Ann. Hist. Comput. 6, 384-400.
    • (1984) Ann. Hist. Comput. , vol.6 , pp. 384-400
    • Trakhtenbrot, B.1
  • 104
    • 1842662529 scopus 로고    scopus 로고
    • 1986. Error correction of algebraic block codes. US Patent Number 4,633,470
    • WELCH, L., AND BERLEKAMP, E. 1986. Error correction of algebraic block codes. US Patent Number 4,633,470.
    • Welch, L.1    Berlekamp, E.2
  • 105
    • 0001843554 scopus 로고
    • On the approximation of maximum satisfiability
    • YANNAKAKIS, M. 1994. On the approximation of maximum satisfiability. J. Algorithms 17, 3 (Nov.), 475-502.
    • (1994) J. Algorithms , vol.17 , Issue.3 NOV. , pp. 475-502
    • Yannakakis, M.1
  • 106
    • 0001327627 scopus 로고    scopus 로고
    • On unapproximable versions of NP-complete problems
    • ZUCKERMAN, D. 1996. On unapproximable versions of NP-complete problems. SIAM J. Comput. 25, 6 (Dec.), 1293-1304.
    • (1996) SIAM J. Comput. , vol.25 , Issue.6 DEC. , pp. 1293-1304
    • Zuckerman, D.1


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