A Survey of Lower Bounds for Satisfiability and Related Problems

Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving non-trivial lower bounds on the computational complexity of satisfiability. In this article, we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the state-of-the-art results and present the underlying arguments in a unified framework.

[1]  Lance Fortnow,et al.  Complexity limitations on quantum computation , 1999, J. Comput. Syst. Sci..

[2]  Dieter van Melkebeek,et al.  A Generic Time Hierarchy with One Bit of Advice , 2007, computational complexity.

[3]  Miklós Ajtai,et al.  A Non-linear Time Lower Bound for Boolean Branching Programs , 2005, Theory Comput..

[4]  Michael J. Fischer,et al.  Separating Nondeterministic Time Complexity Classes , 1978, JACM.

[5]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[6]  William Hesse.  Division Is in Uniform TC0 , 2001, ICALP.

[7]  Lance Fortnow,et al.  Time-space tradeoffs for nondeterministic computation , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[8]  Ryan Williams.  Better time-space lower bounds for SAT and related problems , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[9]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[10]  Dieter van Melkebeek,et al.  A generic time hierarchy for semantic models with one bit of advice , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[11]  Scott Diehl,et al.  Lower Bounds for Swapping Arthur and Merlin , 2007, APPROX-RANDOM.

[12]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[13]  Lance Fortnow,et al.  Time-Space Tradeoffs for Satisfiability , 2000, J. Comput. Syst. Sci..

[14]  Richard Edwin Stearns,et al.  Two-Tape Simulation of Multitape Turing Machines , 1966, JACM.

[15]  Stephen A. Cook,et al.  Short Propositional Formulas Represent Nondeterministic Computations , 1988, Inf. Process. Lett..

[16]  Leonard M. Adleman,et al.  Quantum Computability , 1997, SIAM J. Comput..

[17]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[18]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[19]  W. Hesse.  Division is in Uniform TC 0 , 2001 .

[20]  Iannis Tourlakis.  Time-space lower bounds for SAT on uniform and non-uniform machines , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[21]  А Е Китаев,et al.  Квантовые вычисления: алгоритмы и исправление ошибок@@@Quantum computations: algorithms and error correction , 1997 .

[22]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

[23]  Andrew Chiu,et al.  Division in logspace-uniform NC1 , 2001, RAIRO Theor. Informatics Appl..

[24]  Walter J. Savitch,et al.  Relationships Between Nondeterministic and Deterministic Tape Complexities , 1970, J. Comput. Syst. Sci..

[25]  Michael J. Fischer,et al.  Relations Among Complexity Measures , 1979, JACM.

[26]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computation , 1992, Comb..

[27]  Richard Ryan Williams,et al.  Time-Space Tradeoffs for Counting NP Solutions Modulo Integers , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[28]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[29]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[30]  Umesh V. Vazirani,et al.  Quantum Complexity Theory , 1997, SIAM J. Comput..

[31]  Johan Hå stad.  The Shrinkage Exponent of de Morgan Formulas is 2 , 1998 .

[32]  Walter L. Ruzzo,et al.  Tree-size bounded alternation(Extended Abstract) , 1979, J. Comput. Syst. Sci..

[33]  Andrew Chiu,et al.  Complexity Of Parallel Arithmetic Using The Chinese Remainder Representation , 1995 .

[34]  Manuel Blum,et al.  Algorithms and resource requirements for fundamental problems , 2007 .

[35]  Richard J. Lipton,et al.  Time-space lower bounds for satisfiability , 2005, JACM.

[36]  Noam Nisan,et al.  RL⊆SC , 1992, STOC '92.

[37]  Eric Allender,et al.  Time-space tradeoffs in the counting hierarchy , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[38]  Stephen A. Cook.  A Hierarchy for Nondeterministic Time Complexity , 1973, J. Comput. Syst. Sci..

[39]  Noam Nisan,et al.  Hardness vs. randomness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[40]  Richard J. Lipton,et al.  On the complexity of SAT , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[41]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[42]  Michael E. Saks,et al.  Time-space trade-off lower bounds for randomized computation of decision problems , 2003, JACM.

[43]  Noam Nisan,et al.  Pseudorandomness for network algorithms , 1994, STOC '94.

[44]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[45]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computations , 1990, STOC '90.

[46]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[47]  Dieter van Melkebeek,et al.  Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines , 2006, SIAM J. Comput..

[48]  Emanuele Viola,et al.  On Approximate Majority and Probabilistic Time , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[49]  Russell Impagliazzo,et al.  How to recycle random bits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[50]  D. Sivakumar,et al.  On Quasilinear-Time Complexity Theory , 1995, Theor. Comput. Sci..

[51]  Zvi Galil,et al.  Explicit Constructions of Linear-Sized Superconcentrators , 1981, J. Comput. Syst. Sci..

[52]  Oded Goldreich,et al.  Another proof that bpp?ph (and more) , 1997 .

[53]  John Michael Robson,et al.  An O (T log T) Reduction from RAM Computations to Satisfiability , 1991, Theor. Comput. Sci..

[54]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[55]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[56]  Ravi Kannan.  Towards separating nondeterminism from determinism , 2005, Mathematical systems theory.

[57]  Andrew Chi-Chih Yao,et al.  A Circuit-Based Proof of Toda's Theorem , 1993, Inf. Comput..

[58]  Stanislav Zák,et al.  A Turing Machine Time Hierarchy , 1983, Theor. Comput. Sci..

[59]  Jacobo Torán,et al.  Complexity classes defined by counting quantifiers , 1991, JACM.

[60]  Clemens Lautemann,et al.  BPP and the Polynomial Hierarchy , 1983, Inf. Process. Lett..

[61]  Noam Nisan.  Rl <= Sc , 1994, Comput. Complex..

[62]  Chee-Keng Yap,et al.  Some Consequences of Non-Uniform Conditions on Uniform Classes , 1983, Theor. Comput. Sci..

[63]  Eric Allender,et al.  Uniform constant-depth threshold circuits for division and iterated multiplication , 2002, J. Comput. Syst. Sci..

[64]  Dieter van Melkebeek,et al.  A Quantum Time-Space Lower Bound for the Counting Hierarchy , 2008, Electron. Colloquium Comput. Complex..

[65]  Iannis Tourlakis.  Time-Space Tradeoffs for SAT on Nonuniform Machines , 2001, J. Comput. Syst. Sci..

[67]  D. Melkebeek.  TIME-SPACE LOWER BOUNDS FOR NP-COMPLETE PROBLEMS , 2004 .