## Which Problems Have Strongly Exponential Complexity? (1998)

Venue: | Journal of Computer and System Sciences |

Citations: | 129 - 5 self |

### BibTeX

@ARTICLE{Impagliazzo98whichproblems,

author = {Russell Impagliazzo and Ramamohan Paturi and Francis Zane},

title = {Which Problems Have Strongly Exponential Complexity?},

journal = {Journal of Computer and System Sciences},

year = {1998},

volume = {63},

pages = {512--530}

}

### Years of Citing Articles

### OpenURL

### Abstract

For several NP-complete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of sub-exponential algorithms for these problems. We introduce a generalized reduction which we call Sub-Exponential Reduction Family (SERF) that preserves sub-exponential complexity. We show that CircuitSAT is SERF-complete for all NP-search problems, and that for any fixed k, k-SAT, k-Colorability, k-Set Cover, Independent Set, Clique, Vertex Cover, are SERF--complete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, sub-exponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC 0 ; that is, bounds of the form 2 \Omega\Gamma n) . This problem is even open for depth-3 circuits. In fact, such a bound for depth-3 circuits with even l...

### Citations

8779 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ...reductions: Theorem 2 k-Colorability is SNP -complete under SERF reductions for any ks3 Proof: k-Colorability is in SNP . 3-SAT with parameter m has a strong many-one reduction to 3-colorability. See =-=[9]-=-, p. 962, ex. 36-2. 3-colorability has a strong many one reduction to k-colorability for ks3. Many natural problems like Independent Set are not obviously in SNP . However, they are in a small general... |

1714 | The probabilistic method - Alon, Spencer - 2008 |

959 |
On the uniform convergence of relative frequencies of events to their probabilities
- Vapnik, Chervonenkis
- 1971
(Show Context)
Citation Context ...g the fact that F accepts a large set of inputs where the polynomial is 1. Let A = F \Gamma1 (1), and let l be the least integer such that \Sigma l i=1 \Gamma n i \DeltasjAjs2 ffln . From the work of =-=[26, 35]-=-, we know that there is a set of l variables which are shattered by A. That is, A contains a subset B of size 2 l such that B restricted to those l variables still has size 2 l , so all possible assig... |

735 | Proof Verification and Hardness of Approximation Problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...kely complexity. Other recent work has divided NP-complete problems by their extent of approximability. The extension of NP-completeness theory to a wide variety of approximation problems (See, e.g., =-=[24, 10, 5, 15, 16]-=- and see [6] for a survey) has allowed researchers to compute sharp bounds on how well one should be able to approximate solutions to NP-complete problems. This work has spurred the discovery of new a... |

658 | Some optimal inapproximability results
- H̊astad
(Show Context)
Citation Context ...kely complexity. Other recent work has divided NP-complete problems by their extent of approximability. The extension of NP-completeness theory to a wide variety of approximation problems (See, e.g., =-=[24, 10, 5, 15, 16]-=- and see [6] for a survey) has allowed researchers to compute sharp bounds on how well one should be able to approximate solutions to NP-complete problems. This work has spurred the discovery of new a... |

577 |
Optimization, approximation and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ... commonly studied NPcomplete problems and for all the commonly studied choices of parameters. To be more precise, we show that problems such as k-SAT and k-colorability are complete for the class SNP =-=[24]-=- under a notion of reduction that preserves subexponential complexity. This extends the theory of NP-completeness to give a stratification of problems based on their likely complexity. Other recent wo... |

325 | Polynomial time approximation scheme for Euclidean traveling salesman and other geometric problems
- Arora
- 1998
(Show Context)
Citation Context ...earchers to compute sharp bounds on how well one should be able to approximate solutions to NP-complete problems. This work has spurred the discovery of new approximation algorithms for some problems =-=[4]. While not claiming-=- the same depth or significance for our work, we hope that this paper will further distinguish "easy" from "hard" problems within the ranks of NP-complete problems, and lead to a s... |

290 | Algebraic methods in the theory of lower bounds for boolean circuit complexity
- Smolensky
- 1987
(Show Context)
Citation Context ...iderable progress has been made in understanding the limitations of unbounded fan-in Boolean circuits of bounded depth. The results of Ajtai, Furst, Saxe, Sipser, Yao, Hastad, Razborov, and Smolensky =-=[2, 12, 36, 14, 25, 30]-=-, among others, show that if the size of the circuit is not too large, then any function computed by such circuit must be constant on a large subcube or can be approximated by a small degree polynomia... |

241 |
On the density of families of sets
- Sauer
- 1972
(Show Context)
Citation Context ...g the fact that F accepts a large set of inputs where the polynomial is 1. Let A = F \Gamma1 (1), and let l be the least integer such that \Sigma l i=1 \Gamma n i \DeltasjAjs2 ffln . From the work of =-=[26, 35]-=-, we know that there is a set of l variables which are shattered by A. That is, A contains a subset B of size 2 l such that B restricted to those l variables still has size 2 l , so all possible assig... |

239 | Almost optimal lower bounds for small depth circuits
- Hastad
- 1989
(Show Context)
Citation Context ...iderable progress has been made in understanding the limitations of unbounded fan-in Boolean circuits of bounded depth. The results of Ajtai, Furst, Saxe, Sipser, Yao, Hastad, Razborov, and Smolensky =-=[2, 12, 36, 14, 25, 30]-=-, among others, show that if the size of the circuit is not too large, then any function computed by such circuit must be constant on a large subcube or can be approximated by a small degree polynomia... |

185 |
Approximating clique is almost np-complete
- Feige, Goldwasser, et al.
- 1991
(Show Context)
Citation Context ...kely complexity. Other recent work has divided NP-complete problems by their extent of approximability. The extension of NP-completeness theory to a wide variety of approximation problems (See, e.g., =-=[24, 10, 5, 15, 16]-=- and see [6] for a survey) has allowed researchers to compute sharp bounds on how well one should be able to approximate solutions to NP-complete problems. This work has spurred the discovery of new a... |

176 | The complexity of finite functions - Boppana, Sipser - 1990 |

129 | A probabilistic algorithm for k-SAT and constraint satisfaction problems
- Schöning
- 1999
(Show Context)
Citation Context ... research is supported by NSF grant CCR-9734911 from Theory of Computing Program 1 than others. More recently, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and k-SAT =-=[20, 27, 28, 18, 19, 21, 22, 29]-=-. Similar sequences exist for other problems, such as graph 3-colorability [7, 11]. Is there any complexity-theoretic reason to believe that these improvements could not extend to arbitrarily small co... |

103 |
Graph-theoretic arguments in low-level complexity
- Valiant
- 1977
(Show Context)
Citation Context ...r large k). It is still open to prove a 2 n=2 lower bound for \Sigma k 3 circuits even when k is bounded. One compelling motivation for studying the depth--3 model with limited fan-in is that Valiant =-=[33] show-=-ed that linear--size logarithmic--depth Boolean circuits with bounded fan--in can be computed by depth--3 unbounded fan-in circuits of size O(2 n= log log n ) and bottom fan-in limited by n " for... |

87 | An improved exponential-time algorithm for k-SAT
- Paturi, Pudlák, et al.
(Show Context)
Citation Context ... research is supported by NSF grant CCR-9734911 from Theory of Computing Program 1 than others. More recently, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and k-SAT =-=[20, 27, 28, 18, 19, 21, 22, 29]-=-. Similar sequences exist for other problems, such as graph 3-colorability [7, 11]. Is there any complexity-theoretic reason to believe that these improvements could not extend to arbitrarily small co... |

85 |
1 -formulae on finite structures
- Ajtai
- 1983
(Show Context)
Citation Context ...iderable progress has been made in understanding the limitations of unbounded fan-in Boolean circuits of bounded depth. The results of Ajtai, Furst, Saxe, Sipser, Yao, Hastad, Razborov, and Smolensky =-=[2, 12, 36, 14, 25, 30]-=-, among others, show that if the size of the circuit is not too large, then any function computed by such circuit must be constant on a large subcube or can be approximated by a small degree polynomia... |

67 |
Clique is hard to approximate within n 1\Gammaffl
- Hastad
- 1999
(Show Context)
Citation Context |

64 | Satisfiability coding lemma
- Paturi, Pudlák, et al.
- 1999
(Show Context)
Citation Context ... research is supported by NSF grant CCR-9734911 from Theory of Computing Program 1 than others. More recently, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and k-SAT =-=[20, 27, 28, 18, 19, 21, 22, 29]-=-. Similar sequences exist for other problems, such as graph 3-colorability [7, 11]. Is there any complexity-theoretic reason to believe that these improvements could not extend to arbitrarily small co... |

50 |
bounds on the size of bounded depth networks over a complete basis with logical addition
- Razborov, Lower
- 1987
(Show Context)
Citation Context |

36 | Deciding propositional tautologies: algorithms and their complexity
- Kullmann, Luckhardt
- 1997
(Show Context)
Citation Context |

32 | Clique is hard to approximate within n
- H˚astad
- 1999
(Show Context)
Citation Context ...ikely complexity. Other recent work has divided NP-complete problems by their extent of approximability. The extension of NPcompleteness theory to a wide variety of approximation problems (See, e.g., =-=[24, 10, 4, 15, 16]-=- and see [5] for a survey) has allowed researchers to compute sharp bounds on how well one should be able to approximate solutions to NP-complete problems. This work has spurred the discovery of new a... |

29 |
Solving 3-satisfiability in less than 1.579 n steps
- Schiermeyer
- 1993
(Show Context)
Citation Context |

29 |
Pure Literal Look Ahead: An O(1; 497 n ) 3-Satisfiability Algorithm. Preprint 96. (e-mail: schierme@math.tu-cottbus.de
- Schiermeyer
(Show Context)
Citation Context |

24 |
Zi-formulae on finite structures
- AJTAI
- 1983
(Show Context)
Citation Context ...derable progress has been made in understanding the limitations of unbounded fan-in Boolean circuits of bounded depth. The results of Ajtái, Furst, Saxe, Sipser, Yao, H˚astad, Razborov, and Smolensky =-=[1, 12, 35, 14, 25, 29]-=-, among others, show that if the size of the circuit is not too large, then any function computed by such circuit must be constant on a large subcube or can be approximated by a small degree polynomia... |

18 |
Separating the Polynomial Hierarchy by oracles
- Yao
- 1985
(Show Context)
Citation Context |

17 | Fixed parameter tractability and completeness IV: on completeness for W[P] and PSPACE analogs. Annals of Pure and Applied Logic 73
- Abrahamson, Downey, et al.
- 1995
(Show Context)
Citation Context ...d via fixed parameter complexity. It is shown that the tractability of the class of fixed parameter problems is equivalent to the existence of subexponential-time algorithm for Circuit Satisfiability =-=[1]-=-. 1.1.2 Complexity parameters As mentioned before, we feel that solution size is the most relevant parameter when discussing exact complexities of NP problems. There are, however, a number of other po... |

16 | Proof checking and approximation: Towards tight results
- Bellare
- 1996
(Show Context)
Citation Context ...t work has divided NP-complete problems by their extent of approximability. The extension of NP-completeness theory to a wide variety of approximation problems (See, e.g., [24, 10, 5, 15, 16] and see =-=[6]-=- for a survey) has allowed researchers to compute sharp bounds on how well one should be able to approximate solutions to NP-complete problems. This work has spurred the discovery of new approximation... |

16 |
Worst-case time bounds for coloring and satisfiability problems
- Feder, Motwani
- 1998
(Show Context)
Citation Context ...ly, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and k-SAT [20, 27, 28, 18, 19, 21, 22, 29]. Similar sequences exist for other problems, such as graph 3-colorability =-=[7, 11]-=-. Is there any complexity-theoretic reason to believe that these improvements could not extend to arbitrarily small constants in the exponent ? Is progress on the different problems connected? Can we ... |

13 | Exponential lower bounds for depth 3 boolean circuits
- Paturi, Saks, et al.
- 2000
(Show Context)
Citation Context ... [21] that any \Sigma k 3 circuit for the parity function or the majority function requires\Omega\Gammaq n=k ) gates at level 2, and such bounds are tight for k = O( p n). For the special case k = 2, =-=[23]-=- proved strong exponential lower bounds of the form \Omega\Gammao n\Gammao(n) ) for recognizing the codewords of a good error-correcting code. They show that any 2-CNF accepting a large number of inpu... |

11 | Top– Down Lower Bounds for Depth 3 - H˚astad, Jukna, et al. - 1993 |

11 |
Solving Satisfiability
- Monien, Speckenmeyer
- 1985
(Show Context)
Citation Context |

11 | An introduction to coding theory, 2nd edition - Lint - 1992 |

8 |
Two new upper bounds for
- Hirsch
- 1998
(Show Context)
Citation Context |

3 | The Complexity of Finite Functions, in "The Handbook of Theoretical Computer science", Vol - Boppana, Sipser - 1990 |

1 |
3-Coloring in O(1:3446 n ) time: a no-MIS algorithm
- Beigel, Eppstein
- 1995
(Show Context)
Citation Context ...ly, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and k-SAT [20, 27, 28, 18, 19, 21, 22, 29]. Similar sequences exist for other problems, such as graph 3-colorability =-=[7, 11]-=-. Is there any complexity-theoretic reason to believe that these improvements could not extend to arbitrarily small constants in the exponent ? Is progress on the different problems connected? Can we ... |

1 | What are the least tractable instances - Johnson, Szegedy - 1999 |

1 |
3-Coloring in time Ç � ��� Ò time: a no-MIS algorithm
- Biegel, Epstein
- 1995
(Show Context)
Citation Context ...example, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and �-SAT [20, 27, 28, 18, 19, 21, 22]. Similar sequences exist for other problems, such as graph 3colorability =-=[6, 11]-=-. Is there any complexity reason to believe that these improvements could not extend to arbitrarily small constants in the exponent? Is progress on the different problems connected? It is already know... |

1 |
An Improved Exponential-time Algorithm for
- Paturi, Pudlák, et al.
- 1998
(Show Context)
Citation Context ...e solvability. However, some NP-complete problems have “better” worstcase algorithms than others. For example, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and �-SAT =-=[20, 27, 28, 18, 19, 21, 22]-=-. Similar sequences exist for other problems, such as graph 3colorability [6, 11]. Is there any complexity reason to believe that these improvements could not extend to arbitrarily small constants in ... |

1 |
Solving 3-Satisfiability in less than
- Schiermeyer
- 1993
(Show Context)
Citation Context ...e solvability. However, some NP-complete problems have “better” worstcase algorithms than others. For example, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and �-SAT =-=[20, 27, 28, 18, 19, 21, 22]-=-. Similar sequences exist for other problems, such as graph 3colorability [6, 11]. Is there any complexity reason to believe that these improvements could not extend to arbitrarily small constants in ... |

1 |
Pure Literal Look Ahead: An Ç ���� Ò 3-Satisfiability Algorithm, Workshop on the Satisfiability Problem
- Schiermeyer
- 1996
(Show Context)
Citation Context ...e solvability. However, some NP-complete problems have “better” worstcase algorithms than others. For example, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and �-SAT =-=[20, 27, 28, 18, 19, 21, 22]-=-. Similar sequences exist for other problems, such as graph 3colorability [6, 11]. Is there any complexity reason to believe that these improvements could not extend to arbitrarily small constants in ... |

1 | What are the hardest instances of Max Clique?, manuscript - Szegedy - 1997 |