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

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Venue: | Journal of Computer and System Sciences |

Citations: | 237 - 9 self |

### Citations

10437 | Introduction to algorithms", the
- Cormen, Leiserson, et al.
(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... |

1960 | The Probabilistic Method - Alon, Spencer - 1992 |

1133 |
On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications
- 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... |

792 | Proof verification and hardness of approximation problems
- Arora, Lund, et al.
- 1992
(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... |

748 | Some optimal inapproximability results
- Håstad
- 2001
(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... |

606 |
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... |

390 | Polynomial Time Approximation Schemes 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... |

323 | 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... |

311 |
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... |

273 | Almost optimal lower bounds for small depth circuits
- H˚astad
- 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... |

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

173 |
Approximating Clique is almost NP-complete
- Feige, Goldwasser, et al.
(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... |

156 | 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... |

128 |
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... |

110 | 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... |

86 |
1 -Formulae on Finite Structures
- Ajtái
- 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... |

78 | 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... |

65 |
Clique is hard to approximate within n 1\Gammaffl
- Hastad
(Show Context)
Citation Context |

54 |
Lower bounds on the size of bounded depth networks over a complete basis with logical addition
- Razborov
- 1986
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Citation Context |

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

38 | 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... |

32 |
EI-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... |

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

28 |
Pure literal look ahead: an O(1, 497 n ) 3satisfiability algorithm (extended abstract
- Schiermeyer
- 1996
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Citation Context |

19 | Worst-case time bounds for coloring and satisfiability problems
- Feder, Motwani
(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 ... |

18 |
Separating the Polynomial Hierarchy by Oracles
- Yao
- 1985
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Citation Context |

17 | Fixed Parameter Tractability and Completeness IV: On Completeness for W [P ] and PSPACE Analogs
- 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... |

13 | 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... |

13 | Exponential lower bounds for depth 3 Boolean circuits
- Paturi, Saks, et al.
- 1997
(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... |

12 |
Solving Satisfiability
- Monien, Speckenmeyer
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Citation Context |

12 | J.H.: Introduction to coding theory, 2nd Edition - Lint - 1992 |

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

8 |
Two new upper bounds for
- Hirsch
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Citation Context |

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

2 |
3-Coloring in time O(1:3446 n ) time: a no-MIS algorithm
- Biegel, Epstein
- 1995
(Show Context)
Citation Context ...cently, a series of papers have obtained improved algorithms for Formula-SAT, CNF-SAT, and k-SAT [20, 27, 28, 18, 19, 21, 22]. Similar sequences exist for other problems, such as graph 3-colorability =-=[7, 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 |
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 |

1 | Lower Bounds for Depth 3 - Hirsch |

1 |
Upper
- Kullmann, Luckhardt
- 1997
(Show Context)
Citation Context ...wn that Clique and Partition problems have power index 1=2 (where n is the total problem size). 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]-=-. Similar sequences exist for other problems, such as graph 3-colorability [7, 11]. Is there any complexity reason to believe that these improvements could not extend to arbitrarily small constants in... |

1 |
and their Complexity, submitted to Information and Computation. Tautologies
- Monien, Speckenmeyer
- 1985
(Show Context)
Citation Context ...wn that Clique and Partition problems have power index 1=2 (where n is the total problem size). 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]-=-. Similar sequences exist for other problems, such as graph 3-colorability [7, 11]. Is there any complexity reason to believe that these improvements could not extend to arbitrarily small constants in... |

1 |
Lemma
- Paturi, Pudlák, et al.
- 1997
(Show Context)
Citation Context ...wn that Clique and Partition problems have power index 1=2 (where n is the total problem size). 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]-=-. Similar sequences exist for other problems, such as graph 3-colorability [7, 11]. Is there any complexity reason to believe that these improvements could not extend to arbitrarily small constants in... |

1 |
Bounds on Depth 3 Boolean Circuits
- Papadimitriou, Yannakakis
- 1997
(Show Context)
Citation Context ...en it follows from [21] that any k 3 circuit for the parity function or the majority function requiress(2n=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 forms(2no(n)) for recognizing the codewords of a good error-correcting code. They show that any 2-CNF accepting a large number of inputs must include a ... |

1 |
Journal
- Razborov
- 1991
(Show Context)
Citation Context ...commonly studied NP-complete 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]-=-. Even for those problems that have subexponential algorithms, we show that obtaining more than a polynomial improvement in their complexity is SERF-hard for SNP . This extends the theory of NP-comple... |

1 |
Depth Networks over a Complete Basis with Logical Addition, Mathematische Zametki 41
- Sauer
- 1972
(Show Context)
Citation Context ...erable 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, Håstad, Razborov, and Smolensky =-=[2, 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... |

1 |
Journal
- Schiermeyer
- 1993
(Show Context)
Citation Context ...then prove that almost all degree 2 GF(2) polynomials cannot be constant on any such structure. Consequently, we get that s 2(12)n. Let A f0; 1gn be a set of size at least 2n. From the work of =-=[26, 34]-=-, we know that A contains a subset B of size 2l such that B restricted to restricted to certain l bit positions still has size 2l. In other words, B contains all possible 2l bit combinations in those ... |

1 |
Selected
- Schiermeyer
- 1996
(Show Context)
Citation Context |

1 |
Algorithm, Workshop on the Satisfiability Problem
- Smolensky
(Show Context)
Citation Context |

1 |
Bounds for Boolean Circuit Complexity
- Stearns, Hunt
(Show Context)
Citation Context ...erable 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, Håstad, Razborov, and Smolensky =-=[2, 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... |

1 | Clique?, manuscript. (1997 - Valiant - 1977 |

1 |
Springer–Verlag
- Lint
(Show Context)
Citation Context ...1:644 for large k). It is still open to prove a 2n=2 lower bound for k 3 circuits even when k is bounded. One compelling motivation for studying the depth–3 model with limited fan-in is that Valiant =-=[32]-=- showed that linear–size logarithmic–depth Boolean circuits with bounded fan–in can be computed by depth–3 unbounded fan-in circuits of size O(2n= log logn) and bottom fan-in limited by n" for arbitra... |

1 |
Convergence of Relative Frequencies of Events to their Probabilities, Theory of Probability Applications
- Yao
- 1985
(Show Context)
Citation Context ...then prove that almost all degree 2 GF(2) polynomials cannot be constant on any such structure. Consequently, we get that s 2(12)n. Let A f0; 1gn be a set of size at least 2n. From the work of =-=[26, 34]-=-, we know that A contains a subset B of size 2l such that B restricted to restricted to certain l bit positions still has size 2l. In other words, B contains all possible 2l bit combinations in those ... |

1 | Sigact News 27(1996 - Bellare, Complexitytheorycolumn |

1 | Systems Theory 17(1984),13–28 - Furst, Sipser, et al. |

1 | Theory Ser. A 13(1972),145–147 - Sauer, Combin |

1 | Systems Theory 23(1990),209–225 - Hunt, Math |

1 | Algorithms 7(1986),425–440 - Robson, J |