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Page 2642 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews  
the polynomial-time degrees: the degrees of super sparse sets.  ...  z...) < the polynomial hierarchy is infinite relative to every sparse oracle + the polynomial hierarchy is infinite relative to some sparse oracle, and (2) the polynomial hierarchy equals polynomial space  ... 

Sparse sets in NP-P: EXPTIME versus NEXPTIME

J. Hartmanis, N. Immerman, V. Sewelson
1985 Information and Control  
It is interesting to note that it has been shown by Wilson (1980; Book, Wilson, and Xu, 1981) that there exist oracle sets A such that pA V~ Npa but EXPTIME A = NEXPTIME A.  ...  This paper investigates the structural properties of sets in NP-P and shows that the computational difficulty of lower density sets in NP depends explicitly on the relations between higher deterministic  ...  ACKNOWLEDGMENTS The authors would like to thank Ron Book, Deborah Joseph, Stuart Kurtz, Steve Mahaney, and Yaacov Yesha for helpful discussions and suggestions.  ... 
doi:10.1016/s0019-9958(85)80004-8 fatcat:2spxuego3nbxzo5sgphlt6o45i

On relativized exponential and probabilistic complexity classes

Hans Heller
1986 Information and Control  
Turing reducible to a sparse set (Wilson, C.  ...  The result subsumes several known results about relativized computations: (i) the existence of relativized polynomial hierarchies extending two levels (Long, T., 1978, Dissertation, Purdue Univ., Lafayette  ...  If the polynomial hierarchy PH is ~ Yr-reducible to a sparse set in A~, then PH collapses to A~. Relativizations of Proposition 4 and 5 hold also.  ... 
doi:10.1016/s0019-9958(86)80012-2 fatcat:fo3u5bsczvhudlvodhplvub3ya

A note on sparse oracles for NP

Timothy J. Long
1982 Journal of computer and system sciences (Print)  
Consequently, if NP has either a sparse or co-sparse <T-complete set, then the polynomial-time hierarchy collapses to AC. S.  ...  Mahaney (Proceedings, 21st IEEE Symposium, Foundations of Computer Science, 1980) had previously shown that if NP has a sparse (',complete set, then the polynomial-time hierarchy collapses to A:.  ...  In this paper we shall prove that if S is a sparse oracle for every set in NP, then S is a sparse oracle for every set in the polynomial-time hierarchy (PH).  ... 
doi:10.1016/0022-0000(82)90050-2 fatcat:zi4czk5rhfaptpladdtwnbt6zi

A View of Structural Complexity Theory [chapter]

Ronald V. BOOK, Osamu WATANABE
1993 Current Trends in Theoretical Computer Science  
Acknowledgments Preparation of this paper was supported in part by the National Science Foundation under grant CCR86-11980 and CCR89-13584.  ...  If S is a low set in the polynomial-time hierarchy and NP ⊆ P(S), then the polynomial-time hierarchy extends to only finitely many levels -the level depends on the "degree of lowness" of set S.  ...  Note that if every function that is one-to-one, length-increasing, and polynomial-time computable is also polynomial-time invertible, then A ≡ P 1,li B implies A ≡ P inv B, and thus A ≡ P 1,li B implies  ... 
doi:10.1142/9789812794499_0034 dblp:series/wsscs/Book093 fatcat:jcrj2u7caveejeojt6w6ouzqvq

Page 981 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
In particular, they show that either every sparse set is extended high and the polynomial hierarchy collapses, or no sparse set is extended high and the polynomial hierarchy does not collapse.  ...  It had also been shown that the sparse sets in NP are low, so that any sparse set in NP is high if and only if they are all high, if and only if the polynomial hierarchy collapses.  ... 

On sparse oracles separating feasible complexity classes [chapter]

Juris Hartmanis, Lane Hemachandra
1986 Lecture Notes in Computer Science  
A complete and transparent characterization of oracles that separate NP from P would resolve the long-standing P=?NP question. In this note, we settle a central ease.  ...  Results of this type have the potential to yield deeper insights into the nature of relativization problems and focus our attention on new and interesting classes of languages.  ...  Interestingly, we note that oracles from the advice hierarchy [14] have no effect on the structure of the polynomial hierarchy.  ... 
doi:10.1007/3-540-16078-7_86 fatcat:zdq6gbv5p5aufni22657noonda

Locating P/poly optimally in the extended low hierarchy

J. Köbler
1994 Theoretical Computer Science  
As a consequence of our result, all NP sets in P/poly are relocated from the third X-level L, '9x (Ko and &honing, 1985) to the third O-level Ly,@ of the low hierarchy.  ...  The location of P/poly in EL3 ',@ is optimal since, as shown by Allender and Hemachandra (1992) , there exist sparse sets that are not contained in the next lower level ELt,x.  ...  Later on, Allender and Hemachandra [l] , and Long and Sheu [34] refined the extended low and high hierarchies by introducing intermediate levels based on the A-and O-levels of the polynomial-time hierarchy  ... 
doi:10.1016/0304-3975(94)00016-6 fatcat:dheas3gv4zb6tb425eh4gtxvwq

Boolean operations, joins, and the extended low hierarchy

Lane A. Hemaspaandra, Zhigen Jiang, Jörg Rothe, Osamu Watanabe
1998 Theoretical Computer Science  
We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets.  ...  Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure.  ...  Acknowledgements We thank an anonymous referee for stressing that our results should be interpreted as evidence regarding the unnaturalness of extended lowness as a complexity measure.  ... 
doi:10.1016/s0304-3975(98)00006-1 fatcat:ork2muzluvcybdbcgspzloyrhi

Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity [article]

Tillmann Weisser , Kim-Chuan Toh
2017 arXiv   pre-print
We provide a sparse version of the bounded degree SOS hierarchy BSOS [7] for polynomial optimization problems. It permits to treat large scale problems which satisfy a structured sparsity pattern.  ...  When the sparsity pattern satisfies the running intersection property this Sparse-BSOS hierarchy of semidefinite programs (with semidefinite constraints of fixed size) converges to the global optimum of  ...  Acknowledgement The first author is very thankful to the National University of Singapore and Professor Kim-Chuan Toh for their hospitality and financial support during his stay in Singapore.  ... 
arXiv:1607.01151v3 fatcat:zia57rx3a5cnxbq6zdiobne4yi

Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity

Tillmann Weisser, Jean B. Lasserre, Kim-Chuan Toh
2017 Mathematical Programming Computation  
We provide a sparse version of the bounded degree SOS hierarchy BSOS [6] for polynomial optimization problems. It permits to treat large scale problems which satisfy a structured sparsity pattern.  ...  When the sparsity pattern satisfies the running intersection property this Sparse-BSOS hierarchy of semidefinite programs (with semidefinite constraints of fixed size) converges to the global optimum of  ...  Acknowledgement The first author is very thankful to the National University of Singapore and Professor Kim-Chuan Toh for their hospitality and financial support during his stay in Singapore.  ... 
doi:10.1007/s12532-017-0121-6 fatcat:yeucd5l2tjdo7mpta5murgfi2q

Semidefinite programming relaxation methods for global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets

V. Jeyakumar, S. Kim, G. M. Lee, G. Li
2015 Journal of Global Optimization  
We propose a hierarchy of semidefinite programming (SDP) relaxations for polynomial optimization with sparse patterns over unbounded feasible sets.  ...  We demonstrate that the proposed sparse SDP hierarchy can solve some classes of large scale polynomial optimization problems with unbounded feasible sets using the polynomial optimization solver SparsePOP  ...  Note that, if we setσ l ≡ 0, l = 1, . . . , p, then the hierarchy (4.1) reduces to the known sparse SDP hierarchy proposed in [8, 9] . Theorem 4.1.  ... 
doi:10.1007/s10898-015-0356-6 fatcat:4bhaja56urbkbcyp4otwlu2bmm

Page 5369 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
Summary: “The polynomial time function hierarchy, recently de- veloped by Krentel to study optimization problems in complexity theory, is a function analogue to the polynomial hierarchy and contains many  ...  Such a set is reducible to itself in polynomial time in the normal sense, with the constraint that the machine that implements the self-reducibility makes at most one query to the set, and that query is  ... 

Approximating regions of attraction of a sparse polynomial differential system * [article]

Didier Henrion, Carmen Cardozo, Jean Lasserre
2020 arXiv   pre-print
For this purpose, we combine previous results on non-sparse ROA approximations with sparse semi-algebraic set volume computation.  ...  (ROA) of sparse polynomial differential systems, at the price of solving linear matrix inequalities (LMI) of increasing size.  ...  Note that X is also a compact semialgebraic set.  ... 
arXiv:1911.09500v2 fatcat:zakb64x5rfavddwit4vpdgoxb4

Lower bounds for the low hierarchy

Eric Allender, Lane A. Hemachandra
1992 Journal of the ACM  
We also examine the interrelationships among the levels of the low hierarchies and the classes of sets reducible to or equivalent to sparse and tally sets under different notions of reducibility.  ...  , and relatioized lower bounds on the location of classes in the low hierarchy in I\TP.  ...  Acknowledgment \Ye thank Ron Book for organizing a workshop on computational complexity at Santa Barbara in June, 1988, which led to this collaboration.  ... 
doi:10.1145/147508.147546 fatcat:evjsfwhxz5el3koloccvhkczuq
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