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33
The Extended Low Hierarchy Is an Infinite Hierarchy
, 1992
"... Balc'azar, Book, and Schoning introduced the extended low hierarchy based on the \Sigmalevels of the polynomialtime hierarchy as follows: for k 1, level k of the extended low hierarchy is the set EL P;\Sigma k = fA j \Sigma P k (A) ` \Sigma P k\Gamma1 (A \Phi SAT)g. Allender and Hemachandr ..."
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Balc'azar, Book, and Schoning introduced the extended low hierarchy based on the \Sigmalevels of the polynomialtime hierarchy as follows: for k 1, level k of the extended low hierarchy is the set EL P;\Sigma k = fA j \Sigma P k (A) ` \Sigma P k\Gamma1 (A \Phi SAT)g. Allender and Hemachandra and Long and Sheu introduced refinements of the extended low hierarchy based on the \Delta and \Thetalevels, respectively, of the polynomialtime hierarchy: for k 2, EL P;\Delta k = fA j \Delta P k (A) ` \Delta P k\Gamma1 (A \Phi SAT)g and EL P;\Theta k = fA j \Theta P k (A) ` \Theta P k\Gamma1 (A \Phi SAT)g. In this paper we show that the extended low hierarchy is properly infinite by showing, for k 2, that EL P;\Sigma k ` / EL P;\Theta k+1 ` / EL P;\Delta k+1 ` / EL P;\Sigma k+1 . Our proofs use the circuit lower bound techniques of Hastad and Ko. As corollaries to our constructions, we obtain, for k 2, oracle sets B k , C k , and D k , such that PH(B k ) = \Sigma P k (B k )...
Complexity of the exact domatic number problem and of the exact conveyor flow shop problem
"... Abstract. We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating ..."
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Abstract. We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is exactly one of k given values is complete for BH2k(NP), the 2kth level of the boolean hierarchy over NP. In particular, for k = 1, it is DPcomplete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP = BH2(NP). We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem. Our reductions apply Wagner’s conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP. 1.
Lowness and the Complexity of Sparse and Tally Descriptions
 IN PROC. 3RD INT'L SYMP. ON ALG. AND COMPUT
, 1992
"... We investigate the complexity of obtaining sparse descriptions for sets in various reduction classes to sparse sets. Let A be a set in a certain reduction class Rr(SPARSE). Then we are interested in finding upper bounds for the complexity (relative to A) of sparse sets S such that A 2 Rr (S). By est ..."
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We investigate the complexity of obtaining sparse descriptions for sets in various reduction classes to sparse sets. Let A be a set in a certain reduction class Rr(SPARSE). Then we are interested in finding upper bounds for the complexity (relative to A) of sparse sets S such that A 2 Rr (S). By establishing such upper bounds we are able to derive the lowness of A. In particular, we show that if a set A is in the class R p hd (R p c (SPARSE)) then A is in R p hd (R p c (S)) for a sparse set S 2 NP(A). As a consequence we can locate R p hd (R p c (SPARSE)) in the EL \Theta 3 level of the extended low hierarchy. Since R p hd (R p c (SPARSE)) ' R p b (R p c (SPARSE)) this solves the open problem of locating the closure of sparse sets under bounded truthtable reductions optimally in the extended low hierarchy. Furthermore, we show that for every A 2 R p d (SPARSE) there exists a sparse set S 2 NP(A \Phi SAT)=F\Theta p 2 (A) such that A 2 R p d (S). Based on this we show t...
Algebraic properties for selector functions
 SIAM JOURNAL ON COMPUTING
, 2005
"... The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is ..."
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The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is quadratic [Ko83] and the best current lower bound is linear [HT96]. We prove that every associatively Pselective set is commutatively, associatively Pselective. Using this, we establish an algebraic sufficient condition for the Pselective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all Pselective sets are associatively Pselective then the deterministic advice complexity of the Pselective sets is linear. The weakest previously known sufficient condition was P = NP. We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.
High Sets for NP
 In Advances in Algorithms, Languages, and Complexity
, 1997
"... this paper is a proof that NPhard ..."
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On Pselectivity and Closeness
 Inf. Processing Letters 54
, 1994
"... P/poly, the class of sets with polynomial size circuits, has been the subject of considerable study in complexity theory. Two important subclasses of P/poly are the class of sparse sets [6] and the class of Pselective sets [31]. A large number of results have been proved about both these classes bu ..."
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P/poly, the class of sets with polynomial size circuits, has been the subject of considerable study in complexity theory. Two important subclasses of P/poly are the class of sparse sets [6] and the class of Pselective sets [31]. A large number of results have been proved about both these classes but it has been observed (for example, [17]) that despite their similarity, proofs about one class generally do not translate easily to proofs regarding the other class. In this note, we propose to resolve this asymmetry by investigating the class PSELclose of sets that are polynomially close to Pselective sets; by definition, PSELclose includes both sparse sets and Pselective sets, thereby providing a unifying platform for proving results applicable to both. Intuitively, PSELclose is the class of sets that can in a certain sense be approximated by Pselective sets. We prove several results separating PSELclose from known classes within and including P/poly, and establish its location op...
Selectivity: Reductions, Nondeterminism, and Function Classes
, 1993
"... A set is Pselective [Se179] if there is a polynomialtime semidecision algorithm or the setan algorithm that given any two strings decides which is "more likely" to be in the set. This paper studies two natural generalizations o Pselectivity: the NPselective sets and the sets reducib ..."
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A set is Pselective [Se179] if there is a polynomialtime semidecision algorithm or the setan algorithm that given any two strings decides which is "more likely" to be in the set. This paper studies two natural generalizations o Pselectivity: the NPselective sets and the sets reducible or equivalent to Pselective sets via polynomialtime reductions. We establish a strict hierarchy among the various reductions and equivalences to Pselective sets. We show that the NPselective sets are in (NP coNP)/poly, are extended low, and (those in NP) are Low2; we also show that NPselective sets cannot be NPcomplete unless NP = coNP. By studying more general notions o nondeterministic selectivity, we conclude that all multivalued NP functions have singlevalued NP refinements only if the polynomial hierarchy collapses to its second level.
PolynomialTime SemiRankable Sets
 Special Issue: Proceedings of the 8th International Conference on Computing and Information
, 1995
"... We study the polynomialtime semirankable sets (Psr), the ranking analog of the Pselective sets. We prove that Psr is a strict subset of the Pselective sets, and indeed that the two classes differ with respect to closure under complementation, closure under union with P sets, closure under join ..."
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We study the polynomialtime semirankable sets (Psr), the ranking analog of the Pselective sets. We prove that Psr is a strict subset of the Pselective sets, and indeed that the two classes differ with respect to closure under complementation, closure under union with P sets, closure under join with P sets, and closure under Pisomorphism. While P=poly is equal to the closure of Pselective sets under polynomialtime Turing reductions, we build a tally set that is not polynomialtime reducible to any Psr set. We also show that though Psr falls between the Prankable and the weaklyPrankable sets in its inclusiveness, it equals neither of these classes. Key words: semifeasible sets, Pselectivity, ranking, closure properties, NNT. 1 Introduction In the late 1970s, Selman [Sel79] defined the semifeasible (i.e., Pselective) sets, which are the polynomialtime analog of the Jockusch's [Joc68] semirecursive sets. Recently, there has been an intense renewal of interest in the P...
Boolean operations, joins, and the extended low hierarchy
 Theoretical Computer Science
, 1998
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