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On the Sparse Set Conjecture for Sets with Low Density
, 1995
"... . We study the sparse set conjecture for sets with low density. The sparse set conjecture states that P = NP if and only if there exists a sparse Turing hard set for NP . In this paper we study a weaker variant of the conjecture. We are interested in the consequences of NP having Turing hard set ..."
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. We study the sparse set conjecture for sets with low density. The sparse set conjecture states that P = NP if and only if there exists a sparse Turing hard set for NP . In this paper we study a weaker variant of the conjecture. We are interested in the consequences of NP having Turing hard sets of density f(n), for (unbounded) functions f(n), that are subpolynomial, for example log(n). We establish a connection between Turing hard sets for NP with density f(n) and bounded nondeterminism: We prove that if NP has a Turing hard set of density f(n), then satisfiability is computable in polynomial time with O(log(n) f(n c )) many nondeterministic bits for some constant c. As a consequence of the proof technique we obtain absolute results about the density of Turing hard sets for EXP . We show that no Turing hard set for EXP can have subpolynomial density. On the other hand we show that these results are optimal w.r.t. relativizing computations. For unbounded functions f(...
The Complexity of Generating Test Instances
 IN PROC. STACS'97, LECTURE NOTES IN COMPUTER SCIENCE
, 1997
"... Recently, Watanabe proposed a new framework for testing the correctness and average case behavior of algorithms that purport to solve a given NP search problem efficiently on average. The idea is to randomly generate certified instances in a way that resembles the underlying distribution ¯. We discu ..."
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Recently, Watanabe proposed a new framework for testing the correctness and average case behavior of algorithms that purport to solve a given NP search problem efficiently on average. The idea is to randomly generate certified instances in a way that resembles the underlying distribution ¯. We discuss this approach and show that test instances can be generated for every NP search problem with nonadaptive queries to an NP oracle. Further, we introduce Las Vegas as well as Monte Carlo types of test instance generators and show that these generators can be used to find out whether an algorithm is correct and efficient on average under ¯. In fact, it is not hard to construct Monte Carlo generators for all RP search problems as well as Las Vegas generators for all ZPP search problems. On the other hand, we prove that Monte Carlo generators can only exist for problems in NP " coAM.
A note on the complexity of computing the Smallest FourColoring of Planar Graphs
, 2006
"... We show that computing the lexicographically first fourcoloring for planar graphs is ∆ p 2hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NPhard, and conclude that it is not selfreducible in the sense of Schnorr, assuming P = NP. We discuss this ..."
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We show that computing the lexicographically first fourcoloring for planar graphs is ∆ p 2hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NPhard, and conclude that it is not selfreducible in the sense of Schnorr, assuming P = NP. We discuss this application to nonselfreducibility and provide a general related result.
On Computing the Smallest FourColoring of Planar Graphs and NonSelfReducible Sets in P
, 2006
"... We show that computing the lexicographically first fourcoloring for planar graphs is ∆ p 2hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NPhard, and conclude that it is not selfreducible in the sense of Schnorr, assuming P � = NP. We discuss thi ..."
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We show that computing the lexicographically first fourcoloring for planar graphs is ∆ p 2hard. This result optimally improves upon a result of Khuller and Vazirani who prove this problem NPhard, and conclude that it is not selfreducible in the sense of Schnorr, assuming P � = NP. We discuss this application to nonselfreducibility and provide a general related result. We also discuss when raising a problem’s NPhardness lower bound to ∆ p 2hardness can be valuable.
and
, 2001
"... We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including • In polynomial time, one can find accepting paths of nondeterministic polynomialtime Tu ..."
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We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including • In polynomial time, one can find accepting paths of nondeterministic polynomialtime Turing machines that accept Σ ∗. • Every total multivalued nondeterministic function has a polynomialtime computable refinement. • In polynomial time, one can compute satisfying assignments for any polynomialtime computable set of satisfiable formulae. • In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments. We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse. 1.
Karg, Köbler, and Schuler Complexity of Generating Test Instances (Info) The Chicago Journal of Theoretical Computer Science is abstracted or indexed
, 1999
"... Published one article at a time in L ATEX source form on the Internet. Pagination ..."
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Published one article at a time in L ATEX source form on the Internet. Pagination
A Note on Quadratic Residuosity and UP
"... UP is the class of languages accepted by polynomialtime nondeterministic Turing machines that have at most one accepting path. We show that the quadratic residue problem belongs to UP ∩ coUP. This affirmatively answers an open problem, discussed in Theory of Computational Complexity (Du and Ko, 200 ..."
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UP is the class of languages accepted by polynomialtime nondeterministic Turing machines that have at most one accepting path. We show that the quadratic residue problem belongs to UP ∩ coUP. This affirmatively answers an open problem, discussed in Theory of Computational Complexity (Du and Ko, 2000), of whether the quadratic nonresidue problem is in NP. We generalize to higher powers and show