Generalizations of Opt P to the Polynomial Hierarchy

@article{Krentel1992GeneralizationsOO,
  title={Generalizations of Opt P to the Polynomial Hierarchy},
  author={M. Krentel},
  journal={Theor. Comput. Sci.},
  year={1992},
  volume={97},
  pages={183-198}
}
  • M. Krentel
  • Published 1992
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
Abstract The author defined Opt P as a generalization of NP by considering problems as functions that compute their optimal value. An Opt P function is computed by applying the max (or min) operator to the branches of a nondeterministic machine. In this paper, we show that Opt P has a natural extension to the polynomial hierarchy by considering alternating Turing machines with the max and min operators. We show an equivalence between k alternations of max and min and functions computable with… Expand
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References

SHOWING 1-10 OF 13 REFERENCES
The Complexity of Optimization Problems
  • M. Krentel
  • Computer Science, Mathematics
  • J. Comput. Syst. Sci.
  • 1988
TLDR
It is shown that TRAVELING SALESPERSON and KNAPSACK are complete for OptP, and that CLIQUE and COLORING arecomplete for a subclass of OptP . Expand
The Polynomial-Time Hierarchy
  • L. Stockmeyer
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1976
TLDR
The problem of deciding validity in the theory of equality is shown to be complete in polynomial-space, and close upper and lower bounds on the space complexity of this problem are established. Expand
On the Complexity of Some Two-Person Perfect-Information Games
  • T. Schaefer
  • Computer Science, Mathematics
  • J. Comput. Syst. Sci.
  • 1978
Abstract We present a number of two-person games, based on simple combinatorial ideas, for which the problem of deciding whether the first player can win is complete in polynomial space. ThisExpand
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledgeExpand
NP is as easy as detecting unique solutions
Abstract For every known NP-complete problem, the number of solutions of its instances varies over a large range, from zero to exponentially many. It is therefore natural to ask if the inherentExpand
Simple Local Search Problems That are Hard to Solve
TLDR
It is shown here that several natural, simple local search problems are PLS-complete, and thus just as hard. Expand
An efficient approximation scheme for the one-dimensional bin-packing problem
  • N. Karmarkar, R. Karp
  • Mathematics, Computer Science
  • 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
  • 1982
TLDR
It is proved that the LP relaxation of bin packing, which was solved efficiently in practice by Gilmore and Gomory, has membership in P, despite the fact that it has an astronomically large number of variables. Expand
How easy is local search?
TLDR
A natural class PLS is defined consisting essentially of those local search problems for which local optimality can be verified in polynomial time, and it is shown that there are complete problems for this class. Expand
On Restricting the Access to an NP-Oracle
TLDR
Polynomial time machines having restricted access to an NP oracle are investigated, finding that the class PNP[O(log n)] can be characterized in very different ways. Expand
The Complexity of Computing the Permanent
  • L. Valiant
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1979
Abstract It is shown that the permanent function of (0, 1)-matrices is a complete problem for the class of counting problems associated with nondeterministic polynomial time computations. RelatedExpand
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