The Operators min and max on the Polynomial Hierarchy

@article{Hempel1997TheOM,
  title={The Operators min and max on the Polynomial Hierarchy},
  author={H. Hempel and G. Wechsung},
  journal={Electron. Colloquium Comput. Complex.},
  year={1997},
  volume={4}
}
  • H. Hempel, G. Wechsung
  • Published 1997
  • Computer Science, Mathematics
  • Electron. Colloquium Comput. Complex.
Starting from Krentel's class OptP [Kre88] we define a general maximization operator max and a general minimization operator min for complexity classes and show that there are other interesting optimization classes beside OptP. We investigate the behavior of these operators on the polynomial hierarchy, in particular we study the inclusion structure of the classes max · P, max · NP, max · coNP, min · P, min · NP and min · coNP. Furthermore we prove some very powerful relations regarding the… Expand
The Operators minCh and maxCh on the Polynomial Hierarchy
TLDR
A new acceptance concept for nondeterministic Turing machines with output device which allows a characterization of the complexity class Θ2p = PNP[log] as a polynomial time bounded class and two operators, the so called maxCh- and minCh- operator, respectively which are special types of optimization operators. Expand
On Cluster Machines and Function Classes
We consider a special kind of non-deterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalenceExpand
Uniformly Defining Complexity Classes of Functions
TLDR
A reducibility notion between such families leads to a necessary and sufficient criterion for relativizable inclusion between function classes, which is easily applicable and gets as a consequence e.g. that there are oracles A, B, such that min.PA \(\nsubseteq\) #·NPA, and max.NPB c#·coNPB are known to follow from the corresponding positive inclusions. Expand
Optimization of Unary Costs
We investigate the computational complexity of optimal solutions, when the costs can be bounded by a polynomial, i.e. can be unary coded. Here, we revisit the computation problem OptPlog n] ,Expand
The Complexity of Computing the Size of an Interval
TLDR
The function #DIV that counts the nontrivial divisors of natural numbers is studied, and it is shown that #DIV is the interval size function of a polynomial-time decidable partial order with polynometric-time adjacency checks if and only if primality is inPolynomial time. Expand
Uniform Characterizations of Complexity Classes of Functions
TLDR
A reducibility notion between evaluation schemes leads to a necessary and sufficient criterion for relativizable inclusion between function classes, which is easily applicable and gets as a consequence, e.g., that there is an oracle A, such that min·PA⊈#·NPA (note that no structural consequences are known to follow from the corresponding positive inclusion). Expand
The Complexity of Solving Multiobjective Optimization Problems and its Relation to Multivalued Functions
TLDR
The complexity of multiobjective problems is in general not expressible in terms of sets, and it follows that certain solution notions are not equivalent to NP and NPMVg. Expand
On the connection between interval size functions and path counting
TLDR
This work provides inclusion and separation relations between TotP and interval size counting classes, and defines a new class of interval size functions which strictly contains FP and is strictly contained in TotP under reasonable complexity-theoretic assumptions, and shows that this new class contains hard counting problems. Expand
On Higher Arthur-Merlin Classes
TLDR
A hierarchy theorem is proved for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy is proved. Expand
Acceptor-Definable Counting Classes
TLDR
It is proved that RAP and LAP are are equivalent under the Cook[1] sense with #P and TotP, which implies that all these classes are equally powerful when used as oracles to a polynomial computation, even if only one query is allowed. Expand
...
1
2
...

References

SHOWING 1-10 OF 24 REFERENCES
Generalizations of Opt P to the Polynomial Hierarchy
  • M. Krentel
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1992
TLDR
This paper shows that Opt P has a natural extension to the polynomial hierarchy by considering alternating Turing machines with the max and min operators, and shows an equivalence between k alternations ofmax and min and functions computable with an oracle for the k th level in the poynomial hierarchy. Expand
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 Complexity of Selecting Maximal Solutions
TLDR
This work studies maximality problems from the complexity point of view, and gives characterizations of coNP, DP, ?P2, FPNP||, FNP//OptP log n] and FP?P||2 in terms of subclasses of maximality Problems. Expand
More Complicated Questions About Maxima and Minima, and Some Closures of NP
  • K. Wagner
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1987
TLDR
It is shown that problems defined by more complicated questions about maxima and minima are complete in certain subclasses of the Boolean closure of NP and other classes in the interesting area below the class Δ p 2 of the polynomial-time hierarchy. Expand
Uniformly Defining Complexity Classes of Functions
TLDR
A reducibility notion between such families leads to a necessary and sufficient criterion for relativizable inclusion between function classes, which is easily applicable and gets as a consequence e.g. that there are oracles A, B, such that min.PA \(\nsubseteq\) #·NPA, and max.NPB c#·coNPB are known to follow from the corresponding positive inclusions. Expand
Complexity Classes of Optimization Functions
TLDR
It is shown how these operators translate closure properties from one class to another, how they relate operators on classes of functions and classes of sets, and how they encode classes of maximum, minimum, or median functions into well-studiedclasses of sets. 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
A complexity theory for feasible closure properties
The study of the complexity of sets encompasses two complementary aims: (1) establishing—usually via explicit construction of algorithms-that sets are feasible, and (2) studying the relativeExpand
The Complexity of Finding Middle Elements
TLDR
Two related classes of functions that yield the middle element in the ordered sequence of output values of nondeterministic polynomial time Turing machines are defined and the inclusion structure between these classes is determined. Expand
Some Observations on the Connection Between Counting an Recursion
  • K. Wagner
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1986
TLDR
It turns out that, from the recursion-theoretic point of view, PHCF is an analogue to Kalmar's class E of elementary functions, to the class Pspace of polynomial-space computable functions as well as to theclass P of poynomial-time computable function. Expand
...
1
2
3
...