Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?
Harry Buhrman
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Lance Fortnow
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Michal Kouck
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John D. Rogers
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Nikolay Vereshchagin
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M. Kouck Institute of Mathematics, Academy of Sciences of the Czech Republic
, Prague,
Czech Republic
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L. Fortnow University of Chicago
,
Chicago, USA
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H. Buhrman CWI and University of Amsterdam
,
Amsterdam, The Netherlands
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N. Vereshchagin Lomonosov Moscow State University
,
Moscow, Russia
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J.D. Rogers ( ) DePaul University
,
Chicago, USA
The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued function in deterministic polynomial-time we mean on every input producing one of the possible values of the function on that input. We give a relativized negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomial-time hierarchy is infinite. We also show that relative to this same oracle, P =UP and TFNPNP functions are not computable in polynomial-time with an NP oracle.
1 Introduction
Many problems studied in complexity theory are NP decision problems: for a
polynomial time computable binary relation R(x, y) and a polynomial p, given any string
x find out whether there is a string y of length at most p(|x|) such that R(x, y) holds.
Usually, we are interested not only in finding out whether there is such y but also
in finding such y in the case it exists. Problems of this kind are called NP search
problems.
If an algorithm solves an NP search problem then it certainly solves the
corresponding decision problem, but not the other way around. However, for every NP
search problem S it is easy to construct an NP decision problem D which S reduces
to, using a binary search. Thus every NP complete decision problem is equivalent to
the corresponding NP search problem. Under the assumption that for example double
exponential time does not equal Nondeterministic double exponential time it is easy
to show that there are search problems in NP that are harder than the corresponding
search problem [2].
The situation differs however if we are given the promise that for every x there is
such a y. The class of such search problems was defined by Megiddo and
Papadimitriou [10] and called TFNP, the abbreviation reads Total Functions in NP.
A (multi-valued) function from TFNP is specified by a polynomial time computable
binary relation R(x, y) and a polynomial p such that for every string x there is a
string y of length at most p(|x|) such that R(x, y) holds. It maps x to the set of ys
of length at most p(|x|) such that R(x, y) holds, where in the sequel by a value of
the function on x we mean any of the ys. To compute a TFNP function will mean to
solve the corresponding search problem: given a x find a y of length at most p(|x|)
such that R(x, y) holds. This class of problems includes Factoring, finding a Nash
Equilibrium, finding solutions of Sperners Lemma, finding solutions to Ramsey
theorem, and finding collisions of hash functions.
Note that NP decision problems corresponding to TFNP search problems are
always trivial. On the other hand, TFNP search problems might be hard, as the above
examples show.
Fenner, Fortnow, Naik and Rogers [3] consider the hypothesis, which they called
Q, that for every function in TFNP there is a polynomial-time procedure that will
output a value of that function. That is, Proposition Q states that for every R and
p defining a TFNP-function there is a polynomial time computable function f such
that R(x, f (x)) holds for all x.
If some of the above listed problems cannot be solved in polynomial time then
Proposition Q is false. As all those problems seem to be hard, it is plausible that
Proposition Q is false. In this paper we address the following questions: is there
any evidence that Proposition Q is false, and how does Proposition Q relate to other
similar hypotheses in Complexity Theory?
Fenner et al. showed that Proposition Q is equivalent to a number of different
hypotheses including:
Given an NP machine M with L(M) = , there is a polynomial-time computable
function f such that f (x) is an accepting computation of M(x).
Given an honest onto polynomial-time computable function g there is a
polynomial-time computable function f such that g(f (x)) = x. (A function g(x) is
called honest if there is a polynomial p(n) such that |x| p(|g(x)|) for all x.)
For all polynomial-time computable subsets S of SAT there is a polynomial-time
computable function f such that for all in S, f () is a satisfying assignment
to .
For all NP machines M such that L(M) = SAT, there is a polynomial-time
computable function f such that for every in SAT and accepting path c of M(),
f (, c) is a satisfying assignment of .
Here are (...truncated)