1,528 research outputs found
QIP = PSPACE
We prove that the complexity class QIP, which consists of all problems having
quantum interactive proof systems, is contained in PSPACE. This containment is
proved by applying a parallelized form of the matrix multiplicative weights
update method to a class of semidefinite programs that captures the
computational power of quantum interactive proofs. As the containment of PSPACE
in QIP follows immediately from the well-known equality IP = PSPACE, the
equality QIP = PSPACE follows.Comment: 21 pages; v2 includes corrections and minor revision
Fine-grained Complexity Meets IP = PSPACE
In this paper we study the fine-grained complexity of finding exact and
approximate solutions to problems in P. Our main contribution is showing
reductions from exact to approximate solution for a host of such problems.
As one (notable) example, we show that the Closest-LCS-Pair problem (Given
two sets of strings and , compute exactly the maximum with ) is equivalent to its approximation version
(under near-linear time reductions, and with a constant approximation factor).
More generally, we identify a class of problems, which we call BP-Pair-Class,
comprising both exact and approximate solutions, and show that they are all
equivalent under near-linear time reductions.
Exploring this class and its properties, we also show:
Under the NC-SETH assumption (a significantly more relaxed
assumption than SETH), solving any of the problems in this class requires
essentially quadratic time.
Modest improvements on the running time of known algorithms
(shaving log factors) would imply that NEXP is not in non-uniform
.
Finally, we leverage our techniques to show new barriers for
deterministic approximation algorithms for LCS.
At the heart of these new results is a deep connection between interactive
proof systems for bounded-space computations and the fine-grained complexity of
exact and approximate solutions to problems in P. In particular, our results
build on the proof techniques from the classical IP = PSPACE result
The subpower membership problem for semigroups
Fix a finite semigroup and let be tuples in a direct
power . The subpower membership problem (SMP) asks whether can be
generated by . If is a finite group, then there is a
folklore algorithm that decides this problem in time polynomial in . For
semigroups this problem always lies in PSPACE. We show that the SMP for a full
transformation semigroup on 3 letters or more is actually PSPACE-complete,
while on 2 letters it is in P. For commutative semigroups, we provide a
dichotomy result: if a commutative semigroup embeds into a direct product
of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P;
otherwise it is NP-complete
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