2,204 research outputs found
Hardness of approximation for quantum problems
The polynomial hierarchy plays a central role in classical complexity theory.
Here, we define a quantum generalization of the polynomial hierarchy, and
initiate its study. We show that not only are there natural complete problems
for the second level of this quantum hierarchy, but that these problems are in
fact hard to approximate. Using these techniques, we also obtain hardness of
approximation for the class QCMA. Our approach is based on the use of
dispersers, and is inspired by the classical results of Umans regarding
hardness of approximation for the second level of the classical polynomial
hierarchy [Umans, FOCS 1999]. The problems for which we prove hardness of
approximation for include, among others, a quantum version of the Succinct Set
Cover problem, and a variant of the local Hamiltonian problem with hybrid
classical-quantum ground states.Comment: 21 pages, 1 figure, extended abstract appeared in Proceedings of the
39th International Colloquium on Automata, Languages and Programming (ICALP),
pages 387-398, Springer, 201
The Hardness of Approximation of Euclidean k-means
The Euclidean -means problem is a classical problem that has been
extensively studied in the theoretical computer science, machine learning and
the computational geometry communities. In this problem, we are given a set of
points in Euclidean space , and the goal is to choose centers in
so that the sum of squared distances of each point to its nearest center
is minimized. The best approximation algorithms for this problem include a
polynomial time constant factor approximation for general and a
-approximation which runs in time . At
the other extreme, the only known computational complexity result for this
problem is NP-hardness [ADHP'09]. The main difficulty in obtaining hardness
results stems from the Euclidean nature of the problem, and the fact that any
point in can be a potential center. This gap in understanding left open
the intriguing possibility that the problem might admit a PTAS for all .
In this paper we provide the first hardness of approximation for the
Euclidean -means problem. Concretely, we show that there exists a constant
such that it is NP-hard to approximate the -means objective
to within a factor of . We show this via an efficient reduction
from the vertex cover problem on triangle-free graphs: given a triangle-free
graph, the goal is to choose the fewest number of vertices which are incident
on all the edges. Additionally, we give a proof that the current best hardness
results for vertex cover can be carried over to triangle-free graphs. To show
this we transform , a known hard vertex cover instance, by taking a graph
product with a suitably chosen graph , and showing that the size of the
(normalized) maximum independent set is almost exactly preserved in the product
graph using a spectral analysis, which might be of independent interest
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Hardness of Approximation for Morse Matching
Discrete Morse theory has emerged as a powerful tool for a wide range of
problems, including the computation of (persistent) homology. In this context,
discrete Morse theory is used to reduce the problem of computing a topological
invariant of an input simplicial complex to computing the same topological
invariant of a (significantly smaller) collapsed cell or chain complex.
Consequently, devising methods for obtaining gradient vector fields on
complexes to reduce the size of the problem instance has become an emerging
theme over the last decade. While computing the optimal gradient vector field
on a simplicial complex is NP-hard, several heuristics have been observed to
compute near-optimal gradient vector fields on a wide variety of datasets.
Understanding the theoretical limits of these strategies is therefore a
fundamental problem in computational topology. In this paper, we consider the
approximability of maximization and minimization variants of the Morse matching
problem, posed as open problems by Joswig and Pfetsch. We establish hardness
results for Max-Morse matching and Min-Morse matching. In particular, we show
that, for a simplicial complex with n simplices and dimension , it is
NP-hard to approximate Min-Morse matching within a factor of
, for any . Moreover, using an L-reduction
from Degree 3 Max-Acyclic Subgraph to Max-Morse matching, we show that it is
both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial
complexes of dimension within certain explicit constant factors.Comment: 20 pages, 1 figur
- …