275 research outputs found
Upper and Lower Bounds for Weak Backdoor Set Detection
We obtain upper and lower bounds for running times of exponential time
algorithms for the detection of weak backdoor sets of 3CNF formulas,
considering various base classes. These results include (omitting polynomial
factors), (i) a 4.54^k algorithm to detect whether there is a weak backdoor set
of at most k variables into the class of Horn formulas; (ii) a 2.27^k algorithm
to detect whether there is a weak backdoor set of at most k variables into the
class of Krom formulas. These bounds improve an earlier known bound of 6^k. We
also prove a 2^k lower bound for these problems, subject to the Strong
Exponential Time Hypothesis.Comment: A short version will appear in the proceedings of the 16th
International Conference on Theory and Applications of Satisfiability Testin
On the Equivalence among Problems of Bounded Width
In this paper, we introduce a methodology, called decomposition-based
reductions, for showing the equivalence among various problems of
bounded-width.
First, we show that the following are equivalent for any :
* SAT can be solved in time,
* 3-SAT can be solved in time,
* Max 2-SAT can be solved in time,
* Independent Set can be solved in time, and
* Independent Set can be solved in time, where
tw and cw are the tree-width and clique-width of the instance, respectively.
Then, we introduce a new parameterized complexity class EPNL, which includes
Set Cover and Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and
Independent Set parameterized by path-width are EPNL-complete. This implies
that if one of these EPNL-complete problems can be solved in time,
then any problem in EPNL can be solved in time.Comment: accepted to ESA 201
Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
Obtaining lower bounds for NP-hard problems has for a long time been an
active area of research. Recent algebraic techniques introduced by Jonsson et
al. (SODA 2013) show that the time complexity of the parameterized SAT()
problem correlates to the lattice of strong partial clones. With this ordering
they isolated a relation such that SAT() can be solved at least as fast
as any other NP-hard SAT() problem. In this paper we extend this method
and show that such languages also exist for the max ones problem
(MaxOnes()) and the Boolean valued constraint satisfaction problem over
finite-valued constraint languages (VCSP()). With the help of these
languages we relate MaxOnes and VCSP to the exponential time hypothesis in
several different ways.Comment: This is an extended version of Relating the Time Complexity of
Optimization Problems in Light of the Exponential-Time Hypothesis, appearing
in Proceedings of the 39th International Symposium on Mathematical
Foundations of Computer Science MFCS 2014 Budapest, August 25-29, 201
The parameterized complexity of some geometric problems in unbounded dimension
We study the parameterized complexity of the following fundamental geometric
problems with respect to the dimension : i) Given points in \Rd,
compute their minimum enclosing cylinder. ii) Given two -point sets in
\Rd, decide whether they can be separated by two hyperplanes. iii) Given a
system of linear inequalities with variables, find a maximum-size
feasible subsystem. We show that (the decision versions of) all these problems
are W[1]-hard when parameterized by the dimension . %and hence not solvable
in time, for any computable function and constant
%(unless FPT=W[1]). Our reductions also give a -time lower bound
(under the Exponential Time Hypothesis)
On the Complexity of Hilbert Refutations for Partition
Given a set of integers W, the Partition problem determines whether W can be
divided into two disjoint subsets with equal sums. We model the Partition
problem as a system of polynomial equations, and then investigate the
complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a
given set of integers is not partitionable. We provide an explicit construction
of a minimum-degree certificate, and then demonstrate that the Partition
problem is equivalent to the determinant of a carefully constructed matrix
called the partition matrix. In particular, we show that the determinant of the
partition matrix is a polynomial that factors into an iteration over all
possible partitions of W.Comment: Final versio
A Full Characterization of Quantum Advice
We prove the following surprising result: given any quantum state rho on n
qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of
two-qubit interactions), such that any ground state of H can be used to
simulate rho on all quantum circuits of fixed polynomial size. In terms of
complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which
supersedes the previous result of Aaronson that BQP/qpoly is contained in
PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in
power to untrusted quantum advice combined with trusted classical advice.
Proving our main result requires combining a large number of previous tools --
including a result of Alon et al. on learning of real-valued concept classes, a
result of Aaronson on the learnability of quantum states, and a result of
Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new
ones. The main new tool is a so-called majority-certificates lemma, which is
closely related to boosting in machine learning, and which seems likely to find
independent applications. In its simplest version, this lemma says the
following. Given any set S of Boolean functions on n variables, any function f
in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm
in S, such that each fi is the unique function in S compatible with O(log|S|)
input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines
needed to be changed to preserve our results. The revised definition is more
natural and has the same intuitive interpretation. 2. We needed properties of
Local Hamiltonian reductions going beyond those proved in previous works
(whose results we'd misstated). We now prove the needed properties. See p. 6
for more on both point
A PCP Characterization of AM
We introduce a 2-round stochastic constraint-satisfaction problem, and show
that its approximation version is complete for (the promise version of) the
complexity class AM. This gives a `PCP characterization' of AM analogous to the
PCP Theorem for NP. Similar characterizations have been given for higher levels
of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the
result for AM might be of particular significance for attempts to derandomize
this class.
To test this notion, we pose some `Randomized Optimization Hypotheses'
related to our stochastic CSPs that (in light of our result) would imply
collapse results for AM. Unfortunately, the hypotheses appear over-strong, and
we present evidence against them. In the process we show that, if some language
in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there
exist hard-on-average optimization problems of a particularly elegant form.
All our proofs use a powerful form of PCPs known as Probabilistically
Checkable Proofs of Proximity, and demonstrate their versatility. We also use
known results on randomness-efficient soundness- and hardness-amplification. In
particular, we make essential use of the Impagliazzo-Wigderson generator; our
analysis relies on a recent Chernoff-type theorem for expander walks.Comment: 18 page
Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations
In this article we focus on the parameterized complexity of the
Multidimensional Binary Vector Assignment problem (called \BVA). An input of
this problem is defined by disjoint sets , each
composed of binary vectors of size . An output is a set of disjoint
-tuples of vectors, where each -tuple is obtained by picking one vector
from each set . To each -tuple we associate a dimensional vector by
applying the bit-wise AND operation on the vectors of the tuple. The
objective is to minimize the total number of zeros in these vectors. mBVA
can be seen as a variant of multidimensional matching where hyperedges are
implicitly locally encoded via labels attached to vertices, but was originally
introduced in the context of integrated circuit manufacturing.
We provide for this problem FPT algorithms and negative results (-based
results, [2]-hardness and a kernel lower bound) according to several
parameters: the standard parameter i.e. the total number of zeros), as well
as two parameters above some guaranteed values.Comment: 16 pages, 6 figure
Spatially resolved spectroscopy of monolayer graphene on SiO2
We have carried out scanning tunneling spectroscopy measurements on
exfoliated monolayer graphene on SiO to probe the correlation between its
electronic and structural properties. Maps of the local density of states are
characterized by electron and hole puddles that arise due to long range
intravalley scattering from intrinsic ripples in graphene and random charged
impurities. At low energy, we observe short range intervalley scattering which
we attribute to lattice defects. Our results demonstrate that the electronic
properties of graphene are influenced by intrinsic ripples, defects and the
underlying SiO substrate.Comment: 6 pages, 7 figures, extended versio
Polynomial kernelization for removing induced claws and diamonds
A graph is called (claw,diamond)-free if it contains neither a claw (a
) nor a diamond (a with an edge removed) as an induced subgraph.
Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of
triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex
is in at most two maximal cliques and every edge is in exactly one maximal
clique.
In this paper we consider the parameterized complexity of the
(claw,diamond)-free Edge Deletion problem, where given a graph and a
parameter , the question is whether one can remove at most edges from
to obtain a (claw,diamond)-free graph. Our main result is that this problem
admits a polynomial kernel. We complement this finding by proving that, even on
instances with maximum degree , the problem is NP-complete and cannot be
solved in time unless the Exponential Time
Hypothesis fai
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