12,241 research outputs found
Quantum proofs can be verified using only single qubit measurements
QMA (Quantum Merlin Arthur) is the class of problems which, though
potentially hard to solve, have a quantum solution which can be verified
efficiently using a quantum computer. It thus forms a natural quantum version
of the classical complexity class NP (and its probabilistic variant MA,
Merlin-Arthur games), where the verifier has only classical computational
resources. In this paper, we study what happens when we restrict the quantum
resources of the verifier to the bare minimum: individual measurements on
single qubits received as they come, one-by-one. We find that despite this
grave restriction, it is still possible to soundly verify any problem in QMA
for the verifier with the minimum quantum resources possible, without using any
quantum memory or multiqubit operations. We provide two independent proofs of
this fact, based on measurement based quantum computation and the local
Hamiltonian problem, respectively. The former construction also applies to
QMA, i.e., QMA with one-sided error.Comment: 7 pages, 1 figur
Network correlated data gathering with explicit communication: NP-completeness and algorithms
We consider the problem of correlated data gathering by a network with a sink node and a tree-based communication structure, where the goal is to minimize the total transmission cost of transporting the information collected by the nodes, to the sink node. For source coding of correlated data, we consider a joint entropy-based coding model with explicit communication where coding is simple and the transmission structure optimization is difficult. We first formulate the optimization problem definition in the general case and then we study further a network setting where the entropy conditioning at nodes does not depend on the amount of side information, but only on its availability. We prove that even in this simple case, the optimization problem is NP-hard. We propose some efficient, scalable, and distributed heuristic approximation algorithms for solving this problem and show by numerical simulations that the total transmission cost can be significantly improved over direct transmission or the shortest path tree. We also present an approximation algorithm that provides a tree transmission structure with total cost within a constant factor from the optimal
Polynomial Observables in the Graph Partitioning Problem
Although NP-Complete problems are the most difficult decisional problems, it
is possible to discover in them polynomial (or easy) observables. We study the
Graph Partitioning Problem showing that it is possible to recognize in it two
correlated polynomial observables. The particular behaviour of one of them with
respect to the connectivity of the graph suggests the presence of a phase
transition in partitionability.Comment: 7 pages, 2 figure
Complexity, parallel computation and statistical physics
The intuition that a long history is required for the emergence of complexity
in natural systems is formalized using the notion of depth. The depth of a
system is defined in terms of the number of parallel computational steps needed
to simulate it. Depth provides an objective, irreducible measure of history
applicable to systems of the kind studied in statistical physics. It is argued
that physical complexity cannot occur in the absence of substantial depth and
that depth is a useful proxy for physical complexity. The ideas are illustrated
for a variety of systems in statistical physics.Comment: 21 pages, 7 figure
NP-hardness of the cluster minimization problem revisited
The computational complexity of the "cluster minimization problem" is
revisited [L. T. Wille and J. Vennik, J. Phys. A 18, L419 (1985)]. It is argued
that the original NP-hardness proof does not apply to pairwise potentials of
physical interest, such as those that depend on the geometric distance between
the particles. A geometric analog of the original problem is formulated, and a
new proof for such potentials is provided by polynomial time transformation
from the independent set problem for unit disk graphs. Limitations of this
formulation are pointed out, and new subproblems that bear more direct
consequences to the numerical study of clusters are suggested.Comment: 8 pages, 2 figures, accepted to J. Phys. A: Math. and Ge
Palette-colouring: a belief-propagation approach
We consider a variation of the prototype combinatorial-optimisation problem
known as graph-colouring. Our optimisation goal is to colour the vertices of a
graph with a fixed number of colours, in a way to maximise the number of
different colours present in the set of nearest neighbours of each given
vertex. This problem, which we pictorially call "palette-colouring", has been
recently addressed as a basic example of problem arising in the context of
distributed data storage. Even though it has not been proved to be NP complete,
random search algorithms find the problem hard to solve. Heuristics based on a
naive belief propagation algorithm are observed to work quite well in certain
conditions. In this paper, we build upon the mentioned result, working out the
correct belief propagation algorithm, which needs to take into account the
many-body nature of the constraints present in this problem. This method
improves the naive belief propagation approach, at the cost of increased
computational effort. We also investigate the emergence of a satisfiable to
unsatisfiable "phase transition" as a function of the vertex mean degree, for
different ensembles of sparse random graphs in the large size ("thermodynamic")
limit.Comment: 22 pages, 7 figure
Kernelization Lower Bounds By Cross-Composition
We introduce the cross-composition framework for proving kernelization lower
bounds. A classical problem L AND/OR-cross-composes into a parameterized
problem Q if it is possible to efficiently construct an instance of Q with
polynomially bounded parameter value that expresses the logical AND or OR of a
sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008)
and using a result by Fortnow and Santhanam (STOC 2008) with a refinement by
Dell and van Melkebeek (STOC 2010), we show that if an NP-hard problem
OR-cross-composes into a parameterized problem Q then Q does not admit a
polynomial kernel unless NP \subseteq coNP/poly and the polynomial hierarchy
collapses. Similarly, an AND-cross-composition for Q rules out polynomial
kernels for Q under Bodlaender et al.'s AND-distillation conjecture.
Our technique generalizes and strengthens the recent techniques of using
composition algorithms and of transferring the lower bounds via polynomial
parameter transformations. We show its applicability by proving kernelization
lower bounds for a number of important graphs problems with structural
(non-standard) parameterizations, e.g., Clique, Chromatic Number, Weighted
Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial
kernels with respect to the vertex cover number of the input graphs unless the
polynomial hierarchy collapses, contrasting the fact that these problems are
trivially fixed-parameter tractable for this parameter.
After learning of our results, several teams of authors have successfully
applied the cross-composition framework to different parameterized problems.
For completeness, our presentation of the framework includes several extensions
based on this follow-up work. For example, we show how a relaxed version of
OR-cross-compositions may be used to give lower bounds on the degree of the
polynomial in the kernel size.Comment: A preliminary version appeared in the proceedings of the 28th
International Symposium on Theoretical Aspects of Computer Science (STACS
2011) under the title "Cross-Composition: A New Technique for Kernelization
Lower Bounds". Several results have been strengthened compared to the
preliminary version (http://arxiv.org/abs/1011.4224). 29 pages, 2 figure
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