2,275 research outputs found
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
Computational Difficulty of Global Variations in the Density Matrix Renormalization Group
The density matrix renormalization group (DMRG) approach is arguably the most
successful method to numerically find ground states of quantum spin chains. It
amounts to iteratively locally optimizing matrix-product states, aiming at
better and better approximating the true ground state. To date, both a proof of
convergence to the globally best approximation and an assessment of its
complexity are lacking. Here we establish a result on the computational
complexity of an approximation with matrix-product states: The surprising
result is that when one globally optimizes over several sites of local
Hamiltonians, avoiding local optima, one encounters in the worst case a
computationally difficult NP-hard problem (hard even in approximation). The
proof exploits a novel way of relating it to binary quadratic programming. We
discuss intriguing ramifications on the difficulty of describing quantum
many-body systems.Comment: 5 pages, 1 figure, RevTeX, final versio
Finding Multiple New Optimal Locations in a Road Network
We study the problem of optimal location querying for location based services
in road networks, which aims to find locations for new servers or facilities.
The existing optimal solutions on this problem consider only the cases with one
new server. When two or more new servers are to be set up, the problem with
minmax cost criteria, MinMax, becomes NP-hard. In this work we identify some
useful properties about the potential locations for the new servers, from which
we derive a novel algorithm for MinMax, and show that it is efficient when the
number of new servers is small. When the number of new servers is large, we
propose an efficient 3-approximate algorithm. We verify with experiments on
real road networks that our solutions are effective and attains significantly
better result quality compared to the existing greedy algorithms
Simulation of many-qubit quantum computation with matrix product states
Matrix product states provide a natural entanglement basis to represent a
quantum register and operate quantum gates on it. This scheme can be
materialized to simulate a quantum adiabatic algorithm solving hard instances
of a NP-Complete problem. Errors inherent to truncations of the exact action of
interacting gates are controlled by the size of the matrices in the
representation. The property of finding the right solution for an instance and
the expected value of the energy are found to be remarkably robust against
these errors. As a symbolic example, we simulate the algorithm solving a
100-qubit hard instance, that is, finding the correct product state out of ~
10^30 possibilities. Accumulated statistics for up to 60 qubits point at a slow
growth of the average minimum time to solve hard instances with
highly-truncated simulations of adiabatic quantum evolution.Comment: 5 pages, 4 figures, final versio
Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries
Using exact computations we study the classical hard-core monomer-dimer
models on m x n plane lattice strips with free boundaries. For an arbitrary
number v of monomers (or vacancies), we found a logarithmic correction term in
the finite-size correction of the free energy. The coefficient of the
logarithmic correction term depends on the number of monomers present (v) and
the parity of the width n of the lattice strip: the coefficient equals to v
when n is odd, and v/2 when n is even. The results are generalizations of the
previous results for a single monomer in an otherwise fully packed lattice of
dimers.Comment: 4 pages, 2 figure
Travelling Salesman Problem with a Center
We study a travelling salesman problem where the path is optimized with a
cost function that includes its length as well as a certain measure of
its distance from the geometrical center of the graph. Using simulated
annealing (SA) we show that such a problem has a transition point that
separates two phases differing in the scaling behaviour of and , in
efficiency of SA, and in the shape of minimal paths.Comment: 4 pages, minor changes, accepted in Phys.Rev.
Geometries for universal quantum computation with matchgates
Matchgates are a group of two-qubit gates associated with free fermions. They
are classically simulatable if restricted to act between nearest neighbors on a
one-dimensional chain, but become universal for quantum computation with
longer-range interactions. We describe various alternative geometries with
nearest-neighbor interactions that result in universal quantum computation with
matchgates only, including subtle departures from the chain. Our results pave
the way for new quantum computer architectures that rely solely on the simple
interactions associated with matchgates.Comment: 6 pages, 4 figures. Updated version includes an appendix extending
one of the result
Jarzynski Equality for an Energy-Controlled System
The Jarzynski equality (JE) is known as an exact identity for nonequillibrium
systems. The JE was originally formulated for isolated and isothermal systems,
while Adib reported an JE extended to an isoenergetic process. In this paper,
we extend the JE to an energy-controlled system. We make it possible to control
the instantaneous value of the energy arbitrarily in a nonequilibrium process.
Under our extension, the new JE is more practical and useful to calculate the
number of states and the entropy than the isoenergetic one. We also show
application of our JE to a kind of optimization problems.Comment: 6 pages, 1 figur
On the complexity of computing the -restricted edge-connectivity of a graph
The \emph{-restricted edge-connectivity} of a graph , denoted by
, is defined as the minimum size of an edge set whose removal
leaves exactly two connected components each containing at least vertices.
This graph invariant, which can be seen as a generalization of a minimum
edge-cut, has been extensively studied from a combinatorial point of view.
However, very little is known about the complexity of computing .
Very recently, in the parameterized complexity community the notion of
\emph{good edge separation} of a graph has been defined, which happens to be
essentially the same as the -restricted edge-connectivity. Motivated by the
relevance of this invariant from both combinatorial and algorithmic points of
view, in this article we initiate a systematic study of its computational
complexity, with special emphasis on its parameterized complexity for several
choices of the parameters. We provide a number of NP-hardness and W[1]-hardness
results, as well as FPT-algorithms.Comment: 16 pages, 4 figure
Phase Transition in the Number Partitioning Problem
Number partitioning is an NP-complete problem of combinatorial optimization.
A statistical mechanics analysis reveals the existence of a phase transition
that separates the easy from the hard to solve instances and that reflects the
pseudo-polynomiality of number partitioning. The phase diagram and the value of
the typical ground state energy are calculated.Comment: minor changes (references, typos and discussion of results
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