602 research outputs found
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians
The canonical problem for the class Quantum Merlin-Arthur (QMA) is that of
estimating ground state energies of local Hamiltonians. Perhaps surprisingly,
[Ambainis, CCC 2014] showed that the related, but arguably more natural,
problem of simulating local measurements on ground states of local Hamiltonians
(APX-SIM) is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that
APX-SIM is P^QMA[log]-complete, for P^QMA[log] the class of languages decidable
by a P machine making a logarithmic number of adaptive queries to a QMA oracle.
In this work, we show that APX-SIM is P^QMA[log]-complete even when restricted
to more physical Hamiltonians, obtaining as intermediate steps a variety of
related complexity-theoretic results.
We first give a sequence of results which together yield P^QMA[log]-hardness
for APX-SIM on well-motivated Hamiltonians: (1) We show that for NP, StoqMA,
and QMA oracles, a logarithmic number of adaptive queries is equivalent to
polynomially many parallel queries. These equalities simplify the proofs of our
subsequent results. (2) Next, we show that the hardness of APX-SIM is preserved
under Hamiltonian simulations (a la [Cubitt, Montanaro, Piddock, 2017]). As a
byproduct, we obtain a full complexity classification of APX-SIM, showing it is
complete for P, P^||NP, P^||StoqMA, or P^||QMA depending on the Hamiltonians
employed. (3) Leveraging the above, we show that APX-SIM is P^QMA[log]-complete
for any family of Hamiltonians which can efficiently simulate spatially sparse
Hamiltonians, including physically motivated models such as the 2D Heisenberg
model.
Our second focus considers 1D systems: We show that APX-SIM remains
P^QMA[log]-complete even for local Hamiltonians on a 1D line of 8-dimensional
qudits. This uses a number of ideas from above, along with replacing the "query
Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.Comment: 38 pages, 3 figure
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
An overview on polynomial approximation of NP-hard problems
The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
The next few years will be exciting as prototype universal quantum processors
emerge, enabling implementation of a wider variety of algorithms. Of particular
interest are quantum heuristics, which require experimentation on quantum
hardware for their evaluation, and which have the potential to significantly
expand the breadth of quantum computing applications. A leading candidate is
Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates
between applying a cost-function-based Hamiltonian and a mixing Hamiltonian.
Here, we extend this framework to allow alternation between more general
families of operators. The essence of this extension, the Quantum Alternating
Operator Ansatz, is the consideration of general parametrized families of
unitaries rather than only those corresponding to the time-evolution under a
fixed local Hamiltonian for a time specified by the parameter. This ansatz
supports the representation of a larger, and potentially more useful, set of
states than the original formulation, with potential long-term impact on a
broad array of application areas. For cases that call for mixing only within a
desired subspace, refocusing on unitaries rather than Hamiltonians enables more
efficiently implementable mixers than was possible in the original framework.
Such mixers are particularly useful for optimization problems with hard
constraints that must always be satisfied, defining a feasible subspace, and
soft constraints whose violation we wish to minimize. More efficient
implementation enables earlier experimental exploration of an alternating
operator approach to a wide variety of approximate optimization, exact
optimization, and sampling problems. Here, we introduce the Quantum Alternating
Operator Ansatz, lay out design criteria for mixing operators, detail mappings
for eight problems, and provide brief descriptions of mappings for diverse
problems.Comment: 51 pages, 2 figures. Revised to match journal pape
Approximate solution of NP optimization problems
AbstractThis paper presents the main results obtained in the field of approximation algorithms in a unified framework. Most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes, that is, combinatorial properties of problems and approximation preserving reducibilities. In particular, after reviewing the most important combinatorial characterizations of the classes PTAS and FPTAS, we concentrate on the class APX and, as a concluding result, we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial-time approximation preserving reducibility
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