107 research outputs found
On parallel versus sequential approximation
In this paper we deal with the class NCX of NP Optimization problems that are approximable within constant ratio in NC. This class is the parallel counterpart of the class APX. Our main motivation here is to reduce the study of sequential and parallel approximability to the same framework. To this aim, we first introduce a new kind of NC-reduction that preserves the relative error of the approximate solutions and show that the class NCX has {em complete} problems under this reducibility.
An important subset of NCX is the class MAXSNP, we show that MAXSNP-complete problems have a threshold on the parallel approximation ratio that is, there are positive constants , such that although the problem can be approximated in P within it cannot be approximated in NC within epsilon_2$, unless P=NC. This result is attained by showing that the problem of approximating the value obtained through a non-oblivious local search algorithm is P-complete, for some values of the approximation ratio. Finally, we show that approximating through non-oblivious local search is in average NC.Postprint (published version
The approximability of non-Boolean satisfiability problems and restricted integer programming
AbstractIn this paper we present improved approximation algorithms for two classes of maximization problems defined in Barland et al. (J. Comput. System Sci. 57(2) (1998) 144). Our factors of approximation substantially improve the previous known results and are close to the best possible. On the other hand, we show that the approximation results in the framework of Barland et al. hold also in the parallel setting, and thus we have a new common framework for both computational settings. We prove almost tight non-approximability results, thus solving a main open question of Barland et al.We obtain the results through the constraint satisfaction problem over multi-valued domains, for which we develop approximation algorithms and show non-approximability results. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of probabilistically checkable proofs and multi-prover one-round proof systems
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
Parallelism with limited nondeterminism
Computational complexity theory studies which computational problems can be solved with limited access to resources. The past fifty years have seen a focus on the relationship between intractable problems and efficient algorithms. However, the relationship between inherently sequential problems and highly parallel algorithms has not been as well studied. Are there efficient but inherently sequential problems that admit some relaxed form of highly parallel algorithm? In this dissertation, we develop the theory of structural complexity around this relationship for three common types of computational problems.
Specifically, we show tradeoffs between time, nondeterminism, and parallelizability. By clearly defining the notions and complexity classes that capture our intuition for parallelizable and sequential problems, we create a comprehensive framework for rigorously proving parallelizability and non-parallelizability of computational problems. This framework provides the means to prove whether otherwise tractable problems can be effectively parallelized, a need highlighted by the current growth of multiprocessor systems. The views adopted by this dissertation—alternate approaches to solving sequential problems using approximation, limited nondeterminism, and parameterization—can be applied practically throughout computer science
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
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
Polynomial Kernelizations for MIN F^+Pi_1 and MAX NP
The relation of constant-factor approximability to fixed-parameter tractability and kernelization is a long-standing open question. We prove that two large classes of constant-factor approximable problems, namely~ and~, including the well-known subclass~, admit polynomial kernelizations for their natural decision versions. This extends results of Cai and Chen (JCSS 1997), stating that the standard parameterizations of problems in~ and~ are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al. ICALP 2008)
A note on anti-coordination and social interactions
This note confirms a conjecture of [Bramoull\'{e}, Anti-coordination and
social interactions, Games and Economic Behavior, 58, 2007: 30-49]. The
problem, which we name the maximum independent cut problem, is a restricted
version of the MAX-CUT problem, requiring one side of the cut to be an
independent set. We show that the maximum independent cut problem does not
admit any polynomial time algorithm with approximation ratio better than
, where is the number of nodes, and arbitrarily
small, unless P=NP. For the rather special case where each node has a degree of
at most four, the problem is still MAXSNP-hard.Comment: 7 page
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