In this paper, we introduce a new decision problem associated with lattices, named the Exact Length Vector Problem (ELVP), and prove the NP-completeness of ELVP in the ∞ norm. Moreover, we define two variants of ELVP. The one is a binary variant of ELVP, named the Binary Exact Length Vector Problem (BELVP), and is shown to be NPcomplete in any p norm (1 ≤ p ≤ ∞). The other is a nonnegative vari...

Introduction 0.1 This is a continuously updated catalog of approximability results for NP optimization problems. The compendium will be printed as an appendix of the book Ausiello et al., 1998 that will be published in the end of 1998 or beginning of 1999. You can use WWW forms to report new problems, new results on existing problems or errors. Due to the fact that no NP-complete problem can be...

Using the probability theory-based approach, this paper reveals the equivalence of an arbitrary NPcomplete problem to a problem of checking whether a level set of a specifically constructed harmonic cost function (with all diagonal entries of its Hessian matrix equal to zero) intersects with a unit hypercube in many-dimensional Euclidean space. This connection suggests the possibility that meth...

We perform a mathematical analysis of the classical computational complexity of two genuine quantum-mechanical problems, which are inspired in the calculation of the expected magnetizations and the entanglement between subsystems for a quantum spin system. These problems, which we respectively call SES and SESSP, are specified in terms of pure slightly-entangled quantum states of n qubits, and ...

This 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 ch...

We consider an NP -complete problem SAT01 having a range of remarkable properties. First, it is equivalent to the weighted independent set problem on a graph Γ with vertex weights w, where the required independent set weight equals the maximum possible κ(Γ, w) value m, and hence is decidable by computation of the weighted Lovász number unless α(Γ, w) < θ(Γ, w) = κ(Γ, w) = m. Second, it admits a...

There exist a number of problem classes for which obtaining the exact solution becomes exponentially expensive with increasing problem size. The quadratic assignment problem (QAP) or the travelling salesman problem (TSP) are just two examples of such NP-hard problems. In practice, approximate algorithms are employed to obtain a suboptimal solution, where one must face a trade-off between comput...

An example of a cellular automat on (CA) is given in which the following problems are NP -complete: (i) determining if a given subconfigurati cn s can be gener ated af ter [s] t ime steps, (ii) determining if a given eubccnfigurat ion 8 will rec ur afte r lsi t ime steps, (iii) determining if a given te mporal sequence of state s 8 can be generated in lsi time steps. It is also found that the C...