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Strong NP-Hardness Result for Regularized $L_q$-Minimization Problems with Concave Penalty Functions  [PDF]
Dongdong Ge,Zizhuo Wang,Yinyu Ye,Hao Yin
Mathematics , 2015,
Abstract: In this note, we consider the regularize $L_q$-minimization problem ($q\ge 1$) with a general penalty function. We show that if the penalty function is concave but not linear in a neighborhood of zero, then the optimization problem is strongly NP-hard. This result answers the complexity of many regularized optimization problems studied in the literature. It implies that it is impossible to have a fully polynomial-time approximation scheme (FPTAS) for a large class of regularization problems unless P = NP.
The Frobenius Problem in a Free Monoid  [PDF]
Jui-Yi Kao,Jeffrey Shallit,Zhi Xu
Computer Science , 2007,
Abstract: The classical Frobenius problem is to compute the largest number g not representable as a non-negative integer linear combination of non-negative integers x_1, x_2, ..., x_k, where gcd(x_1, x_2, ..., x_k) = 1. In this paper we consider generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for an analogue of g, depending on the particular measure chosen.
NP-hardness of hypercube 2-segmentation  [PDF]
Uriel Feige
Computer Science , 2014,
Abstract: The hypercube 2-segmentation problem is a certain biclustering problem that was previously claimed to be NP-hard, but for which there does not appear to be a publicly available proof of NP-hardness. This manuscript provides such a proof.
On Percolation and $NP$-Hardness  [PDF]
Daniel Reichman,Igor Shinkar
Computer Science , 2015,
Abstract: We consider the robustness of computational hardness of problems whose input is obtained by applying independent random deletions to worst-case instances. For some classical $NP$-hard problems on graphs, such as Coloring, Vertex-Cover, and Hamiltonicity, we examine the complexity of these problems when edges (or vertices) of an arbitrary graph are deleted independently with probability $1-p > 0$. We prove that for $n$-vertex graphs, these problems remain as hard as in the worst-case, as long as $p > \frac{1}{n^{1-\epsilon}}$ for arbitrary $\epsilon \in (0,1)$, unless $NP \subseteq BPP$. We also prove hardness results for Constraint Satisfaction Problems, where random deletions are applied to clauses or variables, as well as the Subset-Sum problem, where items of a given instance are deleted at random.
Strong NP-hardness of AC power flows feasibility  [PDF]
Daniel Bienstock,Abhinav Verma
Mathematics , 2015,
Abstract: We present a rigorous proof of strong NP-hardness of the AC-OPF problem.
Strong NP-Hardness of the Quantum Separability Problem  [PDF]
Sevag Gharibian
Physics , 2008,
Abstract: Given the density matrix rho of a bipartite quantum state, the quantum separability problem asks whether rho is entangled or separable. In 2003, Gurvits showed that this problem is NP-hard if rho is located within an inverse exponential (with respect to dimension) distance from the border of the set of separable quantum states. In this paper, we extend this NP-hardness to an inverse polynomial distance from the separable set. The result follows from a simple combination of works by Gurvits, Ioannou, and Liu. We apply our result to show (1) an immediate lower bound on the maximum distance between a bound entangled state and the separable set (assuming P != NP), and (2) NP-hardness for the problem of determining whether a completely positive trace-preserving linear map is entanglement-breaking.
An Artificial Neural Networks approach with NP hardness for efficiency approximation
R.Murugadoss,Dr. M.Ramakrishnan
International Journal of Engineering and Technology , 2012,
Abstract: An Artificial Neural Networks approach with NP hardness for efficiency approximation is presented in this paper. The aim of present research is to show that finding the solution and efficiency approximation to the solution of the global optimization problem using NP-hard problem. NP hardness is used as the training algorithm of an artificial neural networks approach for calculating the efficiency approximation. In the ANN initial condition for the input–hidden connection, the left sigmoid function is used as an activation function and for the hidden–output inter connection, the right sigmoid function is used as an activation function.
NP-hardness of polytope M-matrix testing and related problems  [PDF]
Nikos Vlassis
Mathematics , 2012,
Abstract: In this note we prove NP-hardness of the following problem: Given a set of matrices, is there a convex combination of those that is a nonsingular M-matrix? Via known characterizations of M-matrices, our result establishes NP-hardness of several fundamental problems in systems analysis and control, such as testing the instability of an uncertain dynamical system, and minimizing the spectral radius of an affine matrix function.
NP-hardness of the cluster minimization problem revisited  [PDF]
A. B. Adib
Computer Science , 2005, DOI: 10.1088/0305-4470/38/40/001
Abstract: 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.
Raising a Hardness Result  [PDF]
Paolo Liberatore
Computer Science , 2007,
Abstract: This article presents a technique for proving problems hard for classes of the polynomial hierarchy or for PSPACE. The rationale of this technique is that some problem restrictions are able to simulate existential or universal quantifiers. If this is the case, reductions from Quantified Boolean Formulae (QBF) to these restrictions can be transformed into reductions from QBFs having one more quantifier in the front. This means that a proof of hardness of a problem at level n in the polynomial hierarchy can be split into n separate proofs, which may be simpler than a proof directly showing a reduction from a class of QBFs to the considered problem.
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