Terrain Guarding is NP-Hard

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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Optimizing Transmit Sequence and Instrumental Variables Receiver for Dual-Function Complexity System

Complexity ◽

10.1155/2020/4134851 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Yu Yao ◽

Junhui Zhao ◽

Lenan Wu

Keyword(s):

Approximate Solution ◽

Instrumental Variables ◽

Performance Index ◽

Dual Function ◽

Np Hard ◽

Match Filter ◽

Filter Output ◽

Phase Code ◽

Discrete Phase ◽

Simulation Results

This correspondence deals with the joint cognitive design of transmit coded sequences and instrumental variables (IV) receive filter to enhance the performance of a dual-function radar-communication (DFRC) system in the presence of clutter disturbance. The IV receiver can reject clutter more efficiently than the match filter. The signal-to-clutter-and-noise ratio (SCNR) of the IV filter output is viewed as the performance index of the complexity system. We focus on phase only sequences, sharing both a continuous and a discrete phase code and develop optimization algorithms to achieve reasonable pairs of transmit coded sequences and IV receiver that fine approximate the behavior of the optimum SCNR. All iterations involve the solution of NP-hard quadratic fractional problems. The relaxation plus randomization technique is used to find an approximate solution. The complexity, corresponding to the operation of the proposed algorithms, depends on the number of acceptable iterations along with on and the complexity involved in all iterations. Simulation results are offered to evaluate the performance generated by the proposed scheme.

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On the Complexity of Deadlock Recovery

Fundamenta Informaticae ◽

10.3233/fi-1986-9304 ◽

1986 ◽

Vol 9 (3) ◽

pp. 323-342

Author(s):

Joseph Y.-T. Leung ◽

Burkhard Monien

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Arbitrary Number ◽

The Other ◽

Np Hard ◽

Total Cost ◽

Other Hand ◽

Input Parameters ◽

Resource Type

We consider the computational complexity of finding an optimal deadlock recovery. It is known that for an arbitrary number of resource types the problem is NP-hard even when the total cost of deadlocked jobs and the total number of resource units are “small” relative to the number of deadlocked jobs. It is also known that for one resource type the problem is NP-hard when the total cost of deadlocked jobs and the total number of resource units are “large” relative to the number of deadlocked jobs. In this paper we show that for one resource type the problem is solvable in polynomial time when the total cost of deadlocked jobs or the total number of resource units is “small” relative to the number of deadlocked jobs. For fixed m ⩾ 2 resource types, we show that the problem is solvable in polynomial time when the total number of resource units is “small” relative to the number of deadlocked jobs. On the other hand, when the total number of resource units is “large”, the problem becomes NP-hard even when the total cost of deadlocked jobs is “small” relative to the number of deadlocked jobs. The results in the paper, together with previous known ones, give a complete delineation of the complexity of this problem under various assumptions of the input parameters.

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Using Machine Learning for Quantum Annealing Accuracy Prediction

Algorithms ◽

10.3390/a14060187 ◽

2021 ◽

Vol 14 (6) ◽

pp. 187

Author(s):

Aaron Barbosa ◽

Elijah Pelofske ◽

Georg Hahn ◽

Hristo N. Djidjev

Keyword(s):

Machine Learning ◽

Maximum Clique ◽

Classification Model ◽

Maximum Clique Problem ◽

Problem Instance ◽

Np Hard ◽

Machine Learning Classification ◽

Hard Problems ◽

Problem Instances ◽

D Wave

Quantum annealers, such as the device built by D-Wave Systems, Inc., offer a way to compute solutions of NP-hard problems that can be expressed in Ising or quadratic unconstrained binary optimization (QUBO) form. Although such solutions are typically of very high quality, problem instances are usually not solved to optimality due to imperfections of the current generations quantum annealers. In this contribution, we aim to understand some of the factors contributing to the hardness of a problem instance, and to use machine learning models to predict the accuracy of the D-Wave 2000Q annealer for solving specific problems. We focus on the maximum clique problem, a classic NP-hard problem with important applications in network analysis, bioinformatics, and computational chemistry. By training a machine learning classification model on basic problem characteristics such as the number of edges in the graph, or annealing parameters, such as the D-Wave’s chain strength, we are able to rank certain features in the order of their contribution to the solution hardness, and present a simple decision tree which allows to predict whether a problem will be solvable to optimality with the D-Wave 2000Q. We extend these results by training a machine learning regression model that predicts the clique size found by D-Wave.

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Popular Matching in Roommates Setting Is NP-hard

ACM Transactions on Computation Theory ◽

10.1145/3442354 ◽

2021 ◽

Vol 13 (2) ◽

pp. 1-20

Author(s):

Sushmita Gupta ◽

Pranabendu Misra ◽

Saket Saurabh ◽

Meirav Zehavi

Keyword(s):

Computational Complexity ◽

Np Hard ◽

Np Complete ◽

Open Question ◽

Popular Matching

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

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A hybrid metaheuristic for solving asymmetric distance-constrained vehicle routing problem

Computational Social Networks ◽

10.1186/s40649-020-00084-7 ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Ha-Bang Ban ◽

Phuong Khanh Nguyen

Keyword(s):

Vehicle Routing ◽

Vehicle Routing Problem ◽

Natural Extension ◽

Solution Space ◽

Neighborhood Search ◽

Np Hard ◽

Routing Problem ◽

Hybrid Metaheuristic ◽

Asymmetric Distance ◽

Constrained Vehicle Routing

AbstractThe Asymmetric Distance-Constrained Vehicle Routing Problem (ADVRP) is NP-hard as it is a natural extension of the NP-hard Vehicle Routing Problem. In ADVRP problem, each customer is visited exactly once by a vehicle; every tour starts and ends at a depot; and the traveled distance by each vehicle is not allowed to exceed a predetermined limit. We propose a hybrid metaheuristic algorithm combining the Randomized Variable Neighborhood Search (RVNS) and the Tabu Search (TS) to solve the problem. The combination of multiple neighborhoods and tabu mechanism is used for their capacity to escape local optima while exploring the solution space. Furthermore, the intensification and diversification phases are also included to deliver optimized and diversified solutions. Extensive numerical experiments and comparisons with all the state-of-the-art algorithms show that the proposed method is highly competitive in terms of solution quality and computation time, providing new best solutions for a number of instances.

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The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

Journal of Scientific Computing ◽

10.1007/s10915-021-01524-w ◽

2021 ◽

Vol 88 (1) ◽

Author(s):

Antoine Gautier ◽

Matthias Hein ◽

Francesco Tudisco

Keyword(s):

Global Convergence ◽

Convergence Theorem ◽

Matrix Norm ◽

Nonnegative Matrices ◽

Np Hard ◽

Power Method ◽

Contraction Ratio ◽

Matrix Norms ◽

Vector Norms

AbstractWe analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

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Nanolaser-based emulators of spin Hamiltonians

Nanophotonics ◽

10.1515/nanoph-2020-0230 ◽

2020 ◽

Vol 9 (13) ◽

pp. 4193-4198 ◽

Cited By ~ 2

Author(s):

Midya Parto ◽

William E. Hayenga ◽

Alireza Marandi ◽

Demetrios N. Christodoulides ◽

Mercedeh Khajavikhan

Keyword(s):

Large Scale ◽

Optimization Problems ◽

Exchange Interactions ◽

Np Hard ◽

Spin Models ◽

Geometric Frustration ◽

Spin Hamiltonians ◽

Hard Problems ◽

On Chip ◽

Classical Spin Models

AbstractFinding the solution to a large category of optimization problems, known as the NP-hard class, requires an exponentially increasing solution time using conventional computers. Lately, there has been intense efforts to develop alternative computational methods capable of addressing such tasks. In this regard, spin Hamiltonians, which originally arose in describing exchange interactions in magnetic materials, have recently been pursued as a powerful computational tool. Along these lines, it has been shown that solving NP-hard problems can be effectively mapped into finding the ground state of certain types of classical spin models. Here, we show that arrays of metallic nanolasers provide an ultra-compact, on-chip platform capable of implementing spin models, including the classical Ising and XY Hamiltonians. Various regimes of behavior including ferromagnetic, antiferromagnetic, as well as geometric frustration are observed in these structures. Our work paves the way towards nanoscale spin-emulators that enable efficient modeling of large-scale complex networks.

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Dispersing Obnoxious Facilities on a Graph

Algorithmica ◽

10.1007/s00453-021-00800-3 ◽

2021 ◽

Author(s):

Alexander Grigoriev ◽

Tim A. Hartmann ◽

Stefan Lendl ◽

Gerhard J. Woeginger

Keyword(s):

Facility Location ◽

Location Problem ◽

Unit Length ◽

Facility Location Problem ◽

Np Hard ◽

Obnoxious Facilities ◽

Rational Parameter ◽

Polynomially Solvable

AbstractWe study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $$\delta$$ δ from each other. We investigate the complexity of this problem in terms of the rational parameter $$\delta$$ δ . The problem is polynomially solvable, if the numerator of $$\delta$$ δ is 1 or 2, while all other cases turn out to be NP-hard.

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