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Unconditionally energy-stable linear convex splitting algorithm for the L² quasicrystals

COMPUTER PHYSICS COMMUNICATIONS 2024年 295卷

作者： Yang, Junxiang Macau Univ Sci & Technol Fac Innovat Engn Sch Comp Sci & Engn Macau Peoples R China

Quasicrystals have extensive applications in material sciences. In this article, we develop an unconditional energy-dissipation-preserving, temporally second-order accurate, and linear method for solving a L-2-gradient flow-based quasicrystal model. A truncated technique is introduced to regularize the nonlinear terms so that their second derivative is upper bounded. Based on the idea of the convex splitting method, we split free energy into the difference between two convex functionals. The convexity provides an appropriate choice of linear stabilization parameter. A linear and temporally second-order accurate algorithm is constructed utilizing the implicit-explicit Runge-Kutta method. In each time step, we can analytically prove that the proposed scheme satisfies the energy dissipation law. The Fourier spectral method is used to perform spatial discretization. Through numerical experiments, the accuracy, energy stability, and capability of the proposed method for quasicrystals are verified.

关键词： Quasicrystals Unconditional energy stability linear algorithm Fourier spectral method

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A linear algorithm for secure domination in trees

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DISCRETE APPLIED MATHEMATICS 2014年 171卷 15-27页

作者： Burger, A. P. de Villiers, A. P. van Vuuren, J. H. Univ Stellenbosch Dept Logist ZA-7602 Matieland South Africa Univ Stellenbosch Dept Ind Engn ZA-7602 Matieland South Africa

A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X - {v}) U {u} is again a dominating set of G. The secure domination number of G, denoted by gamma(s)(G), is the cardinality of a smallest secure dominating set of G. A linear algorithm is proposed in this paper for finding a minimum secure dominating set and hence the value gamma(s)(T) for a tree T. The algorithm is based on a strategy of repeatedly pruning away pendent spiders of T after having dominated them securely. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Secure domination Graph protection linear algorithm (Wounded) spider Tree

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Transitivity on subclasses of bipartite graphs

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2023年第1期45卷 1-16页

作者： Paul, Subhabrata Santra, Kamal Indian Inst Technol Patna Dept Math Bihta 801106 Bihar India

Let G=(V,E) be a graph where V and E are the vertex and edge sets, respectively. For two disjoint subsets A and B, we say A dominates B if every vertex of B is adjacent to at least one vertex of A. A vertex partition pi={V-1,V-2, horizontexpressionl ellipsis ,V-k} of G is called a transitive k-partition if V-i dominates V-j for all i, j where 1 <= i < j <= k. The maximum integer k for which the above partition exists is called transitivity of G and it is denoted by Tr(G). The MAXIMUM TRANSITIVITY PROBLEM is to find a transitive partition of a given graph with the maximum number of parts. It was known that the decision version of MAXIMUM TRANSITIVITY PROBLEM is NP-complete for chordal graphs, which was proved by Hedetniemi et al. (Discret Math 278:81-108, 2004)]. This paper first strengthens the NP-completeness result by showing that this problem remains NP-complete for perfect elimination bipartite graphs. On the other hand, we propose a linear-time algorithm for finding the transitivity of a given bipartite chain graph. We then characterize graphs with transitivity at least t for any integer t. This result answers two open questions posed by J. T. Hedetniemi and S. T. Hedetniemi (J Combin Math Combin Comput 104:75-91, 2018).

关键词： Transitivity NP-completeness linear algorithm Perfect elimination bipartite graphs Bipartite chain graphs

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algorithmic and complexity aspects of problems related to total restrained domination for graphs

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2023年第5期46卷 1-20页

作者： Yang, Yu Wang, Cai-Xia Xu, Shou-Jun Lanzhou Univ Gansu Ctr Appl Math Sch Math & Stat Lanzhou 730000 Gansu Peoples R China

Let G be a graph with vertex set V and a subset D subset of V. D is a total dominating set of G if every vertex in V is adjacent to a vertex in D. D is a restrained dominating set of G if every vertex in V \D is adjacent to a vertex in D and another vertex in V \D. D is a total restrained dominating set if D is both a total dominating set and a restrained dominating set. The minimum cardinality of total dominating sets (resp. restrained dominating sets, total restrained dominating sets) of G is called the total domination number (resp. restrained domination number, total restrained domination number) of G, denoted by gamma t (G) (resp. gamma r (G), gamma tr (G)). The MINIMUM TOTAL RESTRAINED DOMINATION (MTRD) problem for a graph G is to find a total restrained dominating set of minimum cardinality of G. The TOTAL RESTRAINED DOMINATION DECISION (TRDD) problem is the decision version of the MTRD problem. In this paper, firstly, we show that the TRDD problem is NP-complete for undirected path graphs, circle graphs, S-CB graphs and C-CB graphs, respectively, and that, for a S-CB graph or C-CB graph with n vertices, the MTRD problem cannot be approximated within a factor of (1 -epsilon)lnn for any epsilon > 0 unless N P subset of DT I M E(n(O(loglogn))). Secondly, for a graph G, we prove that the problem of deciding whether gamma r (G) = gamma tr (G) is NP-hard even when G is restricted to planar graphs with maximum degree at most 4, and that the problem of deciding whether gamma t(G) = gamma tr(G) is NP-hard even when G is restricted to planar bipartite graphs with maximum degree at most 5. Thirdly, we show that the MTRD problem is APX-complete for bipartite graphs with maximum degree at most 4. Finally, we design a linear-time algorithm for solving the MTRD problem for generalized series-parallel graphs.

关键词： Total domination number Restrained domination number Total restrained domination number NP-complete APX-complete linear algorithm

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Common Due-Date Problem: linear algorithm for a Given Job Sequence 17

Common Due-Date Problem: Linear Algorithm for a Given Job Se...

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IEEE 17th International Conference on Computational Science and Engineering (CSE)

作者： Laessig, Joerg Awasthi, Abhishek Kramer, Oliver Univ Appl Sci Dept Comp Sci Gorlitz Germany Carl von Ossietzky Univ Oldenburg Dept Comp Sci Oldenburg Germany

ISBN: (纸本)9781479979813

This work presents a study on the general case of the Common Due-Date scheduling problem (CDD). The CDD is a problem of scheduling and sequencing a certain number of jobs with different processing times on a single machine against a common due-date. The objective of the problem is to minimize the total penalty incurred due to earliness or tardiness of the jobs. This work presents a novel property for the CDD which implies that in any given job sequence (also the optimal job sequence) of the CDD the position of the due-date is independent to the processing times of the jobs. Thereafter, we put forward an exact polynomial algorithm to optimize a given job sequence with a run-time complexity of O(n), where n is the number of jobs. Henceforth, we implement our polynomial algorithm in conjunction with a modified Simulated Annealing (SA) algorithm to obtain the optimal or best job sequence. The effectiveness of our approach is evident from our results for the benchmark instances provided in the OR-library.

关键词： computational complexity simulated annealing single machine scheduling CDD OR-library common due-date scheduling problem job processing time job sequence linear algorithm modified SA algorithm modified simulated annealing algorithm optimization polynomial algorithm run-time complexity Equations Heuristic algorithms linear programming Optimal scheduling Schedules Single machine scheduling Vectors job sequence single machine scheduling CDA gene complexity classes Optimal scheduling Heuristic algorithms simulated annealing linear programming jobs

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Efficiently and consistently energy-stable L²-phase-field method for the incompressible ternary fluid problems

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PHYSICA D-NONlinear PHENOMENA 2024年第PartA期470卷

作者： Yang, Junxiang Han, Huan Liu, Shuhong Zuo, Zhigang Macau Univ Sci & Technol Fac Innovat Engn Sch Comp Sci & Engn Taipa Macao Peoples R China Univ Sci & Technol Beijing Sch Energy & Environm Engn Beijing 100083 Peoples R China Tsinghua Univ State Key Lab Hydro Sci & Engn Beijing Peoples R China Tsinghua Univ Dept Energy & Power Engn Beijing 100084 Peoples R China

Ternary incompressible fluid flows extensively exist in atmospheric science, chemical engineering, and energy and power engineering, etc. The phase-field method is popular in multi-phase fluid modeling thanks to its efficient ability of interface capturing. This work aims to develop an energy dissipation law-preserving and temporally second-order accurate algorithm for a ternary phase-field fluid system. To establish a simple energy estimation, an adapted auxiliary variable approach is used to transform the original model into its equivalent form. Later, a second-order backward difference strategy is used to design the fully decoupled and linear time-marching scheme. To improve the consistency between original and modified discrete energy functionals, a practical energy correction technique is presented. We analytically prove the discrete energy dissipation property and show that the energy estimation can be easily established without considering the complex treatments of nonlinear and coupling terms. To facilitate the interested readers, we briefly describe the numerical implementation in each time step. The numerical tests indicate that the proposed method not only has desired accuracy, but also satisfies the energy stability even if a larger time step is used. Moreover, the proposed method can well simulate various ternary fluid phenomena, such as the liquid lens, phase separation, droplet dynamics, Kelvin-Helmholtz instability, and billowing cloud.

关键词： Ternary fluids Phase-field method linear algorithm Energy consistency

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linear construction of a left Lyndon tree

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INFORMATION AND COMPUTATION 2022年第PartB期285卷

作者： Badkobeh, Golnaz Crochemore, Maxime Univ London Goldsmiths New Cross London SE14 6NW England Kings Coll London Informat 30 Aldwych London WC2B 4BG England Univ Gustave Eiffel F-77454 Marne La Vallee France

We extend the left-to-right Lyndon factorisation of a word to the left Lyndon tree construction of a Lyndon word. It yields an algorithm to sort the prefixes of a Lyndon word according to the infinite ordering defined by Dolce et al. (2019). A straightforward variant computes the left Lyndon forest of a word. All algorithms run in linear time on a general alphabet, that is, in the letter-comparison model. (c) 2022 Elsevier Inc. All rights reserved.

关键词： Lyndon tree Infinite ordering Prefix sorting linear algorithm General alphabet

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algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2021年第2期41卷 304-317页

作者： Poureidi, Abolfazl Ghaznavi, Mehrdad Fathali, Jafar Shahrood Univ Technol Fac Math Sci Shahrood Iran

Let G = (V, E) be a graph. A function f : V -> {0, 1, 2} is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex v is an element of V with f (v) = 0 there is a vertex u adjacent to v with f (u) = 2 and {x is an element of V : f (x) = 0} is an independent set. The weight of f is the value f (V) = Sigma(v is an element of V) f (v). An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every v is an element of V with f (v) > 0 there is a vertex u adjacent to v with f (u) > 0. The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by gamma(oiR)(G). Similarly, the outer independent total Roman domination number of G is defined, denoted by gamma(oiR)(G). In this paper, we first show that computing gamma(oiR)(G) (respectively, gamma(oiR)(G) R(G)) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph G = (V, E) we propose an algorithm to compute gamma(oiR)(G) (respectively, gamma(oiR)(G)) in O(vertical bar V vertical bar) time.

关键词： Outer independent Roman dominating function Outer independent total Roman dominating function linear algorithm Proper interval graph NP-hard

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linear state estimation and bad data detection for power systems with RTU and PMU measurements

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IET GENERATION TRANSMISSION & DISTRIBUTION 2020年第23期14卷 5675-5684页

作者： Jovicic, Aleksandar Hug, Gabriela Swiss Fed Inst Technol EEH Power Syst Lab Zurich Switzerland

The reliability of state estimation is of vital importance for secure operation of a power system. The key features of any state estimator are its accuracy, robustness against bad data and the associated computational burden. In order to ensure full system observability and maximise accuracy, a state estimation algorithm needs to take into account all available measurements. A significant number of hybrid estimators that can process both the conventional measurements that have been used in transmission systems for decades, as well as the synchrophasor measurements, can be found in literature. However, these algorithms are predominantly non-linear, thus requiring iterative solving procedures. In this study, a novel linear algorithm is proposed for state estimation including bad data detection of power systems that are monitored both by conventional and synchrophasor measurements. The proposed estimator is based on the linear weighted least square framework. As a result, the computational burden is rendered extremely low since there is no need for an iterative solving procedure, and standardised post-processing tools for bad data detection can be used. To validate the accuracy and robustness of the proposed algorithm, an extensive number of test cases of different sizes are solved and the results are presented and discussed.

关键词： power system state estimation phasor measurement least squares approximations power system security observability data handling power engineering computing synchrophasor measurements linear weighted least square framework bad data detection linear state estimation power system security state estimator system observability hybrid estimators transmission systems linear algorithm RTU measurements PMU measurements

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Optimization on the smallest eigenvalue of grounded Laplacian matrix via edge addition

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THEORETICAL COMPUTER SCIENCE 2023年 980卷

作者： Zhou, Xiaotian Sun, Haoxin Li, Wei Zhang, Zhongzhi Fudan Univ Sch Comp Sci Shanghai Key Lab Intelligent Informat Proc Shanghai 200433 Peoples R China Fudan Univ Acad Engn & Technol Shanghai 200433 Peoples R China Fudan Univ Shanghai Engn Res Inst Blockchains Shanghai 200433 Peoples R China Fudan Univ Res Inst Intelligent Complex Syst Shanghai 200433 Peoples R China

The grounded Laplacian matrix L-s of a graph G = (V,E) with n = | v | nodes and m = |E| edges is a (n -s) x(n -s) submatrix of its Laplacian matrix L, obtained from L by deleting rows and columns corresponding to s = | s | << n ground nodes forming set s subset of V. The smallest eigenvalue of L-s plays an important role in various practical scenarios, such as characterizing the convergence rate of leader-follower opinion dynamics, with a larger eigenvalue indicating faster convergence of opinion. In this paper, we study the problem of adding k << n edges among all the nonexistent edges forming the candidate edge set Q = (v x v)\E, in order to maximize the smallest eigenvalue of the grounded Laplacian matrix. We show that the objective function of the combinatorial optimization problem is monotone but non-submodular. To solve the problem, we first simplify the problem by restricting the candidate edge set Q to be ( S x (V \S))\E, and prove that it has the same optimal solution as the original problem, although the size of set Q is reduced from O(n(2)) to O(n(2)). Then, we propose two greedy approximation algorithms. One is a simple greedy algorithm with an approximation ratio (1 -e(-alpha gamma))/alpha and time complexity O(kn(4)), where gamma and alpha are, respectively, submodularity ratio and curvature, whose bounds are provided for some particular cases. The other is a fast greedy algorithm without approximation guarantee, which has a running time (O) over tilde (km), where (O) over tilde (center dot) suppresses the poly(log..) factors. Numerous experiments on various real networks are performed to validate the superiority of our algorithms, in terms of effectiveness and efficiency.

关键词： Grounded Laplacian Spectral property Graph mining linear algorithm Matrix perturbation Partial derivative

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