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The pairwise flowtime network construction problem

OPERATIONS RESEARCH LETTERS 2021年第6期49卷 902-907页

作者： Averbakh, Igor Univ Toronto Scarborough Dept Management 1265 Mil Trail Toronto ON M1C 1A4 Canada Univ Toronto Rotman Sch Management 105 St George St Toronto ON M5S 3E6 Canada

It is required to find an optimal order of constructing the edges of a network so as to minimize the sum of the weighted connection times of relevant pairs of vertices. Construction can be performed anytime anywhere in the network, with a fixed overall construction speed. The problem is strongly NP-hard even on stars. We present polynomial algorithms for the problem on trees with a fixed number of leaves, and on general networks with a fixed number of relevant pairs. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Flowtime scheduling Network construction planning Connectivity restoration polynomial algorithm

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Fast algorithm for Cyber-Attack Estimation and Attack Path Extraction Using Attack Graphs with AND/OR Nodes

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algorithmS 2024年第11期17卷 504-504页

作者： Levner, Eugene Tsadikovich, Dmitry Holon Inst Technol Fac Sci IL-58102 Holon Israel Bar Ilan Univ Dept Management IL-5290002 Ramat Gan Israel

This paper studies the security issues for cyber-physical systems, aimed at countering potential malicious cyber-attacks. The main focus is on solving the problem of extracting the most vulnerable attack path in a known attack graph, where an attack path is a sequence of steps that an attacker can take to compromise the underlying network. Determining an attacker's possible attack path is critical to cyber defenders as it helps identify threats, harden the network, and thwart attacker's intentions. We formulate this problem as a path-finding optimization problem with logical constraints represented by AND and OR nodes. We propose a new Dijkstra-type algorithm that combines elements from Dijkstra's shortest path algorithm and the critical path method. Although the path extraction problem is generally NP-hard, for the studied special case, the proposed algorithm determines the optimal attack path in polynomial time, O(nm), where n is the number of nodes and m is the number of edges in the attack graph. To our knowledge this is the first exact polynomial algorithm that can solve the path extraction problem for different attack graphs, both cycle-containing and cycle-free. Computational experiments with real and synthetic data have shown that the proposed algorithm consistently and quickly finds optimal solutions to the problem.

关键词： cybersecurity cyber-attack cyclic attack graph attack path AND/OR graph attack time assessment attack path extraction polynomial algorithm

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Multiaffine polynomials over a finite field

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DISCRETE MATHEMATICS AND APPLICATIONS 2021年第6期31卷 421-430页

作者： Selezneva, Svetlana N. Lomonosov Moscow State Univ Moscow Russia

We consider polynomials f (x(1),..., x(n)) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f (x1,..., x(n)) = b coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with respect to the right-hand side b. We obtain the properties of multiaffine polynomials over a finite field. Then we show that checking the multiaffinity with respect to a given right-hand side may be done by an algorithm with polynomial (in terms of the number of variables and summands of the input polynomial) complexity. Besides that, we prove that in case of the positive decision a corresponding system of linear equations may be recovered with complexity which is also polynomial in terms of the same parameters.

关键词： finite field polynomial multiaffinity system of linear equations over a finite field algorithm complexity polynomial algorithm

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polynomial Asymptotically Optimal Coding of Underdetermined Bernoulli Sources of the General Form

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PROBLEMS OF INFORMATION TRANSMISSION 2020年第4期56卷 373-387页

作者： Sholomov, L. A. Russian Acad Sci Fed Res Ctr Comp Sci & Control Moscow Russia Russian Acad Sci Inst Syst Anal Moscow Russia

An underdetermined Bernoulli source generates symbols of a given underdetermined alphabet independently with some probabilities. To each underdetermined symbol there corresponds a set of basic (fully defined) symbols such that it can be substituted (specified) by any of them. An underdetermined source is characterized by its entropy, which is implicitly introduced as a minimum of a certain function and plays a role similar to the Shannon entropy for fully defined sources. Coding of an underdetermined source must ensure, for any sequence generated by the source, recovering some its specification. Coding is asymptotically optimal if the average code length is asymptotically equal to the source entropy. Coding is universal if it does not depend on the probabilities of source symbols. We describe an asymptotically optimal universal coding method for underdetermined Bernoulli sources for which the encoding and decoding procedures can be realized by RAM-programs of almost linear complexity.

关键词： underdetermined source specification entropy of an underdetermined source quasientropy of a word combinatorial entropy of a class underdetermined source coding universal coding polynomial algorithm

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A polynomial algorithm for a lot-sizing problem with backlogging, outsourcing and limited inventory

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COMPUTERS & INDUSTRIAL ENGINEERING 2013年第1期64卷 200-210页

作者： Chu, Chengbin Chu, Feng Zhong, Jinhong Yang, Shanlin Tongji Univ Sch Econ & Management Shanghai Peoples R China Ecole Cent Paris Lab Genie Ind Paris France Univ Eviy Val dEssone Lab IBISC Evry France Hefei Univ Technol Sch Management Hefei Peoples R China

This paper addresses a real-life production planning problem arising in a manufacturer of luxury goods. This problem can be modeled as a single item dynamic lot-sizing model with backlogging, outsourcing and inventory capacity. Setup cost is included in the production cost function, and the production level at each period is unbounded. The holding, backlogging and outsourcing cost functions are assumed to be linear. The backlogging level at each period is also limited. The goal is to satisfy all demands in the planning horizon at minimal total cost. We show that this problem can be solved in O(T-4 logT) time where T is the number of periods in the planning horizon. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Lot-sizing Production planning Dynamic programming polynomial algorithm

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The Implementation of Eulerian Coverings of a Graph for Solving Routing Problems

The Implementation of Eulerian Coverings of a Graph for Solv...

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International-Federation-of-Information-Processing-Working-Group-5.7 (IFIP WG 5.7) International Conference on Advances in Production Management Systems (APMS)

作者： Makarovskikh, Tatiana A. South Ural State Univ 76 Lenin Ave Chelyabinsk 454080 Russia

ISBN: (纸本)9783030859145;9783030859138

In terms of graph theory, the urban transport network can be adequately represented as a graph. All crossroads in the city are considered to be the vertices of the graph, and streets are the edges. Using this model, one can set and effectively solve many routing problems: (1) constructing the routes between given starting and ending points, satisfying the rules of the road (allowed turns at crossroads and a given sequence of driving through the streets), (2) finding the way, passing through all the streets of the city in accordance with the rules of the road and a given sequence of turns at crossroads (an algorithm for constructing an permissible Eulerian chain), (3) constructing a minimal cardinality set of routes running through all the streets of the city (Eulerian cover of any graph with permissible chains). The paper provides a formalization of these problems and effective algorithms for solving them.

关键词： Graphs Trajectory planning polynomial algorithm Transportation control

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Minimal size balanced sets mod p

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SCIENTIFIC AFRICAN 2024年 25卷

作者： Cuong, Nguyen Phu Nedev, Zhivko P. Zungeru, Adamu Murtala Dalhousie Univ Halifax NS Canada Botswana Int Univ Sci & Technol Dept Comp Sci P Bag 0016 Palapye Botswana Botswana Int Univ Sci & Technol Dept Elect Comp & Telecommun Engn P Bag 0016 Palapye Botswana

A nonempty set.. of residues modulo.. is said to be balanced if for each....., there is a.. with 0 1. Thus, the sequence simplifies to.... = 2.. + 1 3 with.. odd. Finally, we prove that if 2.. + 1 3 is prime, then.. ... 详细信息

A nonempty set.. of residues modulo.. is said to be balanced if for each....., there is a.. with 0 <.. =.. /2 such that.. +/-.. mod.. both lie in... We denote the minimum cardinality of a balanced set modulo.. by..(..). Minimal size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets. In this paper, we describe a polynomial algorithm for constructing a minimal size balanced set modulo.., when.. is from two special classes of primes called lucky primes. We prove that lucky primes are all primes among the sequence.... = 2..+3 + (-1).. 3. Then we prove that the numbers.... = 2..+3 + (-1).. 3 are never prime when.. is odd and.. > 1. Thus, the sequence simplifies to.... = 2.. + 1 3 with.. odd. Finally, we prove that if 2.. + 1 3 is prime, then.. must be a prime.

关键词： Minimal size Balanced sets Mod p polynomial algorithm Lucky primes Balanced set modulo Sequences

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Maximum independent sets in subcubic graphs: New results

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THEORETICAL COMPUTER SCIENCE 2020年 846卷 14-26页

作者： Harutyunyan, A. Lampis, M. Lozin, V Monnot, J. Univ Paris 09 Univ PSL CNRS LAMSADE F-75016 Paris France Univ Warwick Math Inst Coventry CV4 7AL W Midlands England

The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We study complexity of the problem on hereditary subclasses of subcubic graphs. Each such subclass can be described by means of forbidden induced subgraphs. In case of finitely many forbidden induced subgraphs a necessary condition for polynomial-time solvability of the problem in subcubic graphs (unless P = NP) is the exclusion of the graph S-i,S-j,S-k, which is a tree with three leaves of distance i, j, k from the only vertex of degree 3. Whether this condition is also sufficient is an open question, which was previously answered only for S-1,S-k,S-k-free subcubic graphs and S-2,S-2,S-2 -free subcubic graphs. Combining various algorithmic techniques, in the present paper we generalize both results and show that the problem can be solved in polynomial time for S-2,S-k,S-k-free subcubic graphs, for any fixed value of k. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Maximum independent set Subcubic graph polynomial algorithm

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Geometric algorithms for Finding a Point in the Intersection of Balls

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AUTOMATION AND REMOTE CONTROL 2020年第5期81卷 869-882页

作者： Lushchakova, I. N. Belarusian State Univ Informat & Radioelect Minsk BELARUS

We consider a problem of finding a point in the intersection of n balls in the Euclidean space E-m. For the case m = 2 we suggest two algorithms of the complexity O(n(2) log n) and O(n(3)) operations, respectively. For the general case we suggest an exact polynomial recursive algorithm which uses the orthogonal transformation of the space E-m.

关键词： intersection of balls approximation a convex set by ellipsoids polynomial algorithm delivery applications of drones configuration of the swarm of drones

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Weak Coverage of a Rectangular Barrier

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algorithmICA 2020年第4期82卷 721-746页

作者： Dobrev, Stefan Kranakis, Evangelos Krizanc, Danny Lafond, Manuel Manuch, Jan Narayanan, Lata Opatrny, Jaroslav Stacho, Ladislav Slovak Acad Sci Inst Math Bratislava Slovakia Carleton Univ Sch Comp Sci Ottawa ON Canada Wesleyan Univ Dept Math & Comp Sci Middletown CT USA Univ Ottawa Dept Math & Stat Ottawa ON Canada Univ British Columbia Dept Comp Sci Vancouver BC Canada Concordia Univ Dept Comp Sci & Software Engn Montreal PQ Canada Simon Fraser Univ Dept Math Burnaby BC Canada

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. Each sensor can monitor a circular area of specific diameter around its position, called the sensor diameter. Sensors are required to move to final locations so that they can there detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak barrier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by any sensor (MinMax). We give an O(n(3/2)) time algorithm for the MinNum problem for sensors of diameter 1 that are initially placed at integer positions;in contrast the problem is shown to be NP-complete even for sensors of diameter 2 that are initially placed at integer positions. We show that the MinSum problem is solvable in O(n log n) time for theManhattan metric and sensors of identical diameter (homogeneous sensors) in arbitrary initial positions, while it is NP-complete for heterogeneous sensors. Finally, we prove that even very restricted homogeneous versions of the MinMax problem are NP-complete.

关键词： Barrier coverage Mobile sensors Optimizing sensor movements polynomial algorithm NP-complete problem

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