As usual lambda(G) denotes the edge-connectivity of the graph G. It was shown in [2] that every graph G contains a spanning (lambda(G) + 1)-partite subgraph H such that lambda(H) = lambda(G) and one can find such a sp...
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As usual lambda(G) denotes the edge-connectivity of the graph G. It was shown in [2] that every graph G contains a spanning (lambda(G) + 1)-partite subgraph H such that lambda(H) = lambda(G) and one can find such a spanning subgraph in polynomial time. We determine the complexity of deciding, for given positive integers r, k whether a graph contains a spanning r-colourable subgraph which is k-edge-connected. We show that the problem is polynomially solvable when r > k and NP-complete otherwise. In fact, combined with the result from [2] above, this means that the problem is polynomially solvable precisely when r is such that every k-edge-connected graph has a spanning r-colourable subgraph which is k-edge-connected. One can show that all graphs whose edge set decomposes into k edge-disjoint spanning trees are 2k-colourable. We consider the problem of deciding whether a given graph G has a collection of k edge-disjoint spanning trees whose union forms an r-colourable spanning subgraph H of G. We show that this problem is polynomially solvable when r > 2k and NP-complete for all other values of r. We also determine the complexity of the analogous problem of deciding whether a digraph D has a collection of k arc-disjoint out-branchings such that the spanning subdigraph formed by the union of the arcs in the branchings is r-colourable.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
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 kno...
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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.
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 s...
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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.
A strongly polynomial algorithm is developed for finding an integer-valued feasible st-flow of a given flow amount, which is decreasingly minimal on a specified subset F of edges in the sense that the largest flow val...
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A strongly polynomial algorithm is developed for finding an integer-valued feasible st-flow of a given flow amount, which is decreasingly minimal on a specified subset F of edges in the sense that the largest flow value on F is as small as possible;within this, the second largest flow value on F is as small as possible;within this, the third largest flow value on F is as small as possible, and so on. A characterization of the set of these st-flows gives rise to an algorithm to compute a cheapest F-decreasingly minimal integer-valued feasible st-flow of given flow amount. Decreasing minimality is a possible formal way to capture the intuitive notion of fairness.
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 ...
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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.
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...
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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.
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, o...
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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.
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.. ...
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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.
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 ...
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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.
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. Fo...
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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.
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