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. Senso...
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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.
The cluster deletion problem (CD) asks for transforming a given graph into a disjoint union of cliques by removing as few edges as possible. CD is among the most studied combinatorial optimization problem and, for gen...
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The cluster deletion problem (CD) asks for transforming a given graph into a disjoint union of cliques by removing as few edges as possible. CD is among the most studied combinatorial optimization problem and, for general graphs, it is NP-hard. In the present paper, we identify a new polynomially solvable CD subproblem. We specifically propose a two-phase polynomial-time algorithm that optimally solves CD on the class of (butterfly,diamond)-free graphs. For this latter class of graphs, our two-phase algorithm provides optimal solutions even for another clustering variant, namely, cluster editing. Then, we propose a 2-optimal CD algorithm dedicated to the super-class of diamond-free graphs. For this class, we also show that CD, when parameterised by the number of deleted edges, admits a quadratic-size kernel. Finally, we report the results of experiments carried out on numerous diamond-free graphs, showing the effectiveness of the proposed approximate algorithm in terms of solution quality.
We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colou...
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We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is NP-complete. We show that for every choice of positive integers r,k there exists a k-strong bipartite tournament, which needs at least r colours in every out-colouring. Our main results are on tournaments and semicomplete digraphs. We prove that, except for the Paley tournament P7, every strong semicomplete digraph of minimum out-degree at least three has a 2-out-colouring. Furthermore, we show that every semicomplete digraph on at least seven vertices has a 2-out-colouring if and only if it has a balanced such colouring, that is, the difference between the number of vertices that receive colour 1 and colour 2 is at most one. In the second half of the paper, we consider the generalization of 2-out-colourings to vertex partitions (V1,V2) of a digraph D so that each of the three digraphs induced by respectively, the vertices of V1, the vertices of V2 and all arcs between V1 and V2, have minimum out-degree k for a prescribed integer k >= 1. Using probabilistic arguments, we prove that there exists an absolute positive constant c so that every semicomplete digraph of minimum out-degree at least 2k+ck has such a partition. This is tight up to the value of c.
We give a transparent combinatorial characterization of the identities satisfied by the Kauffman monoid Our characterization leads to a polynomial time algorithm to check whether a given identity holds in K-3.
We give a transparent combinatorial characterization of the identities satisfied by the Kauffman monoid Our characterization leads to a polynomial time algorithm to check whether a given identity holds in K-3.
The units forms are algebraic expressions that have important role in representation theory of algebras. We identified that existing algorithms have exponential time complexity for weakly nonnegative and weakly positi...
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The units forms are algebraic expressions that have important role in representation theory of algebras. We identified that existing algorithms have exponential time complexity for weakly nonnegative and weakly positive types. In this paper we introduce a polynomial algorithm for the recognition of weakly nonnegative unit forms. The related algorithm identifies hypercritical restrictions in a given unit form, testing every subgraph of 9 vertices of the unit form associated graph. By adding Depth First Search approach, a similar strategy could be used in the recognition of weakly positive unit forms. We also present the most popular methods to decide whether or not a unit form is weakly nonnegative or weakly positive, we analyze their time complexity and we compare the results with our algorithms. (C) 2018 Elsevier B.V. All rights reserved.
Let G = (V, E) be a graph with the vertex set V and the edge set E. We denote by N(nu) the neighborhood of a vertex nu is an element of V. A non-empty set of vertices S subset of V is said to be a defensive alliance i...
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ISBN:
(纸本)9784885523281
Let G = (V, E) be a graph with the vertex set V and the edge set E. We denote by N(nu) the neighborhood of a vertex nu is an element of V. A non-empty set of vertices S subset of V is said to be a defensive alliance if for every vertex nu is an element of S, vertical bar N(nu) boolean AND S vertical bar + 1 >= vertical bar N(nu) boolean AND (V - S)vertical bar. In this paper, we consider a defensive alliance containing a specified vertex in a tree. We show that, for any tree T of order at least three and a vertex x of T, there is a defensive alliance S such that x is an element of S and vertical bar S vertical bar <= inverted right perpendicular vertical bar V-1(T)vertical bar+ 1/2 inverted right perpendicular, where V-1(T) stands for the set of leaves of T. Actually, we give an algorithm which construct a such defensive alliance.
Accompanied with the mushroom growth of communication technology, the correct operation of protocols has been widely concerned and studied in the field of communication. The correctness verification is a difficult pro...
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Accompanied with the mushroom growth of communication technology, the correct operation of protocols has been widely concerned and studied in the field of communication. The correctness verification is a difficult problem due to the state space explosion. In this paper, the event graph model of a communication protocol is established and verification matrix is proposed to verify the correctness of the protocol. It is found out that the correctness of a protocol is equivalent to the nilpotency of the verification matrix on the max-plus algebra framework. The matrix method is constructive and leads to a polynomial algorithm for verifying the correctness of protocols. Some examples about stop-and-wait protocols and handshake protocols are taken to illustrate how the presented results work in practical applications. Copyright (C) 2020 The Authors.
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length 3 with the edges assigned a same color. It is k...
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A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length 3 with the edges assigned a same color. It is known that every edge-colored complete graph without monochromatic triangle always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the exceptions. As a consequence, we partially confirm a structural conjecture given by Bollobas and Erdos (1976) and an algorithmic conjecture given by Gutin and Kim (2009). (C) 2021 Elsevier B.V. All rights reserved.
In the MINIMAL PERMUTATION COMPLETION problem, one is given an arbitrary graph G = (V, E) and the aim is to find a permutation super-graph H = (V, F) defined on the same vertex set and such that F superset of E is inc...
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In the MINIMAL PERMUTATION COMPLETION problem, one is given an arbitrary graph G = (V, E) and the aim is to find a permutation super-graph H = (V, F) defined on the same vertex set and such that F superset of E is inclusion-minimal among all possibilities. The graph H is then called a minimal permutation completion of G. We provide an O(n(2)) incremental algorithm computing such a minimal permutation completion. To the best of our knowledge, this result leads to the first polynomial algorithm for this problem. A preliminary extended abstract of this paper appeared as [4] in the Proceedings of WG 2015. (C) 2018 Elsevier B.V. All rights reserved.
The problem of finding an optimal location of the vertices of a tree network in an assembly space representing a finite set is considered. The optimality criterion is the minimum total cost of location and communicati...
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The problem of finding an optimal location of the vertices of a tree network in an assembly space representing a finite set is considered. The optimality criterion is the minimum total cost of location and communications in all points of this space. Different vertices of the tree can be located in a single point of the assembly space. This problem is well-known as the Weber problem. The representation of the Weber problem as a linear programming problem is given. It is proved that the set of all optimal solutions to the corresponding relaxed Weber problem for the tree network contains an integer solution. This fact can be used to improve the efficiency of algorithms for the problems differing from the Weber problem by the presence of additional constraints: it allows us to find the optimal value of the objective function, which in turn significantly reduces the complexity of calculating the optimal solution itself, e.g., by the branch-and-bound method.
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