Motivated by the connection with the genus of unoriented alternating links, Jin et al. (Acta Math Appl Sin Engl Ser, 2015) introduced the number of maximum state circles of a plane graph G, denoted by , and proved tha...
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Motivated by the connection with the genus of unoriented alternating links, Jin et al. (Acta Math Appl Sin Engl Ser, 2015) introduced the number of maximum state circles of a plane graph G, denoted by , and proved that H is a spanning subgraph of , where e(H), c(H) and v(H) denote the size, the number of connected components and the order of H, respectively. In this paper, we show that for any (not necessarily planar) graph G, can be achieved by the spanning subgraph H of G whose each connected component is a maximal subgraph of G with two edge-disjoint spanning trees. Such a spanning subgraph is proved to be unique and we present a polynomial-time algorithm to find such a spanning subgraph for any graph G.
作者:
Malyshev, D. S.Natl Res Univ
Higher Sch Econ 25-12 Bolshaya Pecherskaya Ulitsa Nizhnii Novgorod 603155 Russia
We show that the weighted coloring problem can be solved for {P-5, banner}-free graphs and for {P-5, dart}-free graphs in polynomialtime on the sum of vertex weights. (C) 2018 Elsevier B.V. All rights reserved.
We show that the weighted coloring problem can be solved for {P-5, banner}-free graphs and for {P-5, dart}-free graphs in polynomialtime on the sum of vertex weights. (C) 2018 Elsevier B.V. All rights reserved.
An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M. The dominating induced matching problem (also known as efficient edge domination) asks whether a gra...
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An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph G contains a dominating induced matching. This problem is generally NP-complete, but polynomial-time solvable for graphs with some special properties. In particular, it is solvable in polynomialtime for claw-free graphs. In the present article, we provide a polynomial-time algorithm to solve the dominating induced matching problem for graphs containing no long claw, that is, no induced subgraph obtained from the claw by subdividing each of its edges exactly once.
Recently, increasing penetration of renewable energy generation has created challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable...
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Recently, increasing penetration of renewable energy generation has created challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, a unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this article, we focus on deriving polynomial-time algorithms for the unit commitment problems with a general convex cost function and piecewise linear cost function, respectively. We refine an O(T-3) time, where T represents the number of time periods, algorithm for the deterministic single-generator unit commitment problem with a general convex cost function and accordingly develop an extended formulation in a higher-dimensional space that can provide an integral solution, in which the physical meanings of the decision variables are described. This means the original problem can be solved as a convex program instead of a mixed-integer convex program. Furthermore, for the case in which the cost function is piecewise linear, by exploring the optimality conditions, we derive more efficient algorithms for both deterministic (i.e., O(T) time) and stochastic (i.e., O(N) time, where N represents the number of nodes in the stochastic scenario tree) single-generator unit commitment problems. We also develop the corresponding extended formulations for both deterministic and stochastic single-generator unit commitment problems that solve the originalmixed-integer linear programs as linear programs. Similarly, physical meanings of the decision variables are explored to show the insights of the new modeling approach.
In this paper, we show that checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations can be implemented in polynomialtime. Multi-linear polynomials o...
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In this paper, we show that checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations can be implemented in polynomialtime. Multi-linear polynomials over GF(2) are often a convenient way to represent Boolean functions. There is a single polynomial for each Boolean function, and the length of the polynomial (i.e. the number of its monomials) which represents a function of n variables can be far less than 2(n). Therefore, in some cases, polynomials are a compressed description of Boolean functions. Besides, polynomial representations of Boolean functions have applications in circuit lower bounds, computational learning, error-correcting codes, cryptography. We construct polynomial-time algorithms for checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations. The considered properties are self-anti-duality (evenness), self-duality, periodicity, 1-invariance (Mobius transform invariance, coincidence). Note that checking each of these properties directly by definition gives, in the general case, only exponential-timealgorithms. The approach to construct our algorithms is the following. Firstly, we prove that if a Boolean function has a certain property then its polynomial has a special structure. And secondly, we check the property by its characterization, cutting negative cases by proven facts. More precisely, we analyze an exponential timealgorithm and prove that if the number of steps of the algorithm exceeds a bound, which is polynomial in the input size, and may be computed in advance, then the input will be necessarily rejected.
Given two graphs, SUBGRAPH ISOMORPHISM is the problem of deciding whether the first graph (the base graph) contains a subgraph isomorphic to the second one (the pattern graph). This problem is NP-complete even for ver...
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Given two graphs, SUBGRAPH ISOMORPHISM is the problem of deciding whether the first graph (the base graph) contains a subgraph isomorphic to the second one (the pattern graph). This problem is NP-complete even for very restricted graph classes such as connected proper interval graphs. Only a few cases are known to be polynomial-time solvable even if we restrict the graphs to be perfect. For example, if both graphs are co-chain graphs, then the problem can be solved in linear time. In this paper, we present a polynomial-time algorithm for the case where the base graphs are chordal graphs and the pattern graphs are co-chain graphs. We also present a linear-timealgorithm for the case where the base graphs are trivially perfect graphs and the pattern graphs are threshold graphs. These results answer some of the open questions of Kijima et al. (2012). To present a complexity contrast, we then show that even if the base graphs are somewhat restricted perfect graphs, the problem of finding a pattern graph that is a chain graph, a co-chain graph, or a threshold graph is NP-complete. (C) 2015 Elsevier B.V. All rights reserved.
A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved in this paper that a cycle base of a permutation group of degree n can be constru...
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A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved in this paper that a cycle base of a permutation group of degree n can be constructed in polynomialtime in n. (C) 2018 Elsevier Inc. All rights reserved.
We consider a communication network where there exist wiretappers who can access a subset of channels, called a wiretap set, which is chosen from a given collection of wiretap sets. The collection of wiretap sets can ...
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We consider a communication network where there exist wiretappers who can access a subset of channels, called a wiretap set, which is chosen from a given collection of wiretap sets. The collection of wiretap sets can be arbitrary. Secure network coding is applied to prevent the source information from being leaked to the wiretappers. In secure network coding, the required alphabet size is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of such coding schemes in terms of computational complexity and storage requirement. In this paper, we develop a systematic graph-theoretic approach for improving Cai and Yeung's lower bound on the required alphabet size for the existence of secure network codes. The new lower bound thus obtained, which depends only on the network topology and the collection of wiretap sets, can be significantly smaller than Cai and Yeung's lower bound. A polynomial-time algorithm is devised for efficient computation of the new lower bound.
A graph G is a B-0-VPG graph if it is the vertex intersection graph of horizontal and vertical paths on a grid. A graph G is a contact B-0-VPG graph if the vertices can be represented by interiorly disjoint horizontal...
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ISBN:
(数字)9783319961514
ISBN:
(纸本)9783319961514;9783319961507
A graph G is a B-0-VPG graph if it is the vertex intersection graph of horizontal and vertical paths on a grid. A graph G is a contact B-0-VPG graph if the vertices can be represented by interiorly disjoint horizontal or vertical paths on a grid and two vertices are adjacent if and only if the corresponding paths touch. In this paper, we present a minimal forbidden induced subgraph characterisation of contact B-0-VPG graphs within the class of chordal graphs and provide a polynomial-time algorithm for recognising these graphs.
Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total domin...
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ISBN:
(纸本)9783319961514;9783319961507
Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-dominating set, in a given graph, are referred to as k-domination, resp. total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from recent work by Kang et al. (2017) that these two families of problems are solvable in time O(vertical bar V(G)vertical bar(6k+4)) in the class of interval graphs. In this work, we develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs. The algorithms run in time O(vertical bar V(G)vertical bar(3k)) for each fixed k >= 1 and are also applicable to the weighted case.
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