In a previous paper we obtained an upper bound for the minimum number of components of a 2-factor in a claw-free graph. This bound is sharp in the sense that there exist infinitely many claw-free graphs for which the ...
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In a previous paper we obtained an upper bound for the minimum number of components of a 2-factor in a claw-free graph. This bound is sharp in the sense that there exist infinitely many claw-free graphs for which the bound is tight. In this paper we extend these results by presenting a polynomial algorithm that constructs a 2-factor of a claw-free graph with minimum degree at least four whose number of components meets this bound. As a byproduct we show that the problem of obtaining a minimum 2-factor (if it exists) is polynomially solvable for a subclass of claw-free graphs. As another byproduct we give a short constructive proof for a result of Ryjacek, Saito and Schelp. (C) 2009 Elsevier B.V. All rights reserved.
The paper deals with the determination of an optimal schedule for the so-called mixed-shop problem when the makespan has to be minimized. In such a problem, some jobs have fixed machine orders (as in the job-shop), wh...
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The paper deals with the determination of an optimal schedule for the so-called mixed-shop problem when the makespan has to be minimized. In such a problem, some jobs have fixed machine orders (as in the job-shop), while the operations of the other jobs may be processed in arbitrary order (as in the open-shop). We prove binary NP-hardness of the preemptive problem with three machines and three jobs (two jobs have fixed machine orders and one may have an arbitrary machine order). We answer all other remaining open questions on the complexity status of mixed-shop problems with the makespan criterion by presenting different polynomial and pseudopolynomial algorithms.
In this paper, we propose a bicriterion objective function for clustering a given set of N entities, which minimizes [alpha d - (1 - alpha)s], where 0 less than or equal to alpha less than or equal to 1, and d and s a...
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In this paper, we propose a bicriterion objective function for clustering a given set of N entities, which minimizes [alpha d - (1 - alpha)s], where 0 less than or equal to alpha less than or equal to 1, and d and s are the diameter and the split of the clustering, respectively. When alpha = 1, the problem reduces to minimum diameter clustering, and when alpha = 0, maximum split clustering. We show that this objective provides an effective way to compromise between the two often conflicting criteria. While the problem is NP-hard in general, a polynomial algorithm with the worst-case time complexity O(N-2) is devised to solve the bipartition version. This algorithm actually gives all the Pareto optimal bipartitions with respect to diameter and split, and it can be extended to yield an efficient divisive hierarchical scheme. An extension of the approach to the objective [alpha(d(1) + d(2)) - 2(1 - alpha)s] is also proposed, where d(1) and d(2) are diameters of the two clusters of a bipartition.
We consider a robust (minmax-regret) version of the problem of selecting p elements of minimum total weight out of a set of In elements with uncertainty in weights of the elements. We present a polynomial algorithm wi...
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We consider a robust (minmax-regret) version of the problem of selecting p elements of minimum total weight out of a set of In elements with uncertainty in weights of the elements. We present a polynomial algorithm with the order of complexity O((min {p, m - p})(2) m) for the case where uncertainty is represented by means of interval estimates for the weights. We show that the problem is NP-hard in the case of an arbitrary finite set of possible scenarios, even if there are only two possible scenarios. This is the first known example of a robust combinatorial optimization problem that is NP-hard in the case of scenario-represented uncertainty bur is polynomially solvable in the case of the interval representation of uncertainty.
An apple A (k) is the graph obtained from a chordless cycle C (k) of length k a parts per thousand yen 4 by adding a vertex that has exactly one neighbor on the cycle. The class of apple-free graphs is a common genera...
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An apple A (k) is the graph obtained from a chordless cycle C (k) of length k a parts per thousand yen 4 by adding a vertex that has exactly one neighbor on the cycle. The class of apple-free graphs is a common generalization of claw-free graphs and chordal graphs, two classes enjoying many attractive properties, including polynomial-time solvability of the maximum weight independent set problem. Recently, Brandstadt et al. showed that this property extends to the class of apple-free graphs. In the present paper, we study further generalization of this class called graphs without large apples: these are (A (k) , A (k+1), . . .)-free graphs for values of k strictly greater than 4. The complexity of the maximum weight independent set problem is unknown even for k = 5. By exploring the structure of graphs without large apples, we discover a sufficient condition for claw-freeness of such graphs. We show that the condition is satisfied by bounded-degree and apex-minor-free graphs of sufficiently large tree-width. This implies an efficient solution to the maximum weight independent set problem for those graphs without large apples, which either have bounded vertex degree or exclude a fixed apex graph as a minor.
Given a graphG = (N, E) and a length functionl: E → ?, the Graphical Traveling Salesman Problem is that of finding a minimum length cycle goingat least once through each node ofG. This formulation has advantages over...
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Given a graphG = (N, E) and a length functionl: E → ?, the Graphical Traveling Salesman Problem is that of finding a minimum length cycle goingat least once through each node ofG. This formulation has advantages over the traditional formulation where each node must be visited exactly once. We give some facet inducing inequalities of the convex hull of the solutions to that problem. In particular, the so-called comb inequalities of Gr?tschel and Padberg are generalized. Some related integer polyhedra are also investigated. Finally, an efficient algorithm is given whenG is a series-parallel graph.
In this note we give a short and relatively simple algorithmic proof of a theorem of Benczur and Frank on covering a symmetric crossing supermodular function with a minimum number of graph edges. Our proof method also...
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In this note we give a short and relatively simple algorithmic proof of a theorem of Benczur and Frank on covering a symmetric crossing supermodular function with a minimum number of graph edges. Our proof method also implies a deficient form of the theorem. (C) 2017 Elsevier B.V. All rights reserved.
Minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension (Grotschel et al., 1988). We provide an alternative, short, and geometrically motivated proof of this resu...
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Minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension (Grotschel et al., 1988). We provide an alternative, short, and geometrically motivated proof of this result. In particular, we present an oracle-polynomial algorithm based on a mixed integer linear optimization oracle. (C) 2014 Elsevier B.V. All rights reserved.
A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstadt, V.B. Le, T. Szymczak. The complexity of some problems related to graph 3-colorability, Discrete Appl...
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A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstadt, V.B. Le, T. Szymczak. The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59-73) that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such that G - S is a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2. check whether G has a stable set S such that G - S is a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G - S is a tree. (c) 2006 Elsevier B.V. All rights reserved.
This paper describes a class of networks called hierarchical networks and shows how reliability can be computed efficiently in a hierarchical network. In particular, it describes an algorithm for the K-terminal proble...
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This paper describes a class of networks called hierarchical networks and shows how reliability can be computed efficiently in a hierarchical network. In particular, it describes an algorithm for the K-terminal problem: computing the probability that a set of k nodes (k > 1) can communicate through the network. The algorithm complexity is polynomial with a small degree. The exact degree depends on the distribution of the nodes that desire to communicate and the external gateways.
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