We consider project scheduling where the project manager's objective is to minimize the time from when an adversary discovers the project until the completion of the project. We analyze the complexity of the probl...
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We consider project scheduling where the project manager's objective is to minimize the time from when an adversary discovers the project until the completion of the project. We analyze the complexity of the problem identifying both polynomially solvable and NP-hard versions of the problem. The complexity of the problem is seen to be dependent on the nature of renewable resource constraints, precedence constraints, and the ability to crash activities in the project. (C) 2014 Elsevier B.V. All rights reserved.
We present a polynomial algorithm for a family of single-machine scheduling problems with mixed variable job processing times, -partite job precedence constraints and the maximum cost criterion, provided that job proc...
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We present a polynomial algorithm for a family of single-machine scheduling problems with mixed variable job processing times, -partite job precedence constraints and the maximum cost criterion, provided that job processing times satisfy certain assumptions.
In this paper, ellipsoidal estimations are used to track the central path of linear programming. A higher-order interior-point algorithm is devised to search the optimizers along the ellipse. The algorithm is proved t...
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In this paper, ellipsoidal estimations are used to track the central path of linear programming. A higher-order interior-point algorithm is devised to search the optimizers along the ellipse. The algorithm is proved to be polynomial with the best complexity bound for all polynomial algorithms and better than the best known bound for higher-order algorithms.
An implicit linear description of the stable matching polytope is provided in terms of the blocker and antiblocker sets of constraints of the matroid-kernel polytope. The explicit identification of both these sets is ...
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An implicit linear description of the stable matching polytope is provided in terms of the blocker and antiblocker sets of constraints of the matroid-kernel polytope. The explicit identification of both these sets is based on a partition of the stable pairs in which each agent participates. Here, we expose the relation of such a partition to rotations. We provide a time-optimal algorithm for obtaining such a partition and establish some new related results;most importantly, that this partition is unique. (C) 2013 Elsevier B.V. All rights reserved.
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.
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 class of polynomially solvable problems of combinatorial optimization is illustrated by the example of the traveling salesman problem. It is proved that this class includes problems for which the structure of initia...
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A class of polynomially solvable problems of combinatorial optimization is illustrated by the example of the traveling salesman problem. It is proved that this class includes problems for which the structure of initial data is specially modeled.
In this paper, we propose interior-point algorithms for P*(kappa)-linear complementarity problem based on a new class of kernel functions. New search directions and proximity measures are defined based on these functi...
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In this paper, we propose interior-point algorithms for P*(kappa)-linear complementarity problem based on a new class of kernel functions. New search directions and proximity measures are defined based on these functions. We show that if a strictly feasible starting point is available, then the new algorithm has O(1 + 2 kappa)root n log n log n mu(0)/epsilon and O(1 + 2 kappa)v n log n mu(0)/epsilon iteration complexity for large-and small-update methods, respectively. These are the best known complexity results for such methods.
Polar, monopolar, and unipolar graphs are defined in terms of the existence of certain vertex partitions. Although it is polynomial to determine whether a graph is unipolar and to find whenever possible a unipolar par...
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Polar, monopolar, and unipolar graphs are defined in terms of the existence of certain vertex partitions. Although it is polynomial to determine whether a graph is unipolar and to find whenever possible a unipolar partition, the problems of recognizing polar and monopolar graphs are both NP-complete in general. These problems have recently been studied for chordal, claw-free, and permutation graphs. polynomial time algorithms have been found for solving the problems for these classes of graphs, with one exception: polarity recognition remains NP-complete in claw-free graphs. In this paper, we connect these problems to edge-coloured homomorphism problems. We show that finding unipolar partitions in general and finding monopolar partitions for certain classes of graphs can be efficiently reduced to a polynomial-time solvable 2-edge-coloured homomorphism problem, which we call the colour-bipartition problem. This approach unifies the currently known results on monopolarity and extends them to new classes of graphs.
Let alpha(G) denote the independence number of graph G, i.e., the size of any maximum independent set of vertices. A graph G is contraction-critical (with respect to alpha) if alpha(G/(xy) over cap) denotes the graph...
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Let alpha(G) denote the independence number of graph G, i.e., the size of any maximum independent set of vertices. A graph G is contraction-critical (with respect to alpha) if alpha(G/(xy) over cap) < alpha(G), for every edge xy is an element of E(G), where G/<(xy)over cap> denotes the graph obtained from G by shrinking the edge xy to a single vertex (xy) over cap and deleting the loop thus formed. Let us denote the class of all such contraction-critical graphs by CCR. First it is shown that all graphs in CCR must be bipartite. Let G = (A, B) be a bipartite graph with bipartition A boolean OR B. It is then shown that (a) if vertical bar A vertical bar < vertical bar B vertical bar, then G is an element of CCR if and only if B is the unique maximum independent set in G if and only if G is 1-expanding, while (b) if vertical bar A vertical bar = vertical bar B vertical bar, then G is an element of CCR if and only if A and B are the only two maximum independent sets in G if and only if G is 1-extendable. It follows that CCR graphs can be recognized in polynomial time. (C) 2014 Elsevier B.V. All rights reserved.
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