Given a set P of n points in the plane, we want to find a simple, not necessarily convex, pentagon 2 with vertices in P of minimum area. We present an algorithm for solving this problem in time O(nT(n)) and space O(n)...
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Given a set P of n points in the plane, we want to find a simple, not necessarily convex, pentagon 2 with vertices in P of minimum area. We present an algorithm for solving this problem in time O(nT(n)) and space O(n), where T(n) is the number of empty triangles in the set. (C) 1997 Elsevier Science B.V. All rights reserved.
A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the initial graph. In this work we study the problem of finding tropical paths in vertex-colored graphs. There are two ver...
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A subgraph of a vertex-colored graph is said to be tropical whenever it contains each color of the initial graph. In this work we study the problem of finding tropical paths in vertex-colored graphs. There are two versions for this problem: the shortest tropical path problem (STPP), i.e., finding a tropical path with the minimum total weight, and the maximum tropical path problem (MTPP), i.e., finding a path with the maximum number of colors possible. We show that both versions of this problems are NP-hard for directed acyclic graphs, cactus graphs and interval graphs. Moreover, we also provide a fixed parameter algorithm for STPP in general graphs and several polynomial-time algorithms for MTPP in specific graphs, including bipartite chain graphs, threshold graphs, trees, block graphs, and proper interval graphs.
Each fixed graph H gives rise to a list H-colouring problem. The complexity of list H-colouring problems has recently been fully classified: if H is a bi-arc graph, the problem is polynomial-time solvable, and otherwi...
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Each fixed graph H gives rise to a list H-colouring problem. The complexity of list H-colouring problems has recently been fully classified: if H is a bi-arc graph, the problem is polynomial-time solvable, and otherwise it is NP-complete. The proof of this fact relies on a forbidden substructure characterization of bi-arc graphs, reminiscent of the theorem of Lekkerkerker and Boland on interval graphs. In this note we show that in fact the theorem of Lekkerkerker and Boland can be derived from the characterization. (c) 2005 Elsevier B.V. All rights reserved.
In this paper, we consider the problem of scheduling jobs on parallel machines with setup times. The setup has to be performed by a single server. The objective is to minimize the schedule length (makespan), as well a...
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In this paper, we consider the problem of scheduling jobs on parallel machines with setup times. The setup has to be performed by a single server. The objective is to minimize the schedule length (makespan), as well as the forced idle time. The makespan problem is known to be NP-hard even for the case of two identical parallel machines. This paper presents a pseudopolynomial algorithm for the case of two machines when all setup times are equal to one. We also show that the more general problem with an arbitrary number of machines is unary NP-hard and analyze some list scheduling heuristics for this problem. The problem of minimizing the forced idle time is known to be unary NP-hard for the case of two machines and arbitrary setup and processing times. We prove unary NP-hardness of this problem even for the case of constant setup times. Moreover, some polynomially solvable cases are given.
Using the notion of modular decomposition we extend the class of graphs on which both the TREEWIDTH and the MINIMUM FILL-IN can be solved in polynomial time. We show that if C is a class of graphs that are modularly d...
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Using the notion of modular decomposition we extend the class of graphs on which both the TREEWIDTH and the MINIMUM FILL-IN can be solved in polynomial time. We show that if C is a class of graphs that are modularly decomposable into graphs that have a polynomial number of minimal separators, or graphs formed by adding a matching between two cliques, then both the TREEWIDTH and the MINIMUM FILL-IN on C can be solved in polynomial time. For the graphs that are modular decomposable into cycles we give algorithms that use respectively 0 (n) and 0 (n(3)) time for TREEWIDTH and MINIMUM FILL-IN.
Given a bidirected graphG and a vectorb of positive integral node-weights, an integer linear program IP is defined on (G, b). IP generalizes the node packing problem on a node-weighted (undirected) graph in the sense ...
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Given a bidirected graphG and a vectorb of positive integral node-weights, an integer linear program IP is defined on (G, b). IP generalizes the node packing problem on a node-weighted (undirected) graph in the sense that it reduces to the latter whenG is undirected. A polynomial time algorithm is given that recognizes whether CP (the linear program obtained by relaxing the integrality constraints of IP) has an integral optimal solution. Also an efficient method for solving the linear programming dual of CP is described.
Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains ...
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Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequence, a new polynomially solvable case for the maximum stable set problem has been detected. (C) 2003 Elsevier Science B.V. All rights. reserved.
In a single item dynamic lot-sizing problem, we are given a time horizon and demand for a single item in every time period. The problem seeks a solution that determines how much to produce and carry at each time perio...
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In a single item dynamic lot-sizing problem, we are given a time horizon and demand for a single item in every time period. The problem seeks a solution that determines how much to produce and carry at each time period, so that we will incur the least amount of production and inventory cost. When the remanufacturing option is included, the input comprises of number of returned products at each time period that can be potentially remanufactured to satisfy the demands, where remanufacturing and inventory costs are applicable. For this problem, we first show that it cannot have a fully polynomial time approximation scheme. We then provide a polynomial time algorithm, when we make certain realistic assumptions on the cost structure.
We consider single-machine scheduling problems with variable job processing times, arbitrary precedence constraints and maximum cost criterion. We show how to solve the problems in polynomial time in the cases when jo...
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We consider single-machine scheduling problems with variable job processing times, arbitrary precedence constraints and maximum cost criterion. We show how to solve the problems in polynomial time in the cases when job processing times are described by functions of the same type or when they are mixed, i.e. some of them are fixed, while the other ones are variable and take into account the effects of learning, ageing or job deterioration.
We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges,...
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We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges, so that the total weight of the partition is minimized. We explore the frontier between polynomial and NP-hard variants of the problem, with respect to the class of the underlying graph, as well as the approximability of NP-hard variants. In particular, we present polynomial algorithms for bounded degree trees and star of chains, as well as an approximation algorithm for bipartite graphs of maximum degree at most twelve which beats the best known approximation ratios.
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