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Many numerical computations reported in the literature show only a small difference between the optimal value of the one-dimensional cutting stock problem (1CSP) and that of the corresponding linear programming relaxation. Moreover, theoretical investigations have proven that this difference is smaller than 2 for a wide range of subproblems of the general 1CSP.
A recently introduced dualization technique for binary linear programs with equality constraints, essentially due to Poljak et al. [13], and further developed in Lemaréchal and Oustry [9], leads to simple alternative derivations of well-known, important relaxations to two well-known problems of discrete optimization: the maximum stable set problem and the maximum vertex cover problem. The resulting relaxation is easily transformed to the well-known Lovász number.
A recently introduced
dualization technique for binary linear programs with equality
constraints, essentially due to Poljak et al. [13],
and further developed in Lemaréchal and Oustry [9], leads
to simple alternative derivations of well-known, important
relaxations to
two well-known problems of discrete optimization: the
maximum stable set problem and the maximum vertex cover problem.
The resulting relaxation is easily transformed
to the well-known Lovász θ number.
AMS Subj. Classification: 90C27, 05C85, 90C59The topic is related to solving the generalized vertex cover problem (GVCP) by genetic
algorithm. The problem is NP-hard as a generalization of well-known vertex cover problem
which was one of the first problems shown to be NP-hard. The definition of the GVCP and
basics of genetic algorithms are described. Details of genetic algorithm and numerical results
are presented in [8]. Genetic algorithm obtained high quality solutions in a short period of
time.
In this paper, we present a new mathematical programming formulation for the euclidean Steiner Tree Problem (ESTP) in . We relax the integrality constrains on this formulation and transform the resulting relaxation, which is convex, but not everywhere differentiable, into a standard convex programming problem in conic form. We consider then an efficient computation of an -optimal solution for this latter problem using interior-point algorithm.
In this paper, we present a
new mathematical programming formulation for the Euclidean Steiner
Tree Problem (ESTP) in ℜ. We relax the integrality
constrains on this formulation and transform the resulting
relaxation, which is convex, but not everywhere differentiable,
into a standard convex programming problem in conic form. We
consider then an efficient computation of an ϵ-optimal
solution for this latter problem using interior-point algorithm.
In this paper we address the two-dimensional knapsack problem with unloading constraints: we have a bin B, and a list L of n rectangular items, each item with a class value in {1,...,C}. The problem is to pack a subset of L into B, maximizing the total profit of packed items, where the packing must satisfy the unloading constraint: while removing one item a, items with higher class values can not block a. We present a (4 + ϵ)-approximation algorithm when the bin is a square. We also present (3 + ϵ)-approximation...
In this paper we present a dual approximation scheme for the class constrained shelf bin packing problem. In this problem, we are given bins of capacity , and items of different classes, each item with class and size . The problem is to pack the items into bins, such that two items of different classes packed in a same bin must be in different shelves. Items in a same shelf are packed consecutively. Moreover, items in consecutive shelves must be separated by shelf divisors of size . In...
In this paper we present a dual approximation scheme for the class
constrained shelf bin packing problem.
In this problem, we are given bins of capacity 1, and n items of
Q different classes, each item e with class ce and size
se. The problem is to pack the items into bins, such that
two items of different classes packed in a same bin must be in
different shelves.
Items in a same shelf are packed consecutively.
Moreover, items in consecutive shelves must be separated by shelf
divisors of size...
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