TY - GEN

T1 - The Minimum Weight In-Tree Cover Problem

AU - Kamiyama, Naoyuki

AU - Katoh, Naoki

PY - 2008/12/1

Y1 - 2008/12/1

N2 - Given a directed graph D = (V,A) with a root s ∈ V such that a non-negative rational weight is associated with each arc in A, we consider the problem for finding a set of minimum weight k spanning in-trees rooted at s which cover A. Here the weight of k spanning in-trees is defined as the sum of weights of all arcs contained in these in-trees. We will show that this problem can be solved in polynomial time. For this, we first consider the set of linear inequalities in ℝA that coincides with the convex hull Pic(D) of a {pipe}A{pipe}-dimensional positive integral vector x such that we can cover A by k spanning in-trees rooted at s such that e ∈ A is contained in xe in-trees where ℝ represents the set of reals. After this, we will show that the separation problem for this polytope can be solved in polynomial time, which implies the polynomial time solvability of the minimum weight in-tree cover problem in conjunction with the ellipsoid method. Furthermore, we will consider the generalization of the minimum in-tree cover problem such that the input directed graph has multiple roots. Although this problem is still open, we give the generalization of the result presented by Vidyasankar [13] which is used to derive the set of linear inequalities which determine Pic(D) to the case of multiple roots.

AB - Given a directed graph D = (V,A) with a root s ∈ V such that a non-negative rational weight is associated with each arc in A, we consider the problem for finding a set of minimum weight k spanning in-trees rooted at s which cover A. Here the weight of k spanning in-trees is defined as the sum of weights of all arcs contained in these in-trees. We will show that this problem can be solved in polynomial time. For this, we first consider the set of linear inequalities in ℝA that coincides with the convex hull Pic(D) of a {pipe}A{pipe}-dimensional positive integral vector x such that we can cover A by k spanning in-trees rooted at s such that e ∈ A is contained in xe in-trees where ℝ represents the set of reals. After this, we will show that the separation problem for this polytope can be solved in polynomial time, which implies the polynomial time solvability of the minimum weight in-tree cover problem in conjunction with the ellipsoid method. Furthermore, we will consider the generalization of the minimum in-tree cover problem such that the input directed graph has multiple roots. Although this problem is still open, we give the generalization of the result presented by Vidyasankar [13] which is used to derive the set of linear inequalities which determine Pic(D) to the case of multiple roots.

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U2 - 10.1007/978-3-540-87477-5_17

DO - 10.1007/978-3-540-87477-5_17

M3 - Conference contribution

AN - SCOPUS:84885011609

SN - 9783540874768

T3 - Communications in Computer and Information Science

SP - 155

EP - 164

BT - Modelling, Computation and Optimization in Information Systems and Management Sciences - Second International Conference, MCO 2008, Proceedings

T2 - 2nd International conference on Modelling, Computation and Optimization in Information Systems and Management Sciences, MCO 2008

Y2 - 8 September 2008 through 10 September 2008

ER -