Weighted k‐cardinality trees: Complexity and polyhedral structure

Abstract

We consider the k-CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with kedges, having minimum weight. Applications of this problem arise in oil-field leasing and facility layout. Although the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k-CARD TREE and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex hull of the integer solutions is studied. 0 7994 by

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

“…Theorem 1: All vectors x and y satisfying (1)(2)(3)(4)(5)(6)(7)(8) are associated with connected subgraphs of G.…”

“…Theorem 2: All vectors x and y satisfying (1)(2)(3)(4) and (6)(7)(8)(9)(10)(11) are associated with elementary cycles of G.…”

“…First of all we characterize all subtrees in G with a given number p n ≤ of nodes, see Fischetti et al (1984). For this we have to consider new constraints as follows.…”

Combining Ant Colony Optimization with Dynamic Programming for Solving the k-Cardinality Tree Problem

“…The k-cardinality tree (KCT) problem-also referred to as the k-minimum spanning tree (k-MST) problem, or just the k-tree problem-is an NP-hard [13] combinatorial optimization problem which generalizes the well-known minimum weight spanning tree problem. In this paper we deal with a generalized problem version in which the given graph G can have both node and edge weights.…”

Ant System for the k-Cardinality Tree Problem

“…Under the heuristic category, an integer programming approach is given in [8], and a Branch and Bound approach is given in [5]. These heuristic algorithms are based on the greedy and dual greedy strategies in addition to dynamic programming technique.…”