Abstract
Finding a minimum spanning tree in a given network is a famous combinatorial optimization problem that appears in different engineering applications. Even though this problem is solvable in polynomial time, having efficient mathematical programming models is important as they can provide insights for formulating larger models that integrate other decisions in more complex applications. In the literature, there are ten different integer and mixed integer linear programming (MILP) models for this problem. They are variants of set packing, cuts, network flow and node level formulations. In addition, this paper introduces an efficient node level MILP model. Comparisons for the eleven models are provided. First, the models are compared in terms of the number of decision variables and the number of constraints. Then, computational comparisons using a commercial MILP solver on sets of randomly generated instances of different sizes are conducted. Results provide evidence that the proposed MILP model is competitive in terms of the computational time needed for proving optimality of generated solutions for instances with up to 50 nodes. Meanwhile, the LP relaxation of a multi-commodity flow MILP model which has integer polyhedron provides stable computational times despite its larger size.
Highlights
For a given undirected, connected network W = (N, A), where N is the set of nodes, A is the set of arcs and la is the length of arc a ∈ A, the minimum spanning tree (MST) is a subset of arcs M ⊂ A
The first aspect to consider in comparing the eleven models presented is the memory size needed by an mixed integer linear programming (MILP) solution algorithm, which is represented by the number of decision variables and the number of constraints
The advantage of the proposed model is attributed to its smaller size, yet the algorithmic techniques used for solving it are not polynomial in time. The latter model is easier to solve using polynomial time algorithms, but due to its larger size, the computational time needed can be larger than the proposed model
Summary
An arc can be unselected to reduce the summation of the selected arcs’ lengths while the involved nodes remain connected This set of arcs does not represent a minimum spanning tree. There are ten different integer and mixed integer linear programming models for the MST problem They are variants of set packing, cuts, network flow and node level formulations. The set packing formulation is based on defining constraints that restrict the cardinality of any selected subset of arcs not to exceed the number of connected nodes less than one. Numerical experiments are conducted using randomly generated instances of various sizes to provide computational comparisons between a selected subset of the models that have polynomial number of constraints in the size of the network represented by the number of nodes.
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