Con-sider the uncapacitated network flow problem.(a) Show that every spanning tree determines a basic solution to the dualproblem

The uncapacitated network flow problem is a type of linear programming problem where the goal is to maximize or minimize the total flow of a product through a network, subject to constraints on the capacities of the arcs and the conservation of flow at the nodes.

The dual problem of the uncapacitated network flow problem is a linear programming problem that is derived from the primal problem by swapping the roles of the objective function coefficients and the right-hand side constants, and swapping the roles of the constraints and the variables.

A spanning tree of a network is a tree that includes every node of the network. In the context of the uncapacitated network flow problem, a spanning tree can be used to determine a basic solution to the dual problem.

Here's how:

1. Start with a spanning tree of the network. This tree will have n-1 arcs, where n is the number of nodes in the network.

2. Assign a dual variable to each node in the network. The value of the dual variable at a node is equal to the sum of the capacities of the arcs entering the node minus the sum of the capacities of the arcs leaving the node.

3. The basic solution to the dual problem is then given by the values of the dual variables at the nodes.

This shows that every spanning tree determines a basic solution to the dual problem. The reason is that the dual variables are determined by the capacities of the arcs in the spanning tree, and these capacities are the basic variables in the dual problem.