Shortest Path Problem - Linear Programming Formulation

Linear Programming Formulation

There is a natural linear programming formulation for the shortest path problem, given below. It is very trivial compared to most other uses of linear programs in discrete optimization, however it illustrates connections to other concepts.

Given a directed graph (V, A) with source node s, target node t, and cost w_ij for each arc (i, j) in A, consider the program with variables x_ij

minimize subject to and for all i,

This LP, which is common fodder for operations research courses, has the special property that it is integral; more specifically, every basic optimal solution (when one exists) has all variables equal to 0 or 1, and the set of edges whose variables equal 1 form an s-t dipath. See Ahuja et al. for one proof, although the origin of this approach dates back to mid-20th century.

The dual for this linear program is

maximize y_t − y_s subject to for all ij, y_j − y_i ≤ w_ij

and feasible duals correspond to the concept of a consistent heuristic for the A* algorithm for shortest paths. For any feasible dual y the reduced costs are nonnegative and A* essentially runs Dijkstra's algorithm on these reduced costs.