Pipeline networks can be viewed as a graph built up of edges (pipeline segments) and nodes (junctions of pipeline segments). Suppose that within this network of pipelines, we want to determine a set of junctions from which we can monitor every pipeline segment. This problem can be modeled as a vertex cover problem in which we want to find a subset of nodes in a graph such that every edge has at least one end point in our subset.

Let's look at a simple example. Below we have a graph representing a simple network of pipelines with seven junctions. Note that each junction is identical in terms of its utility as a monitoring location.

Problem: Given the above set of junctions, which ones should you choose such that you minimize the number selected for monitoring, while still monitoring every segment?

Solution: One possible solution is indicated by the red nodes below.

To run the demo:

python pipelines.py

The program pipelines.py creates a graph using the Python package networkx, and then uses the Ocean software tools to run the minimum_vertex_cover function from within the dwave_networkx package.

A valid vertex cover is a subset of nodes in the graph such that every edge in the graph has at least one end point in the subset. Visual inspection shows that we will need either 1 or 2 in the solution; we will also need 6 or 7; and we will also need 4 or 5. We therefore know that a minimum vertex cover will have at least 3 nodes; it turns out that there must be four.

Gamma, our Lagrange parameter (see Lagrange Multiplier for a discussion of Lagrange multiplier, an analogous concept), weights the constraints in the problem versus the energy term (objective). If the Lagrange parameter is too small relative to the strength of the energy term, the constraints may be violated.

With a Lagrange parameter that is too weak, the solutions are sets which are too small. In the minimum_vertex_cover function from within the dwave_networkx package, the default of the Lagrange parameter is 2. Repeated runs, using the default, produced sets which had size 2 or 3.

Hence, the Lagrange parameter value of 5 was set large enough to (1) balance the energy term and constraints, and to (2) cause the minimum vertex cover conditions to all be satisfied for this problem.

Released under the Apache License 2.0. See LICENSE file.