minimizing total path cost with obstacles and spatial constraints

Question

I have a network of nodes arranged on a 2D Grid. I want to connect pairs of nodes with connections that will then occupy physical space on the 2D grid. The connections are now obstacles themselves and future connections will have to take a path that avoids intersecting them.

I am currently using the A* algorithm and gradually building up the connection. While it finds the shortest path from start to end node, it does not consider the other connections that will need to be made, so the total path cost after connecting all the pairs is not optimal.

Does anyone know whether there is an algorithm that can solve this, or is this a NP complete problem? Any direction on related material would also be appreciated.

My intuition says it is NPC, since the restriction is on the path, but that's just an intuition :| Very interesting question! (+1) — amit, Feb 06 '14 at 14:54
Do you want to solve something like this: https://www.google.com/images?q=circuit+board+auto+routing ? — pentadecagon, Feb 06 '14 at 15:36
How big is the populated grid and how many pairs of connections? — גלעד ברקן, Feb 06 '14 at 16:50
I'm confused by what you mean by "intersecting". Do the paths need to stay on the grid (hopping from one node to the next), or can they weave-around 2D space however they'd like? If the former, how could two edges intersect? — BlueRaja - Danny Pflughoeft, Feb 06 '14 at 17:39

score 0 · Accepted Answer · answered Feb 06 '14 at 23:16

It seems you are trying to find minimum genus graph embedding, where your nodes are the graph vertices and the required connections are its edges. Minimal genus in your case can be interpreted as minimal number of edge crossings needed. Looking for connections of the minimal lengths makes it even harder (or does it?)

Minimum graph genus itself is NP-complete (in particular its decision version - whether a graph can be embedded in a surface of genus less than k). Therefore your problem, if the task would be to find such a path with minimal number of crossings, would be NP-hard as well.

However, if you consider only graphs that are planar, then finding a planar embedding can be done in linear time.

minimizing total path cost with obstacles and spatial constraints

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