Approximation Algorithm for non-intersecting paths in a grid

Question

I recently came across this question and thought I could share it here, since I wasn\'t able to get it.

We are given a 5*5 grid numbered from 1-25, and a set of 5 pa

Answer 1

This problem is essentially the Hamiltonian path/cycle problem problem (since you can connect the end of one path to the start of another, and consider all the five paths as a part of one big cycle). There are no known efficient algorithms for this, as the problem is NP-complete, so you do essentially need to try all possible paths with backtracking (there are fancier algorithms, but they're not much faster).

Your title asks for an approximation algorithm, but this is not an optimization problem - it's not the case that some solutions are better than others; all correct solutions are equally good, and if it isn't correct, then it's completely wrong - so there is no possibility for approximation.

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Edit: The below is a solution to the original problem posted by the OP, which did not include the "all cells must be covered" constraint. I'm leaving it up for those that might face the original problem.

This can be solved with a maximum flow algorithm, such as Edmonds-Karp.

The trick is to model the grid as a graph where there are two nodes per grid cell; one "outgoing" node and one "incoming" node. For each adjacent pair of cells, there are edges from the "outgoing" node in either cell to the "incoming" node in the other cell. Within each cell, there is also an edge from the "incoming" to the "outgoing" node. Each edge has the capacity 1. Create one global source node that has an edge to all the start nodes, and one global sink node to which all end nodes have an edge.

Then, run the flow algorithm; the resulting flow shows the non-intersecting paths.

This works because all flow coming in to a cell must pass through the "internal" edge from the "incoming" to the "ougoing" node, and as such, the flow through each cell is limited to 1 - therefore, no paths will intersect. Also, Edmonds-Karp (and all Floyd-Warshall based flow algorithms) will produce integer flows as long as all capacities are integers.