Ideas for an algorithm for random distribution of circles in a square

Question

I am searching for a concept to distribute circles in a square randomly, so that they dont overlap. All circles are of the same size. The area covered by the circles can be high

Answer 1

Suppose you want n = 200 circles. My suggestion is to pick a number moderately larger than n, say m = 300, and generate that many points at random locations within the square. This set of m points is your candidate set of circle centres. Now produce a graph containing m vertices, one for each point, in which two vertices are linked by an edge if and only if the distance between their points is less than the circle diameter -- it is exactly in this situation that we would be forbidden to include both circles in the solution, because they would overlap.

You can now model the problem of choosing which candidate circle centres should actually become circles as a maximum independent set problem: find a maximum-sized set of vertices in this graph having the property that no two vertices in the set are linked by an edge. This will find a set of circle centres such that no two of their circles would overlap. If this set contains more than n circles, then discard a random selection of circles until just n remain. If fewer than n circles are found, you'll need to increase m and try again.

Unfortunately the maximum independent set problem is NP-hard, so I don't know whether it will be feasible to solve on a 300-vertex graph. (And in turn, I don't know whether 300 random centres will give you enough flexibility to find 200 non-overlapping circles...) But in any case, you would normally solve maximum independent set in one of two ways:

1. Look for a minimum vertex cover, then take every other vertex to be the maximum independent set
2. Produce the complement graph (i.e. the graph in which two vertices are linked by an edge if and only if they are not linked by an edge in the original graph), and then find a maximum clique in this graph

Those Wikipedia pages contain links to papers describing algorithms for these problems that, while still exponential-time, are much faster than a standard "full backtracking" algorithm. A couple of things to bear in mind:

• You don't actually need the provably maximum independent set, just one >= n. So heuristics may be just fine; note the particularly simple heuristic for minimum vertex cover on the Wikipedia page.
• Beware the difference between "maximal" (easy) and "maximum" (hard) cliques/covers/independent sets!