AI - PSO - Choosing the right representations

Question

Let's say I have a set S of N natural numbers and N subsets (S1, S2, ...Sn) of that set. I want to generate 2 subsets D1 and D2 ( D1 + D2 = S, D1 and D2 have no common elements ) so that D1 and D2 does not contain any of those N subsets.

Quick example:

S = 1 2 3 4 5

S1 = 1 4

S2 = 1 2

S3 = 1 2 3

S4 = 1 2 3 4

S5 = 1 2 4

D1 = 1 3 5

D2 = 2 4

My first thought is that the position occupied by a particle will describe the way the elements are chosen (let's say position is an array with N BYTE elements, if position[i] is 1, Set[i] is in D1, 2 in D2, to make it simple).

The fitness of a solution could be N - the number of initial subsets that are included in the solution.

But what will be the velocity? The fact that I can't figure out this part makes me think that maybe I need to represent the position in another way, but again, I can't find something that will over-complicate the situation.

I'm more interested in theoretical answers. What way should I represent the data and why.

I'm new to this PSO thing so any good reads (beginner level) on the subject are appreciated.

Answer 1

As amit suggested might be the case, this is in fact an NP-hard problem. Given a CNF formula, let S be in one-to-one correspondence with the set of all literals (positive and negative) plus one extra pair T and F. Make a set {T, F}. Make one two-element set per variable containing the positive and negative literal for that variable, so that one shares a set with T, and the other, with F. For each clause of the disjunction, make a set with its literals and F. The solutions to the CNF-SAT instance and the instance of this problem are in one-to-one correspondence, by assigning true to all literals sharing the set with T.

If you want to solve this problem, I would suggest a SAT solver, since it's not hard to express it in CNF. If you want to learn about particle swarm optimization, I would suggest a different problem, with a continuous solution space.