Julia - Generate 2-matching odd Set - optimization

Question

In Julia, given a Set{Tuple{Int, Int}} named S of length greater than 3, for instance:
julia> S = Set{Tuple{Int,Int}}([(1, 4), (2, 5), (2, 6), (3, 6)])
Set{Tuple{Int64,Int64}} with 4 elements:
(2, 5)
(3, 6)
(2, 6)
(1, 4)
I want to return a subset T of S of length greater than 3 and odd (3, 5, 7, ...) such that, all first values of the tuples are unique. For instance, I can't have (2, 5) and (2, 6) because first value, 2 will not be unique. The same applies for second values meaning that I can't have (2, 6) and (3, 6).
If it is not possible, returning an empty Set of Tuple is fine.
Finally for the above minimal example the code should return:
julia> T = Set{Tuple{Int,Int}}([(1, 4), (2, 5), (3, 6)])
Set{Tuple{Int64,Int64}} with 3 elements:
(2, 5)
(3, 6)
(1, 4)
I am truly open to any other type of strucutre if you think it is better than Set{Tuple{Int, Int}} :)
I know how I can do it with integer programming. However, I will run this many times with large instances and I would like to know if there is a better way because I deeply think it can be done in polynomial time and perhaps in Julia with clever map or other efficient functions!

Answer 1

What you need is a way to filter the possible combinations of members of a set. So create a filtering function. If the part about an odd [3, 5, 7...] sequence you mentioned applies here, somehow, you might need to add that to the filter logic below:
using Combinatorics
allunique(a) = length(a) == length(unique(a))
slice(tuples, position) = [t[position] for t in tuples]
uniqueslice(tuples, position) = allunique(slice(tuples, position))
is_with_all_positions_unique(tuples) = all(n -> uniqueslice(tuples, n), 1:length(first(tuples)))
Now you can find combinations. With big sets these will explode in number, so make sure to exit when you have enough. You could use Lazy.jl here, or just a function:
function tcombinations(tuples, len, needed)
printed = 0
for combo in combinations(collect(tuples), len)
if is_with_all_positions_unique(combo)
printed += 1
println(combo)
printed >= needed && break
end
end
end
tcombinations(tuples, 3, 4)
[(2, 5), (4, 8), (3, 6)]
[(2, 5), (4, 8), (1, 4)]
[(2, 5), (4, 8), (5, 6)]
[(2, 5), (3, 6), (1, 4)]