Question

I have a problem at hand which can be reduced to something like this :

Assume a bunch of random points in a two-dimension plane X-Y where for each Y, there could be multiple points on X and for each X, there could be multiple points on Y.

Whenever a point (Xi,Yi) is chosen, no other point with X = Xi OR Y = Yi can be chosen. We have to choose the maximum number of points.

Answer 1

This can be reduced to simple maximum flow problem. If you have a point (xi, yi), in graph it should be represented with path from source S to point xi, from xi to yi and from yi to the last node (sink) T.

Note, if we have points (2, 2) and (2, 5), there’s still only one path from S to x2. All paths (edges) have capacity 1.

The flow in this network is the answer.

about general problem
http://en.wikipedia.org/wiki/Max_flow

update
I don’t have graphic editor right now to visualise problem, but you can easily draw example by hand. Let’s say, points are (3, 3) (3, 5) (2, 5)

Then edges (paths) would be
S -> x2, S -> x3
y3 -> T, y5 -> T
x3 -> y3, x3 -> y5, x2 -> y5

Flow: S -> x2 -> y5 -> T and S -> x3 -> y3 -> T
The amount of ‘water’ going from source to sink is 2 and so is the answer.

Also there’s a tutorial about max flow algorithms
http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=maxFlow