91Ó°ÊÓ

1.SET – SPLITING is in NP :

SET – SPLITING is in NP because we can verify in polynomial time that no subsetis monochromatic.

3 SAT ${â\copyright ½}_{\text{p}}$SET - SPLITING :

To prove that the problem is NP complete, we give a polynomial time reduction from S A T to SETSPLITING$\mathrm{Ï•}$.

The splitting is done as follows:

If is satisfiable, consider a satisfying assignment.

If we color all the true literals red, all the false ones are blue, and y blue, then every subset$C_i$ of$S$ has at least one red element (because it is satisfiable and it also contain one blue element y.

In addition, for a given splitting $⟨S,CâŸ\copyright$, we can able to set the literals that are colored differently from y to true.

In the same way, we can able to set the literals that have the same color as y to false.

This concludes that satisfying assignment for $\mathrm{Ï•}$.

Thus, SET - SPLITTING is NP-Complete.