Set splitting problem

In computational complexity theory, the Set Splitting problem is the following decision problem: given a family F of subsets of a finite set S, decide whether there exists a partition of S into two subsets S₁, S₂ such that all elements of F are split by this partition, i.e., none of the elements of F is completely in S₁ or S₂. It is an NP-complete problem.

Variants

The optimization version of this problem is called Max Set Splitting and requires finding the partition which maximizes the number of split elements of F. It is an APX-complete (and NP-hard) problem. When each element of F is restricted to be of cardinality exactly k, the decision variant is called Ek-Set Splitting and the optimization version Max Ek-Set Splitting. For k ≥ 2, the former remains NP complete and the latter APX complete. The Weighted Set Splitting is a variant in which the subsets in F have weights and the objective is to maximize the total weight of the split subsets.

Connection to Other Problems

Set Splitting is special case of the Not-All-Equal Satisfiability problem without negated variables. Additionally, Ek-Set Splitting equals non-monochromatic coloring of k-uniform hypergraphs. For k=2, the optimization variant reduces to the well-known Maximum cut.

Approximability

For k ≥ 4, Ek-Set Splitting is approximation resistant. That is, unless P=NP, there is no polynomial-time (factor) approximation algorithm which does essentially better than a random partition.

Fixed-Parameter Tractability

An alternative formulation of the decision variant is the following: given an integer k, does there exist a partition of S which splits at least k subsets of F? This formulated is fixed-parameter tractable. That is, for every fixed k, there exists a polynomial-time algorithm for solving the problem.