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. Set Splitting is one of Garey & Johnson's classical NP-complete problems.^[1] The problem is sometimes called hypergraph 2-colorability.

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^[2] problem and hence in NPO.

The set k-splitting problem is stated as follows: given S, F, and an integer k, does there exist a partition of S which splits at least k subsets of F? The original formulation is the restricted case with k equal to the cardinality of F. The Set k-Splitting is fixed-parameter tractable, i.e., if k taken to be a fixed parameter, rather than a part of the input, then a polynomial algorithm exists for any fixed k. Dehne, Fellows and Rosamond presented an algorithm that solves it in time ${\displaystyle O(f(k)n^{c})}$ for some function f and constant c.^[3]

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 for k ≥ 2 the latter remains APX complete.^[4] 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.^[5][6]

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 graph coloring of k-uniform hypergraphs. For k=2, the optimization variant reduces to the well-known maximum cut.^[6]

References

1. Garey, Michael R.; Johnson, David S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W.H. Freeman. ISBN 0-7167-1045-5.
2. Petrank, Erez (1994). "The Hardness of Approximation: Gap Location". Computational Complexity. 4 (2). Springer: 133–157. doi:10.1007/BF01202286. S2CID 16433553.
3. Dehne, Frank; Fellows, Michael; Rosamond, Frances (2003). An FPT Algorithm for Set Splitting (PDF). Graph Theoretic Concepts in Computer Science (WG2003), Lecture Notes in Computer Science. Vol. 2880. Springer. pp. 180–191.
4. Lovász, László (1973). Coverings and Colorings of Hypergraphs. 4th Southeastern Conference on Combinatorics, Graph Theory, and Computing.
5. Håstad, Johan (2001). "Some Optimal Inapproximability Results". Journal of the ACM. 48 (4). Association for Computing Machinery: 798–859. doi:10.1145/502090.502098. S2CID 5120748.
6. Guruswami, Venkatesan (2003). "Inapproximability Results for Set Splitting and Satisfiability Problems with no Mixed Clauses". Algorithmica. 38 (3). Springer: 451–469. doi:10.1007/s00453-003-1072-z. S2CID 15541433.