for each possible unordered triple of items. The minimum version of the problem restricted to the tournaments was proven to have polynomial time approximation schemes (PTAS). One can achieve an approximation ratio of 1/3 (in expectation) by ordering the items randomly, and this simple strategy gives the best possible polynomial-time approximation if the unique games conjecture is true. It is also possible to use semidefinite programming or combinatorial methods to find an ordering that satisfies at least half of the triples of any satisfiable instance, in polynomial time. In parameterized complexity, the problem of satisfying as many constraints as possible from a set C of constraints is fixed-parameter tractable when parameterized by the difference q − |C|/3 between the solution quality q found by the parameterized algorithm and the |C|/3 quality guaranteed in expectation by a random ordering. Although not guaranteed to succeed, a greedy heuristic can find solutions to many instances of the betweenness problem arising in practice. Applications One application of betweenness arises in bioinformatics, as part of the process of gene mapping. Certain types of genetic experiments can be used to determine the ordering of triples of genetic markers, but do not distinguish a genetic sequence from its reversal, so the information yielded from such an experiment determines only which one out of three markers is the middle one. The betweenness problem is an abstraction of the problem of assembling a collection of markers into a single sequence given experimental data of this type. The betweenness problem has also been used to model theories of probability, causality, and time. References NP-complete problems Order theory