<P> In the three - party set disjointness problem in communication complexity, three subsets of the integers in some range (1, m) are specified, and three communicating parties each know two of the three subsets . The goal is for the parties to transmit as few bits to each other on a shared communications channel in order for one of the parties to be able to determine whether the intersection of the three sets is empty or nonempty . A trivial m - bit communications protocol would be for one of the three parties to transmit a bitvector describing the intersection of the two sets known to that party, after which either of the two remaining parties can determine the emptiness of the intersection . However, if there exists a protocol that solves the problem with o (m) communication and 2 computation, it could be transformed into an algorithm for solving k - SAT in time O (1.74) for any fixed constant k, violating the strong exponential time hypothesis . Therefore, the strong exponential time hypothesis implies either that the trivial protocol for three - party set disjointness is optimal, or that any better protocol requires an exponential amount of computation . </P> <P> If the exponential time hypothesis is true, then 3 - SAT would not have a polynomial time algorithm, and therefore it would follow that P ≠ NP . More strongly, in this case, 3 - SAT could not even have a quasi-polynomial time algorithm, so NP could not be a subset of QP . However, if the exponential time hypothesis fails, it would have no implication for the P versus NP problem . There exist NP - complete problems for which the best known running times have the form O (2) for c <1, and if the best possible running time for 3 - SAT were of this form, then P would be unequal to NP (because 3 - SAT is NP - complete and this time bound is not polynomial) but the exponential time hypothesis would be false . </P> <P> In parameterized complexity theory, because the exponential time hypothesis implies that there does not exist a fixed - parameter - tractable algorithm for maximum clique, it also implies that W (1) ≠ FPT . It is an important open problem in this area whether this implication can be reversed: does W (1) ≠ FPT imply the exponential time hypothesis? There is a hierarchy of parameterized complexity classes called the M - hierarchy that interleaves the W - hierarchy in the sense that, for all i, M (i) ⊆ W (i) ⊆ M (i + 1); for instance, the problem of finding a vertex cover of size k log n in an n - vertex graph with parameter k is complete for M (1). The exponential time hypothesis is equivalent to the statement that M (1) ≠ FPT, and the question of whether M (i) = W (i) for i> 1 is also open . </P> <P> It is also possible to prove implications in the other direction, from the failure of a variation of the strong exponential time hypothesis to separations of complexity classes . As Williams (2010) shows, if there exists an algorithm A that solves Boolean circuit satisfiability in time 2 / ƒ (n) for some superpolynomially growing function ƒ, then NEXPTIME is not a subset of P / poly . Williams shows that, if algorithm A exists, and a family of circuits simulating NEXPTIME in P / poly also existed, then algorithm A could be composed with the circuits to simulate NEXPTIME problems nondeterministically in a smaller amount of time, violating the time hierarchy theorem . Therefore, the existence of algorithm A proves the nonexistence of the family of circuits and the separation of these two complexity classes . </P>

If p = np does that imply that eth is false justify your answer