One of the (many) hard problems that arises in genome mapping can be formulated in the following abstract way. We are given a set of n markers {μ1, …, μn}–these are positions on a chromosome that we are trying to map–and our goal is to output a linear ordering of these markers. The output should be consistent with a set of k constraints, each specified by a triple (μi, μj, μk), requiring that μj lie between μt and μk in the total ordering that we produce. (Note that this constraint does not specify which of μi or μk should come first in the ordering, only that μj should come between them.)
Now it is not always possible to satisfy all constraints simultaneously, so we wish to produce an ordering that satisfies as many as possible. Unfortunately, deciding whether there is an ordering that satisfies at least k′ of the k constraints is an NP-complete problem (you don't have to prove this.)
Give a constant α> 0 (independent of n) and an algorithm with the following property. If it is possible to satisfy k* of the constraints, then the algorithm produces an ordering of markers satisfying at least αk* of the constraints. Your algorithm may be randomized; in this case it should produce an ordering for which the expected number of satisfied constraints is at least αk*.
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