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We show that two important problems that have applications in computationalbiology are ASP-complete, which implies that, given a solution to a problem, itis NP-complete to decide if another solution exists. We show first that avariation of Betweenness, which is the underlying problem of questions relatedto radiation hybrid mapping, is ASP-complete. Subsequently, we use that resultto show that Quartet Compatibility, a fundamental problem in phylogenetics thatasks whether a set of quartets can be represented by a parent tree, is alsoASP-complete. The latter result shows that Steel's \sc Quartet Challenge, whichasks whether a solution to Quartet Compatibility is unique, is coNP-complete.

The Complexity of Finding Multiple Solutions to Betweenness and Quartet Compatibility