Calculating competitive intransitivity: computational challenges

Intransitive, or 'rock-paper-scissors' competition is compelling because it promotes species coexistence and because recent work suggests it may be common in natural systems. One class of intransitivity indices works by considering s, the minimum number of competitive reversals to convert a given competitive community (i.e., a 'tournament') to a hierarchy. The most straightforward example of such 'reversal-based' indices is Petraitis' index, t = 1 - s/M, where M is the maximum s across all possible n-species tournaments. Using exhaustive searches, we prove that Petraitis' formula for M (and, therefore, t) does not hold for n ≥ 7. Furthermore, the determination of s for even moderate values of n may prove difficult, as the equivalent graph-theoretical problem is NP-hard; there is no known computationally feasible way to compute an exact answer for anything but small values of n, let alone a closed-form solution. Petraitis' t is a valuable index of intransitivity; however, at present its use is limited to relatively species-poor systems. More broadly, reversal-based indices, while intuitive, may be problematic because of this computability issue.