|Abstract:||Candidate sets are typically unbounded in size, but the set of distinct possible optima is inevitably finite. Almost all candidates are 'losers', doomed never to surface because under any ranking there are superior alternatives. Such losers are said to be 'harmonically bounded'. In its most general form, the harmonic bounding of a candidate is a collective effect: under every ranking some member of the bounding set is guaranteed to beat the bounded candidate, but different members may be responsible for defeating it under different rankings (Samek-Lodovici and Prince 1999, ROA-363.). In this paper we focus attention on the space of violation profiles, which is precisely what OT sees of linguistic structure. Suppose, as is typically the case, that the analyst has identified a set of violation profiles. The problem is to determine the regions of violation space that are bounded, simply and collectively, by that set. Concretely, this amounts to characterizing the types of candidates that are bounded by the candidates we have on hand. We show how all bounding effects, no matter how complex, reduce to simple, noncollective bounding when the original set is augmented by 'bounding minima'. We present and justify an algorithm which calculates the bounding minima associated with any set of profiles.