I think you'll have to create a command to do this - there's no way to use a min calculated in Crystal in your selection criteria.
I would use something like this:
SELECT DISTINCT
CallLog.CallID, CallLog.CallType, CallLog.RecvdDate, CallLog.CustID, Asgnmnt.Assignee, CallLog.Priority, CallLog.Category, Asgnmnt.Resolution, CallLog.RecvdTime, Asgnmnt.DateAcknow, Asgnmnt.TimeAcknow, Asgnmnt.WhoAcknow, Subset.RHC
FROM heat.heat.CallLog CallLog
LEFT OUTER JOIN heat.heat.Asgnmnt Asgnmnt ON CallLog.CallID=Asgnmnt.CallID
LEFT OUTER JOIN heat.heat.Subset Subset ON CallLog.CallID=Subset.CallID
WHERE CallLog.Category LIKE 'break%'
AND Subset.RHC LIKE 'LMC%' AND CallLog.Priority='1'
AND Asgnmnt.DateAcknow is not null
AND (Asgnmnt.Resolution='Completed'
OR Asgnmnt.Resolution='Reassigned')
AND Asgnmnt.DateAcknow + Asgnmnt.TimeAcknow =
(Select min(Asgnmnt1.DateAcknow + Asgnmnt.TimeAcknow)
from heat.heat.Asgnmnt Asgnmnt1
where Asgnmnt1.CallID = CallLog.CallID)
ORDER BY CallLog.CallID
-Dell