determining a lower bound price of {HEAD ~ a group} of {TAIL ~ bids} of a subtree of the search tree 
determining a lower bound price of { HEAD ~ a group } of { TAIL ~ bids } of a subtree of the search tree
OTH OTH OTH OTH OTH OTH HEAD HEAD HEAD HEAD HEAD HEAD REL TAIL TAIL TAIL TAIL TAIL OTH OTH OTH OTH OTH OTH OTH