The minimum number of problems to cover all subproblems

Abstract

Th e fo ll ow ing proble m is motiva te d and the n so lved, using the theo ry of sys tems of di s tinct re presentatives . Le t 111 = {I . 2 . . , III}. a nd for each sequ e nce u of di s tinc t me mbers of 111, le t (u ) be th e assoc ia te d subse t. S uppose give n a mathe mati ca l probl e m 1' (5) for e ac h s ubset 5 of 111, and a n a lgo rithm A whi c h whe n a ppli ed to u so lves not on ly 1'( (u» b ut a lso all 1' ( (0») where T is an initi a l segme nt of u. What is the s malles t numbe r of a pplicat ions of A need ed to solve the e ntire e nse mble of prob le ms { p (S):S C ilf }?