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  1. Assignment 8.4.4 (only the first part concerning reduction of the languages in TIME(f(n)))
  2. Show that the following problem is in \Sigma_2 P:
    INSTANCE: A set of clauses S and a variable x
    QUESTION: Is there a truth assignment that is minimal for S and satisfies x?
    (A truth assignment T is minimal for a set of clauses S if T satisfies S and there is no other truth assignment T' which is smaller, i.e., for which T'(y) = true implies T(y) = true for all variables y.)
  3. Show that the following problem 'EXACT COVER BY 4-SETS' is NP-complete:
    INSTANCE: Finite set X with |X|=4q, q an integer, and a collection C of 4-element subsets of X
    QUESTION: Is there a subcollection C' of C such that every element of X occurs in exactly one member of C'?
  4. Show that the following problem 'SET SPLITTING' is NP-complete:
    INSTANCE: Collection C of subsets of a finite set S
    QUESTION: Is there a partitioJf[!J :)Tdz40T YT/NwlqRɊ?xb2N:e*S1`GY :j:A.2'txZ/6'oPfEs:kҸNVYV 6Ͳ齣?㖤Nn&jrչ%͕V5\29mrl]_ iZ"(x3P?Q,3 CB "0"מ'oNCL(=Fq]`{z4žnj߹u#dmn0̦%en"s[a:KC] K.ޟ|Z.}6-3o4 YM.dnƚV^2r_