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CSC 363H                Lecture Summary for Week 11              Spring 2007
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More examples of reductions to SAT, showing that the problem we reduced from is NPC.

 -  INDEPENDENT-SET is NPc:  Already know in NP.  For NP-hardness, use
    reduction from VERTEX-COVER:  Given <G,k>, construct <G',k'> as
    follows: G' = G, k' = n-k.  Clearly this can be done in polytime.
    Also, if G contains a vertex cover of size k, the vertices outside the
    cover form an independent set of size n-k.  Finally, if G' contains an
    independent set of size n-k, the vertices outside the independent set
    form a vertex cover of size k.

 -  CLIQUE is NPc:  Already known in NP.  For NP-hardness, use reduction
    from INDEPENDENT-SET covered last week.
    Note:  textbook uses different reduction from 3SAT.

 -  SUBSET-SUM is NPc:  (see textbook)
    Example of reduction for
    F = (x1 \/ ~x2 \/ ~x4) /\ (x2 \/ ~x3 \/ x1) /\ (~x3 \/ x4 \/ ~x2):
    S = { 1000110,    (corresponding to x1)
          1000000,    (corresponding to ~x1)
          0100010,    (corresponding to x2)
          0100101,    (corresponding to ~x2)
          0010000,    (corresponding to x3)
          0010011,    (corresponding to ~x3)
          0001001,    (corresponding to x4)
          0001100,    (corresponding to ~x4)
          0000100,    (corresponding to C1)
          0000200,    (corresponding to C1)
          0000010,    (corresponding to C2)
          0000020,    (corresponding to C2)
          0000001,    (corresponding to C3)
          0000002 }   (corresponding to C3)
      t = 1111444
    Note: slightly different from book to ensure S contains all distinct
    numbers (book's reduction constructs S with duplicated numbers).