===========================================================================
CSC 373H / L0101        Lecture Summary for Week 13             Winter 2005
===========================================================================

Weighted Set Cover (cont'd):

      . (11.10)  For all S_k, SUM_{s in S_k} c_s <= H(|S_k|) w_k
        (where H(n) = 1 + 1/2 + ... + 1/n = Theta(log n)).

        Proof:  Let d = |S_k| and S_k = {s_1, s_2, ..., s_d},
        in order of coverage by the algorithm.
          * when s_1 is first covered, set used is at least as good as S_k
            so c_{s_1} <= w_k/d (cost per element for S_k)
          * when s_2 is first covered, set used is at least as good as S_k
            so c_{s_2} <= w_k/(d-1) (cost/elem for S_k-{s_1})
          * ...
          * when s_j is first covered, set used is at least as good as S_k
            so c_{s_j} <= w_k/(d-j+1) (cost/elem for S_k-{s_1,...,s_{j-1}})
          * ...
        Total:
        SUM_{s in S_k} c_s <= w_k/d + w_k/(d-1) + ... + w_k/1 = H(d) w_k.

      . Let d* = MAX_{i=1..m} |S_i| (max size of S_i's), C* = optimum set
        cover, and w* = SUM_{i in C*} w_i = optimum weight.

      . (11.11)  SUM_{i in C} w_i <= H(d*) w* = Theta(log n) w*.

        Proof:
          * By (11.10), for each i in C*,
                w_i >= 1/H(|S_i|) * SUM_{s in S_i} c_s
                    >= 1/H(d*) * SUM_{s in S_i} c_s.
          * C* is a cover so
                SUM_{i in C*} SUM_{s in S_i} c_s >= SUM_{s in U} c_s.
          * Hence,
                w* =   SUM   w_i
                     i in C*
                                 1
                   >=   SUM    -----    SUM    c_s
                      i in C*  H(d*)  s in S_i
                        1                   1
                   >= -----   SUM   c_s = -----   SUM   w_i  (by 11.9)
                      H(d*)  s in U       H(d*)  i in C