+----------------------------------------------------------------+
|   The "Local-Ratio Theorem" (Decomposition version)            |
|   for problems of the form:                                    |
|   min/max SUM{W(x): x in of C} s.t. "Constraint(C)"            |
+----------------------------------------------------------------+
|   If   C is r-approximation w.r.t the weight function W1       |
|   and  C is r-approximation w.r.t the weight function W2       |
|   then C is r-approximation w.r.t the weight function W1+W2    |
+----------------------------------------------------------------+

Proof scratch: for min problems (for max, just reverse the <=)
   W1(C) <= r W1^* and W2(C) <= r W2^* implies
   (W1+W2)(C) <= r (W1^* + W2^*)
   the reader only needs to observe that W1^* + W2^* <= (W1+W2)^*





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Denote by (WEIGHT FUNCTION)^* the optimum size w.r.t that weight.