First of all, it should be noted that in the literature such problem is known as "bottleneck multiple subset sum problem" (B-MSSP) and it's a variation of knapsack problem. This problem even with such restrictions is an NP-complete.
  So, the aproximate solution even needed. Precision??? The precision is mystic (based on statement of the problem).

  I don't know how other people solved this problem, but I aproximate solution through aproximation of every pair of subsets. To aproximate a pair of subsets we need to solve B-MSSP just for 2 subsets. This can be solved in O( (|S1|+|S2|)*max(Ai) ) time and O( max(Ai, |S1|+|S2|) ) space complexity.