Find the maximum set whose subset sum is unique for every of its subset. We are given a set of $n$ positive integers.
We assume all of them are bounded by a polynomial of $n$.
We would like to find a subset $S$ of numbers such that
for any $T_1,T_2\subseteq S$, the sum of numbers in $T_1$ is not equal to that of $T_2$.
We want the size of $S$ is maximized.
Clearly, the problem is in NP and can be solved in $n^{O(\log n)}$ time (Since $|S|\leq O(\log n)$).
Is this problem polynomial time solvable?
Is this problem studied before? Any help is appreciated. 
 A: In the modern language, sets with all subset sums pairwise distinct are often called dissociated. Maximal dissociated subsets of a given set are important in Additive Combinatorics and Fourier analysis, although I cannot recall anybody ever addressing the algorithmic aspect. 
Yet another reference you may find useful: http://math.haifa.ac.il/~seva/Papers/DisBases.pdf.

Added: taking into account the interpretation of a maximal dissociated subset of a given set $A$ as a "basis of $A$ over the set $\{-1,0,1\}$", it may be reasonable to try and adjust one of the standard algorithms of finding a maximal independent subset of a given set in a vector space.
A: This has been studied, see 
http://www.mathnet.or.kr/mathnet/kms_tex/978590.pdf
and the somewhat more interesting:
NEWMAN POLYNOMIALS
WITH PRESCRIBED VANISHING AND INTEGER SETS WITH DISTINCT SUBSET SUMS
PETER BORWEIN AND MICHAEL J. MOSSINGHOFF
(available online)
See also the references in:
http://garden.irmacs.sfu.ca/?q=op/sets_with_distinct_subset_sums
None of these answers your question, but I think this is the state of the art...
