  Budget feasible mechanisms, recently initiated by Singer (FOCS 2010), extend
algorithmic mechanism design problems to a realistic setting with a budget
constraint. We consider the problem of designing truthful budget feasible
mechanisms for general submodular functions: we give a randomized mechanism
with approximation ratio $7.91$ (improving the previous best-known result 112),
and a deterministic mechanism with approximation ratio $8.34$. Further we study
the knapsack problem, which is special submodular function, give a $2+\sqrt{2}$
approximation deterministic mechanism (improving the previous best-known result
6), and a 3 approximation randomized mechanism. We provide a similar result for
an extended knapsack problem with heterogeneous items, where items are divided
into groups and one can pick at most one item from each group.
  Finally we show a lower bound of approximation ratio of $1+\sqrt{2}$ for
deterministic mechanisms and 2 for randomized mechanisms for knapsack, as well
as the general submodular functions. Our lower bounds are unconditional, which
do not rely on any computational or complexity assumptions.
