Stack the Stones

Jayesh recently went on a trip to Japan and fell in love with the ancient Japanese art of stone stacking/rock balancing. Mesmerized by it, he now wants to try it out himself. He bought $N$ rocks each having a width $W_i$ and cost $C_i$. While returning he met a priest who told him the ancient trick of balancing the stones - "an arrangement with $k$ stones will be balanced only if more than half of