Given a sequence of decreasing integers, i.e., $a_1 \geq a_2 \geq \cdots \geq a_T $ and a positive real $k\geq 1$, find a subset $S$ such that$$\max_{S\subseteq \{1,\ldots,T\}} \sum_{i\in S} a_i$$$$s.t., \sum_{i\in S/\{t\}} a_i \leq k \cdot a_t,$$where $a_t$ represents the smallest one of subset $S$. Note that $a_t$ is the smallest of set $S$ and may be different for different set $S$. I am wondering whether this problem is still NP-complete. Any comments or suggestions will be very appreciated.
Variant of Subset Sum Problem with Changing Bound
np hardness;partition problem;subset sum