# Complexity of a scheduling problem with a fixed left bound of jobs

Consider the following scheduling problem. We have a finite set of jobs $$\mathcal{J}= \{j_1, j_2, ..., j_n\}$$ to do. Every job $$j_i$$ has its own value $$c_i$$ (the amount of money we are paid for finishing the respective job before the deadline), processing time $$t_i$$ and deadline time $$b_i$$. We can perform only one job in a moment with no interruptions. Our goal is to plan the order of performing the jobs with the highest total value of finished jobs before the deadline.

Formally, we have sets $$S=\{1, 2, \ldots, n\}, C=\{c_1, c_2, \ldots, c_n\}, T=\{t_1, t_2, \ldots, t_n\}, B=\{b_1, b_2, \ldots, b_n\}$$. We are looking for the subset $$S_1 \subseteq S$$ with an order $$\pi$$ on its elements such that $$\{\pi_1, \pi_2, \ldots \pi_k\} = S_1$$ and the following restrictions are performed: $$\forall l \in \{1, 2, \ldots, k\}: b_{\pi_l} \geqslant \sum_{i=1}^{l} t_{\pi_i}$$ $$\sum_{i=1}^k c_{\pi_i} \longrightarrow \max$$

I'm interested in the complexity of this task. It's easy to see that this problem is $$NP$$-hard if parameters are given in a binary representation (because in the case of all equal parameters $$b_i$$ we can get the Knapsack problem). So my questions are:

1. Does this task belong to $$P$$ if numbers are given in unary form?
2. What approximation algorithms are known for this task, are there any PTAS or FPTAS?