Correctness proof of a greedy approximation algorithm

Question

How do I prove the correctness of this algorithm?

Answer 1

Desicion of Algorithms : $ T_{1},T_{2},....,T_{i}$ , cost $ C = C_{1},C_2,....,C_i$

$ \frac{Algo}{Opt} = \frac{C_1 + C_2 + ... + C_i}{(C_1/n_1)*n}$

and $ 4C_1 \geq C_k ,\forall 1 \leq k \leq m$

$ \implies \frac{C_1 + C_2 + ... + C_i}{(C_1/n_1) * n} \leq \frac{4iC_1}{(C_1/n_1)*n} = \frac{4in_1}{n}$

consider $ \frac{C_1}{n_1} \leq \frac{C_k}{n_k} \leq \frac{4C_1}{n_k} \implies n_k \leq 4n_1 $

$ \implies n_1 + n_2 + ... + n_{i-1} \leq n \implies 4(i - 1)n_1 \leq n \implies \frac{4in1}{n} \leq \frac{4n_1 + n}{n}$

$\frac{Algo}{Opt} \leq \frac{4in_1}{n} \leq \frac{4n_1 + n}{n} \leq \frac{4n_1 + n_1}{n_1} \leq 5 $