An Almost Tight Lower Bound for the Scheduling Problem to Meet Two Min-Sum Objectives

Abstract

In this note, we provide an almost tight lower bound for the scheduling problem to meet two min-sum objectives considered by Angel et al. in Oper. Res. Lett. 35(1): 69–73, 2007.

1 Previous results

Angel et al. [1] recently investigated the following bi-criteria scheduling problem 1∥{∑C _j ,∑w _j C _j } via the simultaneous approximation approach invented by Stein and Wein [2] and obtained the following result:

Theorem 1

(Angel et al. [1])

For the bi-criteria schedule problem 1∥(∑C _j ,∑w _j C _j ) with n jobs, (i) there exists a $(1+\frac{1}{r}, 1+r )$-approximation schedule for any r>0; and (ii) there exists an instance such that no $(1+\frac{1}{r}, 1+\frac{r-1}{2r+1})$-approximation schedule exists for r>1.

The lower bound result above was improved later by Yan [3]:

Theorem 2

(Yan [3])

For the bi-criteria schedule problem 1∥(∑C _j ,∑w _j C _j ) with n jobs, there exists an instance such that no $(1+\frac{1}{r}, 1+\frac{r-1}{1.5+\sqrt{2r}} )$-approximation schedule exists for any r>1.

Note that the second term of the lower bound results in Theorems 1 and 2 are respectively in the order of Ω(1) and $\varOmega(\sqrt{r})$.

In this note, we improve the lower bound further to obtain an almost tight lower bound up to a constant factor, namely in the order of Ω(r).

2 Our Results

Theorem 3

For the bi-criteria schedule problem 1∥(∑C _j ,∑w _j C _j ) with n jobs, there exists an instance such that no $(1+\frac{1}{r}, 1+\frac{1}{2}r-\epsilon )$-approximate schedule exists for any r>0 and ϵ>0.

Proof

Let k be a positive integer such that $k>\frac{1}{r}$, for any given r>0. Consider the following instance: there are n>k jobs with processing times

and with weights w ₁=⋯=w _n−1=0 and w _n =1. Let π _ℓ (ℓ=1,⋯,n) be the schedule with job order corresponding to the permutation such that job p _n is on the ℓth position, namely π _ℓ =(1,⋯,ℓ−1,n,ℓ,⋯,n−1). By the choice of the processing times p’s, evidently π _n and π ₁ are the optimal schedules for the two objectives ∑C _j and ∑w _j C _j , respectively. For schedule π _ℓ , we have

Note that f and g are strictly respectively decreasing and increasing functions of ℓ.

By the choice of the processing times p’s, for the first objective we have

$$\frac{f(n-k)}{f(n)}=\frac{\sum_{j=1}^n C_j(\pi_{n-k})}{\sum_{j=1}^n C_j(\pi_n)}=1+\frac{1}{r}, $$

implying that for all 1⩽ℓ<n−k:

$$\frac{f(\ell)}{f(n)}>1+\frac{1}{r}. $$

Therefore for each schedule π _ℓ (ℓ=n−k+1,⋯,n), we have that

$$\frac{f(\ell)}{f(n)}=\frac{\sum_{j=1}^n C_j(\pi_{\ell })}{\sum_{j=1}^nC_j(\pi_n)}< 1+\frac{1}{r}, $$

because f(ℓ) is strictly decreasing with ℓ.

However, for these schedules, the smallest approximation ratio with respect to the second objective is equal to

$$\min_{\ell=n-k+1,\cdots, n}\frac{g(\ell)}{g(1)} =\frac{g(n-k+1)}{g(1)}=\frac{n-k+p_n}{p_n}= 1+\frac{1}{\frac{1}{n-k} +\frac{n(n+1)/2}{(n-k)(rk-1)}}:=R(n), $$

where the first equality follows from that g(ℓ) is increasing with ℓ. The last quantity R(n) is a concave function of n and achieves its maximum when $n^{*}=k+\sqrt{k^{2}+(2r+1)k-2}$. Although n ^∗ may not be an integer, we can find a lower bound of R(n ^∗) as follows

which is an increasing function of k, asymptotically attaining its supreme $1+\frac{1}{2}r$, when k→∞. Therefore there exists k large enough (and hence n ^∗) such that, for any ϵ>0:

$$R\bigl(n^*\bigr)\geqslant R\bigl(\bigl\lceil n^* \bigr\rceil\bigr)\geqslant 1+ \frac{1}{2}r-\epsilon. $$

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Together with the upper bound result in Theorem 1 [1], we actually have

Corollary 1

For the bi-criteria schedule problem 1∥(∑C _j ,∑w _j C _j ) with n jobs, any $(1+\frac{1}{r}, 1+ar-\epsilon)$-approximation schedule must satisfy $\frac{1}{2}\leqslant a\leqslant 1$, where r,ϵ>0.