Malleable scheduling beyond identical machines

Abstract

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. In this setting, jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S with respect to j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than $\frac{e}{e-1}$, unless $P = NP$. On the positive side, we present polynomial-time algorithms with approximation ratios $\frac{2e}{e-1}$ for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding. They result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of $1+\varphi$ for unrelated speeds ($\varphi$ is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms for a variant where we determine the effective speed of a set of allocated machines based on the $L_p$ norm of their speeds.

Appendix

1.1 A slight improvement by optimizing the threshold

Recall that in both algorithms for the unrelated machines case, the threshold for deciding whether a job j is assigned to p(j) or to T(j) is set to 1/2. While this is the optimal choice for the simple 4-approximate rounding scheme, we can achieve a slightly better bound for our improved algorithm by optimizing this threshold accordingly. Let $\beta \in (0,1)$ be the threshold such that a job j is assigned to p(j) when $x_{p(j),j} \ge \beta$, or to the set T(j), when $\sum _{i \in T(j)} x_{i,j} > 1 - \beta$. Formally, let ${{\,\mathrm{{\mathcal {J}}}\,}}^{(1)} := \{j \in {{\,\mathrm{{\mathcal {J}}}\,}}{{\,\mathrm{\,|\,}\,}}x_{p(j),j} \ge \beta \}$ be the set of jobs that are assigned to their parent machines and ${{\,\mathrm{{\mathcal {J}}}\,}}^{(2)} := {{\,\mathrm{{\mathcal {J}}}\,}}\setminus {{\,\mathrm{{\mathcal {J}}}\,}}^{(1)}$ the rest of the jobs. For $j \in {{\,\mathrm{{\mathcal {J}}}\,}}^{(2)}$ and $\theta \in [0, 1]$ define $S_j(\theta ) := \{ i \in T(j) {{\,\mathrm{\,|\,}\,}}1 - \frac{\ell _i}{C} \ge \theta \}$. Choose $\theta _j$ so as to minimize $2(1 - \theta _j)C + f_j(S_j(\theta _j))$ (note that this minimizer can be determined by trying out at most |T(j)| different values for $\theta _j$). We then assign each job in $j \in J^{(2)}$ to the machine set $S_j(\theta _j)$.

Recall that for any $i \in {{\,\mathrm{{\mathcal {M}}}\,}}$ there is at most one $j \in J^{(2)}$ with $i \in T(j)$. If $i \notin S_j(\theta _j)$, then load of machine i is bounded by $\frac{1}{\beta } \ell _i \le \frac{1}{\beta } C$, where $\ell _i$ as defined in Sect. 3.1. If $i \in S_j(\theta _j)$, then the load of machine i is bounded by

$$\begin{aligned} \max _{i' \in S_j(\theta _j)}\Big \{\beta ^{-1} \ell _{i} + f_j(S_j(\theta _j)) \Big \}&\le \beta ^{-1}(1 - \theta _j)C \nonumber \\&\quad + f_j(S_j(\theta _j)), \end{aligned}$$

(7)

where the inequality comes from the fact that $1 - \frac{\ell _{i'}}{C} \ge \theta _j$ for all $i' \in S_{\theta _j}$.

We now fix $\beta$ to be the unique solution of $\beta ^{-1}+1 = \frac{e^{\frac{1}{\beta } - 1}}{\beta (e^{\frac{1}{\beta } - 1} - 1)}$ in the interval [0, 1]. The following proposition gives an upper bound on the RHS of (4) as a result of our filtering technique.

Proposition 12

For each $j \in {{\,\mathrm{{\mathcal {J}}}\,}}^{(2)}$, there is a $\theta \in [0,1]$ with $\beta ^{-1}(1- \theta )C + f_j(S_j(\theta )) \le \frac{e^{\frac{1}{\beta } - 1}}{\beta (e^{\frac{1}{\beta } - 1} - 1)}C$.

Proof

We first assume that for all $i \in T(j)$, it is the case that $s_{i,j} \le \gamma _j$ and, thus, $r_{i,j} = \gamma _j$. In the opposite case, where there exists some $i' \in T(j)$ such that $s_{i,j} > \gamma _j$, by choosing $\theta = 0$, then (4) can be upper bounded by $\beta ^{-1}C + f_j(S_j(0)) = \beta ^{-1}C + f_j(T(j)) \le \left( \beta ^{-1}+1\right) C$. In that case, the proposition follows directly by the fact that $\beta ^{-1}+1 = \frac{e^{\frac{1}{\beta } - 1}}{\beta (e^{\frac{1}{\beta } - 1} - 1)}$, by our choice of $\beta$.

Define $\alpha := \frac{e^{\frac{1}{\beta } - 1}}{\beta (e^{\frac{1}{\beta } - 1} - 1)}$. We show that there is a $\theta \in [0, 1]$ with $\sigma _j(S_j(\theta )) \ge \frac{\gamma _j f_j(\gamma _j)}{(\alpha + \beta ^{-1} \theta -\beta ^{-1})C}$. Then $f_j(S_j(\theta )) \le (\alpha + \beta ^{-1} \theta -\beta ^{-1}) C$ by Fact 5, implying the lemma.

Define the function $g: [0,1] \rightarrow {\mathbb {R}}_{+}$ by $g(\theta ) := \sigma _j (S_j(\theta ))$. It is easy to see g is non-increasing integrable and that

$$\begin{aligned} \int _{0}^{1} g(\theta ) d\theta = \sum _{i \in T(j)} s_{i,j} (1 - \frac{\ell _i}{C}). \end{aligned}$$

Now assume by contradiction that $g(\theta ) < \frac{\gamma _j f_j(\gamma _j)}{(\alpha + \beta ^{-1} \theta - \beta ^{-1})C}$ for all $\theta \in [0, 1]$. Note that $\ell _i + \frac{\gamma _j f_j(\gamma _j)}{s_{i,j}} x_{i,j} \le C$ for every $i \in T(j)$ by constraints (2). Hence $\frac{f_j(\gamma _j)\gamma _j}{C} x_{i,j} \le s_{i,j}(1 - \frac{\ell _i}{C})$ for all $i \in T(j)$. Summing over all $i \in T(j)$ and using the fact that $\sum _{i \in T(j)} x_{i,j} \ge 1 - \beta$ because $j \in J^{(2)}$, we get

$$\begin{aligned} (1- \beta )\frac{f_j(\gamma _j)\gamma _j}{C}&\le \sum _{i \in T(j)} s_{i,j} (1 - \frac{\ell _i}{C})\\&= \int _{0}^{1} g(\theta ) d\theta \\&< \frac{f_j(\gamma _j)\gamma _j}{C} \int _{0}^{1} \frac{1}{\alpha + \beta ^{-1} \theta - \beta ^{-1}} d\theta , \end{aligned}$$

where the last inequality uses the assumption that $g(\theta ) < \frac{\gamma _j f_j(\gamma _j)}{(\alpha + \beta ^{-1} \theta _j - \beta ^{-1})C}$ for all $\theta \in [0,1]$. By simplifying the above inequality, we get the contradiction

$$\begin{aligned} \frac{1 - \beta }{\beta } < \int _{\alpha -\beta ^{-1}}^{\alpha } \frac{1}{\lambda } d\lambda = \ln (\frac{\alpha }{\alpha -\beta ^{-1}}) = \frac{1 - \beta }{\beta }. \end{aligned}$$

$\square$

Therefore, by choosing $\alpha = \inf _{\beta \in (0,1)} \{ \frac{e^{\frac{1}{\beta } - 1}}{\beta (e^{\frac{1}{\beta } - 1} - 1)} \} \approx 3.14619$⁵, with threshold $\beta \approx 0.465941$, we can prove the following theorem.

Theorem 12

There exists a polynomial-time 3.1461-approximation algorithm for the problem of scheduling malleable jobs on unrelated machines.