Inapproximability Results for Scheduling with Interval and Resource Restrictions

TL;DR

This paper establishes new hardness results for scheduling problems with interval and resource restrictions, proving the non-existence of PTAS and tight approximation bounds under P≠NP.

Contribution

It provides the first negative results for the existence of PTAS in interval restriction scheduling and tight approximation bounds for resource-restricted scheduling with multiple resources.

Findings

No PTAS for interval restrictions unless P=NP.

Approximation ratio lower bounds of 48/47 and 1.5 for resource restrictions with 2 and 4 resources.

Results extend to Santa Claus variants maximizing minimal processing time.

Abstract

In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions. In the case of interval restrictions the machines can be totally ordered such that jobs are eligible on consecutive machines. We resolve the open question of whether the problem admits a polynomial time approximation scheme (PTAS) in the negative (unless P=NP). There are several special cases of this problem known to admit a PTAS. Furthermore, we consider a variant with resource restriction where each machine has capacities and each job demands for a fixed number of resources. A job is eligible on a machine if its demand is at most the capacity of the machine for each resource. For one resource, this problem is known to admit a PTAS, for two, the case of interval restrictions is contained, and in general, the problem is closely related to unrelated scheduling with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives.

Full text

Inapproximability Results for Scheduling with Interval and Resource Restrictions

Marten Maack

  

Klaus Jansen

For the case of interval restrictions—where the machines can be totally ordered such that jobs are eligible on consecutive machines—we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known.

Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than $48/47\approx 1.02$ or $1.5$ for scheduling with resource restrictions with $2$ or $4$ resources, respectively, unless P$=$NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives.

1 Introduction

Consider the restricted assignment problem: Given a set of machines $\mathcal{M}$ and a set of jobs $\mathcal{J}$ each with a processing time or size $p_{j}$ and a subset of eligible machines $\mathcal{M}(j)\subseteq\mathcal{M}$, the goal is to find a schedule $\sigma:\mathcal{J}\rightarrow\mathcal{M}$ with $\sigma(j)\in\mathcal{M}(j)$ for each job $j$ and minimizing the makespan $C_{\max}(\sigma)=\max_{i\in\mathcal{M}}\sum_{j\in\sigma^{-1}(i)}p_{j}$.

In a seminal work, Lenstra, Shmoys and Tardos [LenstraST90] presented a $2$-approximation for restricted assignment and also showed that there is no polynomial time approximation algorithm with rate smaller than $1.5$ for the problem, unless P=NP. Closing this gap is a prominent open problem in approximation and scheduling theory [SchuurmanW99, WS11book]. If there are no restrictions, i.e., $\mathcal{M}(j)=\mathcal{M}$ for each job $j$, we have the classical problem of makespan minimization on identical parallel machines (machine scheduling) which is already strongly NP-hard. On the other hand, machine scheduling is well-known to admit a polynomial time approximation scheme (PTAS) due to a classical result by Hochbaum and Shmoys [HochbaumS87identische]. In recent years, the approximability of special cases of restricted assignment has been intensively studied (see, e.g., [ChakrabartyS16, EbenlendrKS14, HuangO16, JansenR18GrapbBalancing]) with one line of research focusing on the existence of approximation schemes (see, e.g., [EpsteinL11, JansenMS17, MuratoreSW10, OLL08]). The present work seeks to contribute in this research direction.

Interval Restrictions.

Arguably one of the most natural variants of the restricted assignment problem is the case of scheduling with interval restrictions (RAI). In this variant, the machines are totally ordered and each job is eligible on consecutive machines. More precisely, we have $\mathcal{M}=\{M_{1},\dots,M_{m}\}$, and for each job $j$ we have $\mathcal{M}(j)=\{M_{\ell},\dots,M_{r}\}$ for some indices $\ell,r\in[m]$. Several special cases of RAI are known to admit a PTAS: the hierarchical case [OLL08], where for each job the interval of eligible machines starts with the first machine; the nested case [MuratoreSW10, EpsteinL11], where $\mathcal{M}(j)\subseteq\mathcal{M}(j^{\prime})$, $\mathcal{M}(j^{\prime})\subseteq\mathcal{M}(j)$ or $\mathcal{M}(j)\cap\mathcal{M}(j^{\prime})=\emptyset$ for each pair of jobs $(j,j^{\prime})$; and the inclusion-free case [Schwarz10Thesis, KhodamoradiKRS16], where $\mathcal{M}(j)\subseteq\mathcal{M}(j^{\prime})$ implies that $j$ and $j^{\prime}$ share either their first or last eligible machine. Furthermore, for general RAI, a $2-2/(\max_{j\in\mathcal{J}}p_{j})$-approximation due to Schwarz [Schwarz10Thesis] is known (assuming integral processing times); and the special case with two distinct processing times is even polynomial time solvable [WangS16]. Note that the problem has also been studied in the context of online algorithms (see [LeeLP13survey, LeungL16update]).

The question of whether there is a PTAS for RAI has been posed by several authors [Khodamoradi16Thesis, Schwarz10Thesis, WangS16]. As the main result of the present work, we resolve this question in the negative:

Theorem 1.1**.**

*There is no PTAS for scheduling with interval restrictions unless P$=$NP. 111There is a paper [KhodamoradiKRS13] claiming to have found a PTAS for

Problem 1.2**.**

RAI. However, according to [PTASflawed], the result is not correct and the authors published a revised version of the paper [KhodamoradiKRS16] claiming a less general result, namely, a PTAS for the inclusion-free case.

Resource Restrictions.

The second variant considered in this work, is the problem of scheduling with resource restrictions with $R$ resources (RAR$(R)$). Herein, a set $\mathcal{R}$ or $R$ (renewable) resources is given, each machine $i$ has a resource capacity $c_{r}(i)$ and each job $j$ has a resource demand $d_{r}(j)$ for each $r\in\mathcal{R}$. Job $j$ is eligible on machine $i$ if $d_{r}(j)\leq c_{r}(i)$ for each resource $r$. For $R=1$, the problem is equivalent to the mentioned hierarchical case and has been studied intensively [LeungL08survey, LeungL16update]. Furthermore, it is not hard to see that

Problem 1.3**.**

RAIis properly placed between RAR$(1)$ and RAR$(2)$ (see LABEL:sec:resource) and hence there is a close relationship between the two problems. For arbitrary $R$, the problem was mentioned in a work by Bhaskara et el. [BhaskaraKTW13] under the name of geometrically restricted scheduling222The demands $d(j)$ and capacities $c(i)$ may be interpreted as points in $R$-dimensional space. but to the best of our knowledge it has not been further studied up to now. There is, however, a close relationship to the low rank version of makespan minimization on unrelated parallel machines (unrelated scheduling) introduced in [BhaskaraKTW13]. In the problem of unrelated scheduling, the processing time of each job is dependent on the machine it is scheduled on, that is, a processing time matrix $(p_{ij})_{j\in\mathcal{J},i\in\mathcal{M}}$ is given in the input. Unrelated scheduling is a classical problem, and the 2-approximation by Lensta et al. [LenstraST90] was actually formulated for this problem. Restricted assignment can be seen as a special case of unrelated scheduling by setting $p_{ij}=p_{j}$ for $i\in\mathcal{M}(j)$ and $p_{ij}=\infty$ otherwise. In the rank $D$ version of unrelated scheduling (LRS$(D)$), the processing time matrix has a rank of at most $D$, or, equivalently [CMYZ17], we may assume that there are $D$-dimensional size vectors $s(j)$ for each job $j$ and speed vectors $v(i)$ for each machine $i$ such that $p_{ij}=\sum_{k=1}^{D}s_{k}(j)\cdot v_{k}(i)$. Considering the latter definition, scheduling with resource restrictions may intuitively be seen as the restricted assignment equivalent of low rank unrelated scheduling. It is not hard to see that formally already for RAR$(1)$ instances the processing time matrix can have rank $|\mathcal{M}|$. However, LRS$(D)$ includes approximations of any RAR$(D-1)$ instance with arbitrary precision (see LABEL:sec:resource for details). The case with $D=1$ of LRS$(D)$ is equivalent with the classical problem of makespan minimization on uniformly related machines and well known to admit a PTAS [HochbaumS88uniforme]. Bhaskara et el. [BhaskaraKTW13] gave a quasi-polynomial time approximation scheme (QPTAS) for $D=2$, and showed that there is no PTAS or approximation algorithm with rate smaller than $1.5$ for $D\geq 4$ or $D\geq 7$ respectively. The latter two results have been improved from $D=4$ to $D=3$ by Chen et al. [CMYZ17] and from $D=7$ to $D=4$ by Chen, Ye and Zhang [ChenYZ14]. We present similar inapproximability results for scheduling with resource restrictions:

Theorem 1.4**.**

There is no approximation algorithm with rate less than $48/47\approx 1.02$ or $1.5$ for scheduling with resource restrictions with $2$ or $4$ resources, respectively, unless P$=$NP.

Santa Claus.

The problems of restricted assignment and unrelated scheduling are also studied with the reverse objective of maximizing the minimal machine load $\min_{i\in\mathcal{M}}\sum_{j\in\sigma^{-1}(i)}p_{ij}$. Usually these variants are described in a more game theoretical context with players instead of machines, goods instead of jobs, and values instead of processing times, and sometimes unrelated scheduling with the reverse objective is called the Santa Claus problem. In this paper, we will mostly stick to the scheduling notation but denote the variants of the considered problems with reverse objective as the Santa Claus version of the respective problem.

For the Santa Claus version of the restricted assignment problem a 13-approximation due to Annamalai, Kalaitzis and Svensson [AnnamalaiKS17] is known, which has been improved to a rate of $6+\varepsilon$ by both Cheng and Mao [ChengM18] and Davies, Rothvoss and Zhang [DRZ18]. PTAS results are known for the case without restrictions [Woeginger97] and the inclusion-free interval case [KhodamoradiKRS16].

Our results can be directly transferred to the Santa Claus versions of the respective problems:

Theorem 1.5**.**

Unless P$=$NP, there is no PTAS for the Santa Claus version of scheduling with interval restrictions and no approximation algorithm with rate less than $47/46$ or $2$ for the Santa Claus version of scheduling with resource restrictions with $2$ or $4$ resources, respectively.

Paper structure.

*In the remainder of this section, we discuss some further related literature and present preliminary considerations needed throughout the paper. In LABEL:sec:interval, we present our results for *