Scheduling Malleable Jobs Under Topological Constraints

Abstract

Bleuse et al. (EuroPar 2018) introduced a general model for interference-aware scheduling in large scale parallel platforms. They considered two different types of communications: the flows induced by data exchanges during computations and the flows related to Input/Output operations. Rather than taking into account these communications explicitly, they restrict the possible allocations of a job by external topological constraints. In their work, jobs are considered to be rigid: a job requires a specific number of machines in order to be executed. Here, we first adopt the same framework for the platform and the aforementioned topological constraints. We show that there is no polynomial time approximation algorithm under the rigid setting with ratio smaller than 3/2, unless P = NP. Then, we focus on the malleable setting. We show that in the proportional-malleable setting, where the work of every job remains constant independently of the number of machines on which it is executed, the scheduling problem remains NPhard even in the uniform case, where the maximum number of machines is the same for all the jobs. Then, we propose a 2-approximation algorithm for this case. Furthermore, we present an approximation algorithm solving the more general case where the maximum number of machines is job-dependent and the work of the jobs is increasing with respect to the number of used machines, due to the communication overhead.

Assigning and Scheduling Generalized Malleable Jobs under Subadditive or Submodular Processing Speeds

“…In computing, e.g., modern heterogeneous parallel systems are typically consist of CPUs, GPUs, and I/O nodes, featuring different architectures and memory restrictions (Bleuse et al, 2017). To make optimal use of such systems, scheduling algorithms need to take the different architectures and the topology of the underlying communication network into consideration (Bampis et al, 2020). Another example is workforce assignment in production, where the assumption of identically-skilled workers is inappropriate in environments characterized by short-term contracts and an absence of standardized training (Nakade and Nishiwaki, 2008), in sheltered work centers for the disabled (Borba and Ritt, 2014), or in teams combining highly-trained specialized workers (Walter and Zimmermann, 2016).…”

A Constant-Factor Approximation for Generalized Malleable Scheduling under $M^\natural$-Concave Processing Speeds

“…In the construction of the assignment T, the jobs are split in two sets J (1) and J (2) such that each machine each machine processes at most one job from J (2) and each j ∈ J (2) is assigned to a set of machines with f j (T j ) ≤ 2C [9, Proposition 9], -and each job j ∈ J (1) is assigned to a single machine i with x ij ≥ 1 2 . Moreover j∈J (1) :i∈Tj f j ({i}) ≤ 2 • j∈J (1) :i∈Tj f j ({i})x ij for each machine i ∈ M [9, Proposition 8].…”

“…Moreover j∈J (1) :i∈Tj f j ({i}) ≤ 2 • j∈J (1) :i∈Tj f j ({i})x ij for each machine i ∈ M [9, Proposition 8]. As a consequence, f j ({i}) ≤ 2C and no machine is assigned more than two jobs from J (1) . Moreover, if a machine i is assigned two jobs from j 1 , j 2 ∈ J (1) , then x ij1 , x ij2 ≥ 1 2 together with (1) implies x ij = 0 for all other jobs.…”

“…As a consequence, f j ({i}) ≤ 2C and no machine is assigned more than two jobs from J (1) . Moreover, if a machine i is assigned two jobs from j 1 , j 2 ∈ J (1) , then x ij1 , x ij2 ≥ 1 2 together with (1) implies x ij = 0 for all other jobs. Thus a machine can be assigned at most two jobs (either two jobs from J (1) , or one jobs from J (1) and one from J (2) ) and g j (T j ) = 1/f j (T j ) ≥ 1 2C for all j ∈ J.…”

“…Moreover, if a machine i is assigned two jobs from j 1 , j 2 ∈ J (1) , then x ij1 , x ij2 ≥ 1 2 together with (1) implies x ij = 0 for all other jobs. Thus a machine can be assigned at most two jobs (either two jobs from J (1) , or one jobs from J (1) and one from J (2) ) and g j (T j ) = 1/f j (T j ) ≥ 1 2C for all j ∈ J. Translated back to the instance of Santa Claus, we conclude that in T no item is assigned more than two time and every agent receives a utility of u j (T j ) ≥ g j (T j ) ≥ 1 2C = V 2 .…”