arXiv subjects

David Stalfa

Publications and source records attributed to David Stalfa.

6 recordsLinked to original sources

arXiv2023

Scheduling Splittable Jobs on Configurable Machines

Motivated by deep neural network applications, we study the problem of scheduling splittable jobs (e.g., neural network inference tasks) on configurable machines (e.g., multi-instance GPUs). We are given $n$ jobs and a set $C$ of configurations (e.g, representing ways to configure a GPU) consisting of multisets of blocks (e.g., representing GPU instances). A schedule consists of a set of machines, each assigned some configuration in $C$ with each block in the configuration assigned to process one job. The amount of a job's demand that is satisfied by a given block is an arbitrary function of the job and block. The objective is to satisfy all demands on as few machines as possible. We provide a tight logarithmic approximation algorithm for this problem in the general setting, an asymptotic $(2 + \varepsilon)$-approximation with $O(1)$ input configurations for arbitrary $\varepsilon > 0$, and a polynomial time approximation scheme when both the number and size of configurations are $O(1)$.

cs.DS↗

arXiv2022

Scheduling under Non-Uniform Job and Machine Delays

We study the problem of scheduling precedence-constrained jobs on heterogenous machines in the presence of non-uniform job and machine communication delays. We are given as input $n$ unit size precedence-ordered jobs and $m$ related machines such that machine $i$ can execute up to $m_i$ jobs at a time. Each machine $i$ has an in-delay $\rho^{\mathrm{in}}_i$ and out-delay $\rho^{\mathrm{out}}_i$. Likewise, each job $v$ has an in-delay $\rho^{\mathrm{in}}_v$ and out-delay $\rho^{\mathrm{out}}_v$. In a schedule, job $v$ may be executed on machine $i$ at time $t$ if each predecessor $u$ of $v$ is completed on $i$ before time $t$ or on any machine $j$ before time $t - (\rho^{\mathrm{in}}_i + \rho^{\mathrm{out}}_j + \rho^{\mathrm{out}}_u + \rho^{\mathrm{in}}_v)$. The goal is to construct a schedule that minimizes makespan. We consider schedules that allow duplication of jobs as well as schedules which do not. When duplication is allowed, we provide an asymptotic $\mathrm{polylog}(n)$-approximation algorithms both when duplication is allowed and when it is not. We also obtain a true $\mathrm{polylog}(n)$-approximation for symmetric machine and job delays. These are the first polylogarithmic approximation algorithms for scheduling with non-uniform communication delays. We also consider a more general model, where the delay can be an arbitrary function of the job and the machine executing it: job $v$ can be executed on machine $i$ at time $t$ if all of $v$'s predecessors are executed on $i$ by time $t-1$ or on any machine by time $t - \rho_{v,i}$. We present an approximation-preserving reduction from the Unique Machines Precedence-constrained Scheduling (UMPS) problem, first defined in [DKRSTZ22], to this job-machine delay model. The reduction entails logarithmic hardness for this delay setting, as well as polynomial hardness if the conjectured hardness of UMPS holds.

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arXiv2021

Improved Bounds for Scheduling Flows under Endpoint Capacity Constraints

We study flow scheduling under node capacity constraints. We are given capacitated nodes and an online sequence of jobs, each with a release time and a demand to be routed between two nodes. A schedule specifies which jobs are routed in each step, guaranteeing that the total demand on a node in any step is at most its capacity. A key metric in this scenario is response time: the time between a job's release and its completion. Prior work shows no un-augmented algorithm is competitive for average response time, and that a constant factor competitive ratio is achievable with augmentation exceeding 2 (Dinitz-Moseley Infocom 2020). For maximum response time, the best known result is a 2-competitive algorithm with a augmentation 4 (Jahanjou et al SPAA 2020). We improve these bounds under various response time objectives. We show that, without resource augmentation, the best competitive ratio for maximum response time is $\Omega(n)$, where $n$ is the number of nodes. Our Proportional Allocation algorithm uses $(1+\varepsilon)$ resource augmentation to achieve a $(1/\varepsilon)$-competitive ratio in the setting with general demands and capacities, and splittable jobs. Our Batch Decomposition algorithm is $2$-competitive (resp., optimal) for maximum response time using resource augmentation 2 (resp., 4) in the setting with unit demands and capacities, and unsplittable jobs. We also derive bounds for the simultaneous approximation of average and maximum response time metrics.

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arXiv2020

Scheduling Flows on a Switch to Optimize Response Times

We study the scheduling of flows on a switch with the goal of optimizing metrics related to the response time of the flows. The input to the problem is a sequence of flow requests on a switch, where the switch is represented by a bipartite graph with a capacity on each vertex (or port), and a flow request is an edge with associated demand. In each round, a subset of edges can be scheduled subject to the constraint that the total demand of the scheduled edges incident on any vertex is at most the capacity of the vertex. Previous work has essentially settled the complexity of metrics based on {\em completion time}. The objective of average or maximum {\em response time}, however, is much more challenging. We present approximation algorithms for flow scheduling over a switch to optimize response time based metrics. For the average response time metric, whose NP-hardness follows directly from past work, we present an offline $O(1 + O(\log(n))/c)$ approximation algorithm for unit flows, assuming that the port capacities of the switch can be increased by a factor of $1 + c$, for any given positive integer $c$. For the maximum response time metric, we first establish that it is NP-hard to achieve an approximation factor of better than 4/3 without augmenting capacity. We then present an offline algorithm that achieves {\em optimal maximum response time}, assuming the capacity of each port is increased by at most $2 d_{max} - 1$, where $d_{max}$ is the maximum demand of any flow. Both algorithms are based on linear programming relaxations. We also study the online version of flow scheduling using the lens of competitive analysis, and present preliminary results along with experiments that evaluate the performance of fast online heuristics.

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arXiv2020

Scheduling Precedence-Constrained Jobs on Related Machines with Communication Delay

We consider the problem of scheduling $n$ precedence-constrained jobs on $m$ uniformly-related machines in the presence of an arbitrary, fixed communication delay $ρ$. We consider a model that allows job duplication, i.e. processing of the same job on multiple machines, which, as we show, can reduce the length of a schedule (i.e., its makespan) by a logarithmic factor. Our main result is an $O(\log m \log ρ/ \log \log ρ)$-approximation algorithm for minimizing makespan, assuming the minimum makespan is at least $ρ$. Our algorithm is based on rounding a linear programming relaxation for the problem, which includes carefully designed constraints capturing the interaction among communication delay, precedence requirements, varying speeds, and job duplication. Our result builds on two previous lines of work, one with communication delay but identical machines (Lepere, Rapine 2002) and the other with uniformly-related machines but no communication delay (Chudak, Shmoys 1999). We next show that the integrality gap of our mathematical program is $Ω(\sqrt{\log ρ})$. Our gap construction employs expander graphs and exploits a property of robust expansion and its generalization to paths of longer length. Finally, we quantify the advantage of duplication in scheduling with communication delay. We show that the best schedule without duplication can have makespan $Ω(ρ/\log ρ)$ or $Ω(\log m/\log\log m)$ or $Ω(\log n/\log \log n)$ times that of an optimal schedule allowing duplication. Nevertheless, we present a polynomial time algorithm to transform any schedule to a schedule without duplication at the cost of a $O(\log^2 n \log m)$ factor increase in makespan. Together with our makespan approximation algorithm for schedules allowing duplication, this also yields a polylogarithmic-approximation algorithm for the setting where duplication is not allowed.

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arXiv2018

Maximum Area Axis-Aligned Square Packings

Given a point set $S=\{s_1,\ldots , s_n\}$ in the unit square $U=[0,1]^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in $S$. It is shown that area$(R(S))\geq \frac12$ for every finite set $S\subset U$, and this bound is the best possible. The region $R(S)$ can be computed in $O(n\log n)$ time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.

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