arXiv subjects

Jakub Tarnawski

Publications and source records attributed to Jakub Tarnawski.

31 records · Page 2Linked to original sources

arXiv2021

On the Hardness of Scheduling With Non-Uniform Communication Delays

In the scheduling with non-uniform communication delay problem, the input is a set of jobs with precedence constraints. Associated with every precedence constraint between a pair of jobs is a communication delay, the time duration the scheduler has to wait between the two jobs if they are scheduled on different machines. The objective is to assign the jobs to machines to minimize the makespan of the schedule. Despite being a fundamental problem in theory and a consequential problem in practice, the approximability of scheduling problems with communication delays is not very well understood. One of the top ten open problems in scheduling theory, in the influential list by Schuurman and Woeginger and its latest update by Bansal, asks if the problem admits a constant factor approximation algorithm. In this paper, we answer the question in negative by proving that there is a logarithmic hardness for the problem under the standard complexity theory assumption that NP-complete problems do not admit quasi-polynomial time algorithms. Our hardness result is obtained using a surprisingly simple reduction from a problem that we call Unique Machine Precedence constraints Scheduling (UMPS). We believe that this problem is of central importance in understanding the hardness of many scheduling problems and conjecture that it is very hard to approximate. Among other things, our conjecture implies a logarithmic hardness of related machine scheduling with precedences, a long-standing open problem in scheduling theory and approximation algorithms.

cs.DS↗

arXiv2020

Efficient Algorithms for Device Placement of DNN Graph Operators

Modern machine learning workloads use large models, with complex structures, that are very expensive to execute. The devices that execute complex models are becoming increasingly heterogeneous as we see a flourishing of domain-specific accelerators being offered as hardware accelerators in addition to CPUs. These trends necessitate distributing the workload across multiple devices. Recent work has shown that significant gains can be obtained with model parallelism, i.e, partitioning a neural network's computational graph onto multiple devices. In particular, this form of parallelism assumes a pipeline of devices, which is fed a stream of samples and yields high throughput for training and inference of DNNs. However, for such settings (large models and multiple heterogeneous devices), we require automated algorithms and toolchains that can partition the ML workload across devices. In this paper, we identify and isolate the structured optimization problem at the core of device placement of DNN operators, for both inference and training, especially in modern pipelined settings. We then provide algorithms that solve this problem to optimality. We demonstrate the applicability and efficiency of our approaches using several contemporary DNN computation graphs.

cs.LG↗

arXiv2020

Fairness in Streaming Submodular Maximization: Algorithms and Hardness

Submodular maximization has become established as the method of choice for the task of selecting representative and diverse summaries of data. However, if datapoints have sensitive attributes such as gender or age, such machine learning algorithms, left unchecked, are known to exhibit bias: under- or over-representation of particular groups. This has made the design of fair machine learning algorithms increasingly important. In this work we address the question: Is it possible to create fair summaries for massive datasets? To this end, we develop the first streaming approximation algorithms for submodular maximization under fairness constraints, for both monotone and non-monotone functions. We validate our findings empirically on exemplar-based clustering, movie recommendation, DPP-based summarization, and maximum coverage in social networks, showing that fairness constraints do not significantly impact utility.

cs.LG↗

arXiv2020

A Constant-Factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP). Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is a reduction to Subtour Partition Cover, an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. We first show that any algorithm for Subtour Partition Cover can be turned into an algorithm for ATSP while only losing a small constant factor in the performance guarantee. Next, we present a reduction from general ATSP instances to structured instances, on which we then solve Subtour Partition Cover, yielding our constant-factor approximation algorithm for ATSP.

cs.DS↗

arXiv2020

Hierarchy-Based Algorithms for Minimizing Makespan under Precedence and Communication Constraints

We consider the classic problem of scheduling jobs with precedence constraints on a set of identical machines to minimize the makespan objective function. Understanding the exact approximability of the problem when the number of machines is a constant is a well-known question in scheduling theory. Indeed, an outstanding open problem from the classic book of Garey and Johnson asks whether this problem is NP-hard even in the case of 3 machines and unit-length jobs. In a recent breakthrough, Levey and Rothvoss gave a $(1+ε)$-approximation algorithm, which runs in nearly quasi-polynomial time, for the case when job have unit lengths. However, a substantially more difficult case where jobs have arbitrary processing lengths has remained open. We make progress on this more general problem. We show that there exists a $(1+ε)$-approximation algorithm (with similar running time as that of Levey and Rothvoss) for the non-migratory setting: when every job has to be scheduled entirely on a single machine, but within a machine the job need not be scheduled during consecutive time steps. Further, we also show that our algorithmic framework generalizes to another classic scenario where, along with the precedence constraints, the jobs also have communication delay constraints. Both of these fundamental problems are highly relevant to the practice of datacenter scheduling.

cs.DS↗

arXiv2020

Scheduling with Communication Delays via LP Hierarchies and Clustering

We consider the classic problem of scheduling jobs with precedence constraints on identical machines to minimize makespan, in the presence of communication delays. In this setting, denoted by $\mathsf{P} \mid \mathsf{prec}, c \mid C_{\mathsf{max}}$, if two dependent jobs are scheduled on different machines, then at least $c$ units of time must pass between their executions. Despite its relevance to many applications, this model remains one of the most poorly understood in scheduling theory. Even for a special case where an unlimited number of machines is available, the best known approximation ratio is $2/3 \cdot (c+1)$, whereas Graham's greedy list scheduling algorithm already gives a $(c+1)$-approximation in that setting. An outstanding open problem in the top-10 list by Schuurman and Woeginger and its recent update by Bansal asks whether there exists a constant-factor approximation algorithm. In this work we give a polynomial-time $O(\log c \cdot \log m)$-approximation algorithm for this problem, where $m$ is the number of machines and $c$ is the communication delay. Our approach is based on a Sherali-Adams lift of a linear programming relaxation and a randomized clustering of the semimetric space induced by this lift.

cs.DS↗

arXiv2018

Beyond $1/2$-Approximation for Submodular Maximization on Massive Data Streams

Many tasks in machine learning and data mining, such as data diversification, non-parametric learning, kernel machines, clustering etc., require extracting a small but representative summary from a massive dataset. Often, such problems can be posed as maximizing a submodular set function subject to a cardinality constraint. We consider this question in the streaming setting, where elements arrive over time at a fast pace and thus we need to design an efficient, low-memory algorithm. One such method, proposed by Badanidiyuru et al. (2014), always finds a $0.5$-approximate solution. Can this approximation factor be improved? We answer this question affirmatively by designing a new algorithm SALSA for streaming submodular maximization. It is the first low-memory, single-pass algorithm that improves the factor $0.5$, under the natural assumption that elements arrive in a random order. We also show that this assumption is necessary, i.e., that there is no such algorithm with better than $0.5$-approximation when elements arrive in arbitrary order. Our experiments demonstrate that SALSA significantly outperforms the state of the art in applications related to exemplar-based clustering, social graph analysis, and recommender systems.

cs.LG↗

arXiv2017

Streaming Robust Submodular Maximization: A Partitioned Thresholding Approach

We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm's memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-GREEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance.

cs.DS↗

arXiv2017

Constant Factor Approximation for ATSP with Two Edge Weights

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.

cs.DS↗

arXiv2017

The Matching Problem in General Graphs is in Quasi-NC

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in $O(\log^3 n)$ time on $n^{O(\log^2 n)}$ processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.

cs.CC↗

arXiv2017

Active Learning and Proofreading for Delineation of Curvilinear Structures

Many state-of-the-art delineation methods rely on supervised machine learning algorithms. As a result, they require manually annotated training data, which is tedious to obtain. Furthermore, even minor classification errors may significantly affect the topology of the final result. In this paper we propose a generic approach to addressing both of these problems by taking into account the influence of a potential misclassification on the resulting delineation. In an Active Learning context, we identify parts of linear structures that should be annotated first in order to train a classifier effectively. In a proofreading context, we similarly find regions of the resulting reconstruction that should be verified in priority to obtain a nearly-perfect result. In both cases, by focusing the attention of the human expert on potential classification mistakes which are the most critical parts of the delineation, we reduce the amount of required supervision. We demonstrate the effectiveness of our approach on microscopy images depicting blood vessels and neurons.

cs.CV↗

arXiv2016

Unrelated Machine Scheduling of Jobs with Uniform Smith Ratios

We consider the classic problem of scheduling jobs on unrelated machines so as to minimize the weighted sum of completion times. Recently, for a small constant $\varepsilon >0 $, Bansal et al. gave a $(3/2-\varepsilon)$-approximation algorithm improving upon the natural barrier of $3/2$ which follows from independent randomized rounding. In simplified terms, their result is obtained by an enhancement of independent randomized rounding via strong negative correlation properties. In this work, we take a different approach and propose to use the same elegant rounding scheme for the weighted completion time objective as devised by Shmoys and Tardos for optimizing a linear function subject to makespan constraints. Our main result is a $1.21$-approximation algorithm for the natural special case where the weight of a job is proportional to its processing time (specifically, all jobs have the same Smith ratio), which expresses the notion that each unit of work has the same weight. In addition, as a direct consequence of the rounding, our algorithm also achieves a bi-criteria $2$-approximation for the makespan objective. Our technical contribution is a tight analysis of the expected cost of the solution compared to the one given by the Configuration-LP relaxation - we reduce this task to that of understanding certain worst-case instances which are simple to analyze.

cs.DS↗

arXiv2015

Fast Generation of Random Spanning Trees and the Effective Resistance Metric

We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected $\tilde{O}(m^{4/3})$ time. This improves over the best previously known bound of $\min(\tilde{O}(m\sqrt{n}),O(n^ω))$ -- that follows from the work of Kelner and Mądry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] -- whenever the input graph is sufficiently sparse. At a high level, our result stems from carefully exploiting the interplay of random spanning trees, random walks, and the notion of effective resistance, as well as from devising a way to algorithmically relate these concepts to the combinatorial structure of the graph. This involves, in particular, establishing a new connection between the effective resistance metric and the cut structure of the underlying graph.

cs.DS↗