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Naveen Garg

Publications and source records attributed to Naveen Garg.

14 recordsLinked to original sources
arXiv2026

Stochastic Load Balancing with Machine Reservations

We introduce a novel variant of stochastic load balancing that enables a quantitative tradeoff between the practical benefits of non-adaptive policies and their performance limitations. Our model describes a solution in two stages. In the first stage, given only job-size distributions, we reserve a set of at most k machines for each job (a k-reservation). In the second stage, after observing job-size realizations, we assign each job to one of its reserved machines (a consistent assignment). The goal is to minimize the expected makespan. If k=1, we get the standard stochastic load balancing problem of finding a non-adaptive assignment with minimum expected makespan. If k is equal to the number of machines, then we obtain an all-powerful omniscient optimum that can tailor the assignment arbitrarily to the job-size realizations. We give a number of results that quantify this tradeoff. Most saliently, we show that in the setting of identical machines, a 2-reservation suffices to achieve a constant-factor approximation to the omniscient optimum, establishing a "power-of-two-choices" result for stochastic load balancing. We also show that this no longer holds true in the more challenging setting of related machines. Nonetheless, we give a number of positive algorithmic results for this setting: a true O(log m/log log m)-approximation; a bicriteria O(1)-approximation by reserving twice as many machines per job relative to an optimal k-reservation; and a 2-reservation whose cost is within a constant factor of what the adaptive optimum can achieve.

cs.DS↗
arXiv2024

Fully Dynamic $k$-Clustering with Fast Update Time and Small Recourse

In the dynamic metric $k$-median problem, we wish to maintain a set of $k$ centers $S \subseteq V$ in an input metric space $(V, d)$ that gets updated via point insertions/deletions, so as to minimize the objective $\sum_{x \in V} \min_{y \in S} d(x, y)$. The quality of a dynamic algorithm is measured in terms of its approximation ratio, "recourse" (the number of changes in $S$ per update) and "update time" (the time it takes to handle an update). The ultimate goal in this line of research is to obtain a dynamic $O(1)$ approximation algorithm with $\tilde{O}(1)$ recourse and $\tilde{O}(k)$ update time. Dynamic $k$-median is a canonical example of a class of problems known as dynamic $k$-clustering, that has received significant attention in recent years. To the best of our knowledge, however, previous papers either attempt to minimize the algorithm's recourse while ignoring its update time, or minimize the algorithm's update time while ignoring its recourse. For dynamic $k$-median, we come arbitrarily close to resolving the main open question on this topic, with the following results. (I) We develop a new framework of randomized local search that is suitable for adaptation in a dynamic setting. For every $\epsilon > 0$, this gives us a dynamic $k$-median algorithm with $O(1/\epsilon)$ approximation ratio, $\tilde{O}(k^{\epsilon})$ recourse and $\tilde{O}(k^{1+\epsilon})$ update time. This framework also generalizes to dynamic $k$-clustering with $\ell^p$-norm objectives, giving similar bounds for the dynamic $k$-means and a new trade-off for dynamic $k$-center. (II) If it suffices to maintain only an estimate of the value of the optimal $k$-median objective, then we obtain a $O(1)$ approximation algorithm with $\tilde{O}(k)$ update time. We achieve this result via adapting the Lagrangian Relaxation framework to the dynamic setting.

cs.DS↗
arXiv2024

Equitable Core Imputations for Max-Flow, MST and $b$-Matching Games

We study fair allocation of profit (or cost) for three central problems from combinatorial optimization: Max-Flow, MST and $b$-matching. The essentially unequivocal choice of solution concept for this purpose would be the core, because of its highly desirable properties. However, recent work [Vaz24] observed that for the assignment game, an arbitrary core imputation makes no fairness guarantee at the level of individual agents. To rectify this deficiency, special core imputations, called equitable core imputations, were defined - there are two such imputations, leximin and leximax - and efficient algorithms were given for finding them. For all three games, we start by giving examples to show that an arbitrary core imputation can be excessively unfair to certain agents. This led us to seek equitable core imputations for our three games as well. However, the ubiquitous tractable vs intractable schism separates the assignment game from our three games, making our task different from that of [Vaz24]. As is usual in the presence of intractability, we resorted to defining the Owen set for each game and algorithmically relating it to the set of optimal dual solutions of the underlying combinatorial problem. We then give polynomial time algorithms for finding equitable imputations in the Owen set. The motivation for this work is two-fold: the emergence of automated decision-making, with a special emphasis on fairness, and the plethora of industrial applications of our three games.

cs.GT↗
arXiv2020

Dual Half-integrality for Uncrossable Cut Cover and its Application to Maximum Half-Integral Flow

Given an edge weighted graph and a forest $F$, the $\textit{2-edge connectivity augmentation problem}$ is to pick a minimum weighted set of edges, $E'$, such that every connected component of $E'\cup F$ is 2-edge connected. Williamson et al. gave a 2-approximation algorithm (WGMV) for this problem using the primal-dual schema. We show that when edge weights are integral, the WGMV procedure can be modified to obtain a half-integral dual. The 2-edge connectivity augmentation problem has an interesting connection to routing flow in graphs where the union of supply and demand is planar. The half-integrality of the dual leads to a tight 2-approximate max-half-integral-flow min-multicut theorem.

cs.DS↗
arXiv2020

Integer Plane Multiflow Maximisation : Flow-Cut Gap and One-Quarter-Approximation

In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-cut-gap. Planarity means here that the union of the supply and demand graph is planar. We first prove that there exists a multiflow of value at least half of the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer one of value at least half of the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing again at most half of the value, in polynomial time, achieving a $1/4$-approximation algorithm for maximum integer multiflows in the plane, and an integer-flow-cut gap of $8$.

cs.DS↗
arXiv2019

Parallel Machine Scheduling to Minimize Energy Consumption

Given n jobs with release dates, deadlines and processing times we consider the problem of scheduling them on m parallel machines so as to minimize the total energy consumed. Machines can enter a sleep state and they consume no energy in this state. Each machine requires Q units of energy to awaken from the sleep state and in its active state the machine can process jobs and consumes a unit of energy per unit time. We allow for preemption and migration of jobs and provide the first constant approximation algorithm for this problem.

cs.DS↗
arXiv2019

DropConnect Is Effective in Modeling Uncertainty of Bayesian Deep Networks

Deep neural networks (DNNs) have achieved state-of-the-art performances in many important domains, including medical diagnosis, security, and autonomous driving. In these domains where safety is highly critical, an erroneous decision can result in serious consequences. While a perfect prediction accuracy is not always achievable, recent work on Bayesian deep networks shows that it is possible to know when DNNs are more likely to make mistakes. Knowing what DNNs do not know is desirable to increase the safety of deep learning technology in sensitive applications. Bayesian neural networks attempt to address this challenge. However, traditional approaches are computationally intractable and do not scale well to large, complex neural network architectures. In this paper, we develop a theoretical framework to approximate Bayesian inference for DNNs by imposing a Bernoulli distribution on the model weights. This method, called MC-DropConnect, gives us a tool to represent the model uncertainty with little change in the overall model structure or computational cost. We extensively validate the proposed algorithm on multiple network architectures and datasets for classification and semantic segmentation tasks. We also propose new metrics to quantify the uncertainty estimates. This enables an objective comparison between MC-DropConnect and prior approaches. Our empirical results demonstrate that the proposed framework yields significant improvement in both prediction accuracy and uncertainty estimation quality compared to the state of the art.

cs.LG↗
arXiv2019

On Fair Division of Indivisible Items

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of $e^{1/{e}} \approx 1.445$. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of $2 + \eps$

cs.DS↗
arXiv2019

Non-clairvoyant Precedence Constrained Scheduling

We consider the online problem of scheduling jobs on identical machines, where jobs have precedence constraints. We are interested in the demanding setting where the jobs sizes are not known up-front, but are revealed only upon completion (the non-clairvoyant setting). Such precedence-constrained scheduling problems routinely arise in map-reduce and large-scale optimization. In this paper, we make progress on this problem. For the objective of total weighted completion time, we give a constant-competitive algorithm. And for total weighted flow-time, we give an $O(1/ε^2)$-competitive algorithm under $(1+ε)$-speed augmentation and a natural ``no-surprises'' assumption on release dates of jobs (which we show is necessary in this context). Our algorithm proceeds by assigning {\em virtual rates} to all the waiting jobs, including the ones which are dependent on other uncompleted jobs, and then use these virtual rates to decide on the actual rates of minimal jobs (i.e., jobs which do not have dependencies and hence are eligible to run). Interestingly, the virtual rates are obtained by allocating time in a fair manner, using a Eisenberg-Gale-type convex program (which we can also solve optimally using a primal-dual scheme). The optimality condition of this convex program allows us to show dual-fitting proofs more easily, without having to guess and hand-craft the duals. We feel that this idea of using fair virtual rates should have broader applicability in scheduling problems.

cs.DS↗
arXiv2018

Constant Factor Approximation Algorithm for Weighted Flow Time on a Single Machine in Pseudo-polynomial time

In the weighted flow-time problem on a single machine, we are given a set of n jobs, where each job has a processing requirement p_j, release date r_j and weight w_j. The goal is to find a preemptive schedule which minimizes the sum of weighted flow-time of jobs, where the flow-time of a job is the difference between its completion time and its released date. We give the first pseudo-polynomial time constant approximation algorithm for this problem. The running time of our algorithm is polynomial in n, the number of jobs, and P, which is the ratio of the largest to the smallest processing requirement of a job. Our algorithm relies on a novel reduction of this problem to a generalization of the multi-cut problem on trees, which we call the Demand Multi-Cut problem. Even though we do not give a constant factor approximation algorithm for the Demand Multi-Cut problem on trees, we show that the specific instances of Demand Multi-Cut obtained by reduction from weighted flow-time problem instances have more structure in them, and we are able to employ techniques based on dynamic programming. Our dynamic programming algorithm relies on showing that there are near optimal solutions which have nice smoothness properties, and we exploit these properties to reduce the size of DP table.

cs.DS↗
arXiv2016

A Human Computer Interaction Solution for Radiology Reporting: Evaluation of the Factors of Variation

The purpose of this research is to evaluate the human and technical factors required to create a human-computer interface (HCI) for a structured reporting solution based on eye-gaze and speech signals. Gaze and speech signals from radiologists acquired during simulated image interpretation and dictation sessions were analyzed to determine a) variation of temporal relationship between eye gaze and speech in a dictation environment, and b) variation in eye movements for a particular image interpretation task among radiologists. Knowledge of these factors provides information regarding the complexity of the image interpretation or dictation task, and provides information that can be used to design a HCI for use in diagnostic radiology. Our ultimate goal is to use these data to create an HCI to automate the generation of a particular type of structured radiology report. Our data indicate that the a) temporal relationships between eye gaze and speech and b) scan paths substantially vary among radiologists, thus implying that an HCI system based on eye gaze and speech for automating the capture of data for structured reporting processes should be customized for each user. The image resolution and layout, image content, and order of targets during an image interpretation session are not relevant factors to consider when designing an HCI. Our findings can be applied to the design of other HCI solutions for radiological applications that involve visual inspection and verbal descriptions of image findings.

cs.HC↗
arXiv2014

Rejecting Jobs to Minimize Load and Maximum Flow-time

Online algorithms are usually analyzed using the notion of competitive ratio which compares the solution obtained by the algorithm to that obtained by an online adversary for the worst possible input sequence. Often this measure turns out to be too pessimistic, and one popular approach especially for scheduling problems has been that of "resource augmentation" which was first proposed by Kalyanasundaram and Pruhs. Although resource augmentation has been very successful in dealing with a variety of objective functions, there are problems for which even a (arbitrary) constant speedup cannot lead to a constant competitive algorithm. In this paper we propose a "rejection model" which requires no resource augmentation but which permits the online algorithm to not serve an epsilon-fraction of the requests. The problems considered in this paper are in the restricted assignment setting where each job can be assigned only to a subset of machines. For the load balancing problem where the objective is to minimize the maximum load on any machine, we give $O(\log^2 1/\eps)$-competitive algorithm which rejects at most an $\eps$-fraction of the jobs. For the problem of minimizing the maximum weighted flow-time, we give an $O(1/\eps^4)$-competitive algorithm which can reject at most an $\eps$-fraction of the jobs by weight. We also extend this result to a more general setting where the weights of a job for measuring its weighted flow-time and its contribution towards total allowed rejection weight are different. This is useful, for instance, when we consider the objective of minimizing the maximum stretch. We obtain an $O(1/\eps^6)$-competitive algorithm in this case. Our algorithms are immediate dispatch, though they may not be immediate reject. All these problems have very strong lower bounds in the speed augmentation model.

cs.DS↗
arXiv2002

On-Line End-to-End Congestion Control

Congestion control in the current Internet is accomplished mainly by TCP/IP. To understand the macroscopic network behavior that results from TCP/IP and similar end-to-end protocols, one main analytic technique is to show that the the protocol maximizes some global objective function of the network traffic. Here we analyze a particular end-to-end, MIMD (multiplicative-increase, multiplicative-decrease) protocol. We show that if all users of the network use the protocol, and all connections last for at least logarithmically many rounds, then the total weighted throughput (value of all packets received) is near the maximum possible. Our analysis includes round-trip-times, and (in contrast to most previous analyses) gives explicit convergence rates, allows connections to start and stop, and allows capacities to change.

cs.DS↗