arXiv subjects

Nathaniel Kell

Publications and source records attributed to Nathaniel Kell.

4 recordsLinked to original sources

arXiv2021

Approximations for Allocating Indivisible Items with Concave-Additive Valuations

We study a general allocation setting where agent valuations are concave additive. In this model, a collection of items must be uniquely distributed among a set of agents, where each agent-item pair has a specified utility. The objective is to maximize the sum of agent valuations, each of which is an arbitrary non-decreasing concave function of the agent's total additive utility. This setting was studied by Devanur and Jain (STOC 2012) in the online setting for divisible items. In this paper, we obtain both tight multiplicative and additive approximations in the offline setting for indivisible items. Our approximations depend on novel parameters that measure the local multiplicative/additive curvatures of each agent valuation, which we show correspond directly to the integrality gap of the natural assignment convex program of the problem. Furthermore, we extend our additive guarantees to obtain constant multiplicative approximations for Asymmetric Nash Welfare Maximization when agents have smooth valuations (Fain et al. EC 2018, Fluschnik et al. AAAI 2019). This algorithm also yields an interesting tatonnement-style interpretation, where agents adjust uniform prices and items are assigned according to maximum weighted bang-per-buck ratios.

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arXiv2017

Online Load Balancing for Related Machines

In the load balancing problem, introduced by Graham in the 1960s (SIAM J. of Appl. Math. 1966, 1969), jobs arriving online have to be assigned to machines so to minimize an objective defined on machine loads. A long line of work has addressed this problem for both the makespan norm and arbitrary $\ell_q$-norms of machine loads. Recent literature (e.g., Azar et al., STOC 2013; Im et al., FOCS 2015) has further expanded the scope of this problem to vector loads, to capture jobs with multi-dimensional resource requirements in applications such as data centers. In this paper, we completely resolve the job scheduling problem for both scalar and vector jobs on related machines, i.e., where each machine has a given speed and the time taken to process a job is inversely proportional to the speed of the machine it is assigned on. We show the following results. For scalar scheduling, we give a constant competitive algorithm for optimizing any $\ell_q$-norm for related machines. The only previously known result was for the makespan norm. For vector scheduling, there are two natural variants for vector scheduling, depending on whether the speed of a machine is dimension-dependent or not. We show a sharp contrast between these two variants, proving that they are respectively equivalent to unrelated machines and identical machines for the makespan norm. We also extend these results to arbitrary $\ell_q$-norms of the machine loads. No previous results were known for vector scheduling on related machines.

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arXiv2016

Online Budgeted Allocation with General Budgets

We study the online budgeted allocation (also called ADWORDS) problem, where a set of impressions arriving online are allocated to a set of budget-constrained advertisers to maximize revenue. Motivated by connections to Internet advertising, several variants of this problem have been studied since the seminal work of Mehta, Saberi, Vazirani, and Vazirani (FOCS 2005). However, this entire body of work focuses on a single budget for every advertising campaign, whereas in order to fully represent the actual agenda of an advertiser, an advertising budget should be expressible over multiple tiers of user-attribute granularity. A simple example is an advertising campaign that is constrained by an overall budget but is also accompanied by a set of sub-budgets for each target demographic. In such a contract scheme, an advertiser can specify their true user-targeting goals, allowing the publisher to fulfill them through relevant allocations. In this paper, we give a complete characterization of the ADWORDS problem for general advertising budgets. In the most general setting, we show that, unlike in the single-budget ADWORDS problem, obtaining a constant competitive ratio is impossible and give asymptotically tight upper and lower bounds. However for our main result, we observe that in many real-world scenarios (as in the above example), multi-tier budgets have a laminar structure, since most relevant consumer or product classifications are hierarchical. For laminar budgets, we obtain a competitive ratio of e/(e-1) in the small bids case, which matches the best known ADWORDS result for single budgets. Our algorithm has a primal-dual structure and generalizes the primal-dual analysis for single- budget ADWORDS first given by Buchbinder, Jain, and Naor (ESA 2007).

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arXiv2015

Tight Bounds for Online Vector Scheduling

Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and $L_r$ norms of loads are among the most popular objectives considered. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J of Comp. 2006), vector scheduling is a generalization of classical load balancing, where every job has a vector load instead of a scalar load. In this paper, we resolve the online complexity of the vector scheduling problem and its important generalizations. Our main results are: -For identical machines, we show that the optimal competitive ratio is $Θ(\log d / \log \log d)$ by giving an online lower bound and an algorithm with an asymptotically matching competitive ratio. The lower bound is technically challenging, and is obtained via an online lower bound for the minimum mono-chromatic clique problem using a novel online coloring game and randomized coding scheme. -For unrelated machines, we show that the optimal competitive ratio is $Θ(\log m + \log d)$ by giving an online lower bound that matches a previously known upper bound. Unlike identical machines, however, extending these results, particularly the upper bound, to general $L_r$ norms requires new ideas. In particular, we use a carefully constructed potential function that balances the individual $L_r$ objectives with the overall (convexified) min-max objective to guide the online algorithm and track the changes in potential to bound the competitive ratio.

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