# Nitish Korula

Nitish is a Research Scientist at Google. His research interests lie in algorithm design, especially for optimization problems where it is hard to find optimal solutions. In particular, he is interested in approximation and online algorithms, combinatorial optimization, graph theory, and algorithmic game theory.

Before joining Google, Nitish received his Ph.D.in Computer Science at the University of Illinois, and his B.E. from Birla Institute of Technology & Science (BITS), Pilani.

Authored Publications

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ONLINE SUBMODULAR WELFARE MAXIMIZATION: GREEDY BEATS 1/2 IN RANDOM ORDER

SIAM Journal on Computing, 47(3) (2018), pp. 1056-1086

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In the submodular welfare maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents. Each agent ell has a valuation function v_ell, where v_ell(S) denotes the welfare obtained by this agent if she receives the set of items S. The functions v_ell are all submodular; as is standard, we assume that they are monotone and v_ell(∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, . . . , Sm in order to maximize the social welfare, defined as \Sum_ell v_ell(S_ell). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondr´ak’s recent (1 − 1/e)-approximation algorithm. In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users/impressions they receive. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary/worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm’s performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: In the random order model, the adversary can construct a worst-case set of items and valuations but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n, which is barely better than the ratio for the adversarial order. Obtaining a competitive ratio of 1/2 + Ω(1) for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least 0.505 for online SWM in the random order model. This is the first result showing a competitive ratio bounded above 1/2 in the random order model, even for special cases such as the weighted matching or budgeted allocation problem (without the so-called large capacity assumptions). For special cases of submodular functions including weighted matching, weighted coverage functions, and a broader class of “second-order supermodular” functions, we provide a different analysis that gives a competitive ratio of 0.51. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. In addition to our new competitive ratios for online SWM, we make two further contributions: First, we define the classes of second-order modular, supermodular, and submodular functions, which are likely to be of independent interest in submodular optimization.
Second, we obtain an improved competitive ratio via a technique we refer to as gain linearizing,
which may be useful in other contexts: Essentially, we linearize the submodular function by dividing the gain of an optimal solution into gain from individual elements, compare the algorithm’s gain when it assigns an element to the optimal solution’s gain from the element, and, crucially, bound the extent to which assigning elements can affect the potential gain of other elements.
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Whole-Page Optimization and Submodular Welfare Maximization with Online Bidders

Nikhil R. Devanur

Zhiyi Huang

ACM Trans. Economics and Comput. 4(3) (2016)

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In the context of online ad serving, display ads may appear on different types of webpages, where each page includes several ad slots and therefore multiple ads can be shown on each page. The set of ads that can be assigned to ad slots of the same page needs to satisfy various prespecified constraints including exclusion constraints, diversity constraints, and the like. Upon arrival of a user, the ad serving system needs to allocate a set of ads to the current webpage respecting these per-page allocation constraints. Previous slot-based settings ignore the important concept of a page and may lead to highly suboptimal results in general. In this article, motivated by these applications in display advertising and inspired by the submodular welfare maximization problem with online bidders, we study a general class of page-based ad allocation problems, present the first (tight) constant-factor approximation algorithms for these problems, and confirm the performance of our algorithms experimentally on real-world datasets.
A key technical ingredient of our results is a novel primal-dual analysis for handling free disposal, which updates dual variables using a “level function” instead of a single level and unifies with previous analyses of related problems. This new analysis method allows us to handle arbitrarily complicated allocation constraints for each page. Our main result is an algorithm that achieves a 1 &minus frac 1 e &minus o(1)-competitive ratio. Moreover, our experiments on real-world datasets show significant improvements of our page-based algorithms compared to the slot-based algorithms.
Finally, we observe that our problem is closely related to the submodular welfare maximization (SWM) problem. In particular, we introduce a variant of the SWM problem with online bidders and show how to solve this problem using our algorithm for whole-page optimization.
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Linking Users Across Domains with Location Data: Theory and Validation

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Chistopher Riederer

Yunsung Kim

Augustin Chaintreau

WWW (2016) (to appear)