We present the first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph-which undergoes both edge insertions and deletions-in polylogarithmic time. Our algorithm is randomized and, per update, takes O(log 2 ∆ · log 2 n) expected time. Furthermore, the algorithm can be adjusted to have O(log 2 ∆ · log 4 n) worst-case update-time with high probability. Here, n denotes the number of vertices and ∆ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC'18] who presented an algorithm with O(m 3/4 ) update-time (and thus broke the natural Ω(m) barrier) where m denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had O(min{ √ n, m 1/3 }) update-time [Assadi et al. SODA'19].Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time. * A preliminary version of this paper is

On online platforms, consumers face an abundance of options that are displayed in the form of a position ranking. Only products placed in the first few positions are readily accessible to the consumer, and she needs to exert effort to access more options. For such platforms, we develop a two-stage sequential search model where, in the first stage, the consumer sequentially screens positions to observe the preference weight of the products placed in them and forms a consideration set. In the second stage, she observes the additional idiosyncratic utility that she can derive from each product and chooses the highest-utility product within her consideration set. For this model, we first characterize the optimal sequential search policy of a welfare-maximizing consumer. We then study how platforms with different objectives should rank products. We focus on two objectives: (i) maximizing the platform’s market share and (ii) maximizing the consumer’s welfare. Somewhat surprisingly, we show that ranking products in decreasing order of their preference weights does not necessarily maximize market share or consumer welfare. Such a ranking may shorten the consumer’s consideration set due to the externality effect of high-positioned products on low-positioned ones, leading to insufficient screening. We then show that both problems—maximizing market share and maximizing consumer welfare—are NP-complete. We develop novel near-optimal polynomial-time ranking algorithms for each objective. Further, we show that, even though ranking products in decreasing order of their preference weights is suboptimal, such a ranking enjoys strong performance guarantees for both objectives. We complement our theoretical developments with numerical studies using synthetic data, in which we show (1) that heuristic versions of our algorithms that do not rely on model primitives perform well and (2) that our model can be effectively estimated using a maximum likelihood estimator. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.

Mixed strategies are often evaluated based on the expected payoff that they guarantee. This is not always desirable. In this paper, we consider games for which maximizing the expected payoff deviates from the actual goal of the players. To address this issue, we introduce the notion of a (u, p)-maxmin strategy which ensures receiving a minimum utility of u with probability at least p. We then give approximation algorithms for the problem of finding a (u, p)-maxmin strategy for these games.The first game that we consider is Colonel Blotto, a well-studied game that was introduced in 1921. In the Colonel Blotto game, two colonels divide their troops among a set of battlefields. Each battlefield is won by the colonel that puts more troops in it. The payoff of each colonel is the weighted number of battlefields that she wins. We show that maximizing the expected payoff of a player does not necessarily maximize her winning probability for certain applications of Colonel Blotto. For example, in presidential elections, the players' goal is to maximize the probability of winning more than half of the votes, rather than maximizing the expected number of votes that they get. We give an exact algorithm for a natural variant of continuous version of this game. More generally, we provide constant and logarithmic approximation algorithms for finding (u, p)-maxmin strategies.We also introduce a security game version of Colonel Blotto which we call auditing game. It is played between two players, a defender and an attacker. The goal of the defender is to prevent the attacker from changing the outcome of an instance of Colonel Blotto. Again, maximizing the expected payoff of the defender is not necessarily optimal.

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