Online Bipartite Matching with random user arrival is a fundamental problem in the online advertisement ecosystem. Over the last 30 years, many algorithms and impossibility results have been developed for this problem. In particular, the latest impossibility result was established by Manshadi, Oveis Gharan and Saberi in 2011. Since then, several algorithms have been published in an effort to narrow the gap between the upper and the lower bounds on the competitive ratio. In this paper we show that no algorithm can achieve a competitive ratio better than 1−e^{1-e}=0.82062..., improving upon the 0.823 upper bound presented in Manshadi, Oveis Gharan and Saberi (2011). Our construction is simple to state, accompanied by a fully analytic proof, and yields a competitive ratio bound intriguingly similar to 1−e^{-1}, the optimal competitive ratio for the fully adversarial Online Bipartite Matching problem. Although the tightness of our upper bound remains an open question, we show that our construction is extremal in a natural class of instances.

We present a novel approach for LDA (Latent Dirichlet Allocation) topic reconstruction. The main technical idea is to show that the distribution over the documents generated by LDA can be transformed into a distribution for a much simpler generative model in which documents are generated from {\em the same set of topics} but have a much simpler structure: documents are single topic and topics are chosen uniformly at random. Furthermore, this reduction is approximation preserving, in the sense that approximate distributions-- the only ones we can hope to compute in practice-- are mapped into approximate distribution in the simplified world. This opens up the possibility of efficiently reconstructing LDA topics in a roundabout way. Compute an approximate document distribution from the given corpus, transform it into an approximate distribution for the single-topic world, and run a reconstruction algorithm in the uniform, single topic world-- a much simpler task than direct LDA reconstruction. Indeed, we show the viability of the approach by giving very simple algorithms for a generalization of two notable cases that have been studied in the literature, $p$-separability and Gibbs sampling for matrix-like topics.

When a frequently-accessed cache item expires, multiple requests to that item can trigger a cache miss and start regenerating that same item at the same time. This phenomenon, known as cache stampede, severely limits the performance of databases and web servers. A natural countermeasure to this issue is to let the processes that perform such requests to randomly ask for a regeneration before the expiration time of the item. In this paper we give optimal algorithms for performing such probabilistic early expirations. Our algorithms are theoretically optimal and have much better performances than other solutions used in real-world applications.

Multi-fold cross-validation is an established practice to estimate the error rate of a learning algorithm. Quantifying the variance reduction gains due to cross-validation has been challenging due to the inherent correlations introduced by the folds. In this work we introduce a new and weak measure of stability called \emphloss stability and relate the cross-validation performance to loss stability; we also establish that this relationship is near-optimal. Our work thus quantitatively improves the current best bounds on cross-validation.

Greedy algorithms are practitioners' best friends - they are intuitive, simple to implement, and often lead to very good solutions. However, implementing greedy algorithms in a distributed setting is challenging since the greedy choice is inherently sequential, and it is not clear how to take advantage of the extra processing power.

The present work deals with the problem of information acquisition in a strategic networked environment. To study this problem, Kleinberg and Raghavan (FOCS 2005) introduced the model of query incentive networks, where the root of a binomial branching process wishes to retrieve an information -- known by each node independently with probability 1/n -- by investing as little as possible. The authors considered fixed-payment contracts in which every node strategically chooses an amount to offer its children paid upon information retrieval to convince them to seek the information in their subtrees. Kleinberg and Raghavan discovered that the investment needed at the root exhibits an unexpected threshold behavior that depends on the branching parameter b. For b>2, the investment is linear in the expected distance to the closest information (logarithmic in n, the rarity of the information), while, for 1<b<2, it becomes exponential in the same distance (i.e., polynomial in n). Arcaute et al. (EC 2007) later observed the same threshold behavior for arbitrary Galton-Watson branching processes.

Over half a century old and showing no signs of aging, k -means remains one of the most popular data processing algorithms. As is well-known, a proper initialization of k -means is crucial for obtaining a good final solution. The recently proposed k -means++ initialization algorithm achieves this, obtaining an initial set of centers that is provably close to the optimum solution. A major downside of the k -means++ is its inherent sequential nature, which limits its applicability to massive data: one must make k passes over the data to find a good initial set of centers. In this work we show how to drastically reduce the number of passes needed to obtain, in parallel, a good initialization. This is unlike prevailing efforts on parallelizing k -means that have mostly focused on the post-initialization phases of k -means. We prove that our proposed initialization algorithm k -means|| obtains a nearly optimal solution after a logarithmic number of passes, and then show that in practice a constant number of passes suffices. Experimental evaluation on real-world large-scale data demonstrates that k -means|| outperforms k -means++ in both sequential and parallel settings.

The secretary problem lies at the core of mechanism design for online auctions. In this work we study the generalization of the classical secretary problem in a setting where there is only a partial order between the elements and the goal of the algorithm is to return one of the maximal elements of the poset. This is equivalent to the auction setting where the seller has a multidimensional objective function with only a partial order among the outcomes. We obtain an algorithm that succeeds with probability at least k-k/(k-1)((1 + log k1/(k-1))k - 1), where k is the number of maximal elements in the poset and is the only information about the poset that is known to the algorithm; the success probability approaches the classical bound of 1/e as k -> 1. On the other hand, we prove an almost matching upper bound of k-1/(k-1) on the success probability of any algorithm for this problem; this upper bound holds even if the algorithm knows the complete structure of the poset.

The k-means algorithm is a well-known method for partitioning n points that lie in the d-dimensional space into k clusters. Its main features are simplicity and speed in practice. Theoretically, however, the best known upper bound on its running time (i.e., n O(kd)) is, in general, exponential in the number of points (when kd=Ω(n/log n)). Recently Arthur and Vassilvitskii (Proceedings of the 22nd Annual Symposium on Computational Geometry, pp. 144–153, 2006) showed a super-polynomial worst-case analysis, improving the best known lower bound from Ω(n) to $2^{\varOmega (\sqrt{n})}$ with a construction in $d=\varOmega (\sqrt{n})$ dimensions. In Arthur and Vassilvitskii (Proceedings of the 22nd Annual Symposium on Computational Geometry, pp. 144–153, 2006), they also conjectured the existence of super-polynomial lower bounds for any d≥2. Our contribution is twofold: we prove this conjecture and we improve the lower bound, by presenting a simple construction in the plane that leads to the exponential lower bound 2Ω(n).

We consider list coloring problems for graphs $\mathcal{G}$ of girth larger than $c\log_{\Delta-1}n$, where n and $\Delta\geq3$ are, respectively, the order and the maximum degree of $\mathcal{G}$, and c is a suitable constant. First, we determine that the edge and total list chromatic numbers of these graphs are $\chi'_l(\mathcal{G})=\Delta$ and $\chi”_l(\mathcal{G})=\Delta+1$. This proves that the general conjectures of Bollobás and Harris [Graphs Combin., 1 (1985), pp. 115–127], Behzad [The total chromatic number, in Combinatorial Mathematics and Its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, pp. 1–8], Vizing [Diskret. Analiz., 3 (1964), pp. 25–30], and Juvan, Mohar, and Škrekovski [Combin. Probab. Comput., 7 (1998), pp. 181–188] hold for this particular class of graphs. Moreover, our proofs exhibit a certain degree of “locality,” which we exploit to obtain an efficient distributed algorithm able to compute both kinds of optimal list colorings. Also, using an argument similar to one of Erdös, we show that our algorithm can compute k-list vertex colorings of graphs having girth larger than $c\log_{k-1}n$.

The k-means algorithm is a well-known method for partitioning n points that lie in the d-dimensional space into k clusters. Its main features are simplicity and speed in practice. Theoretically, however, the best known upper bound on its running time (i.e. O(nkd)) is, in general, exponential in the number of points (when kd=Ω(n log n)). Recently, Arthur and Vassilvitskii [2] showed a super-polynomial worst-case analysis, improving the best known lower bound from Ω(n) to 2Ω(√n) with a construction in d=Ω(√n) dimensions. In [2] they also conjectured the existence of super-polynomial lower bounds for any d≥ 2. Our contribution is twofold: we prove this conjecture and we improve the lower bound, by presenting a simple construction in the plane that leads to the exponential lower bound 2Ω(n).