We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to $1$. Specifically, we show that any algorithm that computes set cover exactly using $({1 \over 2\delta}-1)$ passes must use $\tilde{\Omega}(mn^{\delta})$ space in the regime of $m=O(n)$. Furthermore, we consider the problem in the geometric setting where the elements are points in $\mathbb{R}^2$ and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal $\tilde{O}(n)$ space to find a logarithmic approximation in $O(1/\delta)$ passes. Finally, we show that any randomized one-pass algorithm that distinguishes between covers of size 2 and 3 must use a linear (i.e., $\Omega(mn)$) amount of space. This is the first result showing that a randomized, approximate algorithm cannot achieve a space bound that is sublinear in the input size. This indicates that using multiple passes might be necessary in order to achieve sub-linear space bounds for this problem while guaranteeing small approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201

Abstract. We develop the first streaming algorithm and the first two-party communication protocol that uses a constant number of passes/rounds and sublinear space/communication for logarithmic approximation to the classic Set Cover problem. Specifically, for n elements and m sets, our algorithm/protocol achieves a space bound of O(m · n δ log 2 n log m) using O(4 1/δ ) passes/rounds while achieving an approximation factor of O(4 1/δ log n) in polynomial time (for δ = Ω(1/ log n)). If we allow the algorithm/protocol to spend exponential time per pass/round, we achieve an approximation factor of O(4 1/δ ). Our approach uses randomization, which we show is necessary: no deterministic constant approximation is possible (even given exponential time) using o(mn) space. These results are some of the first on streaming algorithms and efficient two-party communication protocols for approximation algorithms. Moreover, we show that our algorithm can be applied to multi-party communication model.

It is well established that extracting and annotating occurrences of entities in a collection of unstructured text documents with their concepts improve the effectiveness of answering queries over the collection. However, it is very resource intensive to create and maintain large annotated collections. Since the available resources of an enterprise are limited and/or its users may have urgent information needs, it may have to select only a subset of relevant concepts for extraction and annotation. We call this subset a conceptual design for the annotated collection. In this paper, we introduce the problem of cost effective conceptual design, where given a collection, a set of relevant concepts, and a fixed budget, one likes to find a conceptual design that improves the effectiveness of answering queries over the collection the most. We prove that the problem is generally NP-hard in the number of relevant concepts and propose two efficient approximation algorithms to solve the problem: Approximate Popularity Maximization (APM for short) and Approximate Annotation-benefit Maximization (AAM for short). We show that if there is not any constraints regrading the overlap of concepts, APM is a fully polynomial time approximation scheme. We also prove that if the relevant concepts are mutually exclusive, APM has a constant approximation ratio and AAM is a fully polynomial time approximation scheme. Our empirical results using Wikipedia collection and a search engine query log validate the proposed formalization of the problem and show that APM and AAM efficiently compute conceptual designs. They also indicate that in general APM delivers the optimal conceptual designs if the relevant concepts are not mutually exclusive. Also, if the relevant concepts are mutually exclusive, the conceptual designs delivered by AAM improve the effectiveness of answering queries over the collection more than the solutions provided by APM.

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We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities.On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an α-approximate cover using O(m(n/k) 1/(α−1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is Ω(m(n/k) 1/(2α) ), even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k.On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1 + ε)-approximation algorithm with O(mn/kε 2 ) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing a lower bound of Ω(mn/k) query complexity.Our lower-bound results follow by carefully designing two distributions of instances that are hard to distinguish. In particular, our first lower bound involves a probabilistic construction of a certain set system with a minimum set cover of size αk, with the key property that a small number of "almost uniformly distributed" modifications can reduce the minimum set cover size down to k. Thus, these modifications are not detectable unless a large number of queries are asked. We believe that our probabilistic construction technique might find applications to lower bounds for other combinatorial optimization problems.