Towards Tight Bounds for the Streaming Set Cover Problem

Abstract

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

Polynomial pass lower bounds for graph streaming algorithms

“…A closely related line of work to graph streaming algorithms that have received a significant attention in recent years is on streaming algorithms for submodular optimization and in particular set cover and maximum coverage [9,11,12,26,39,41,50,54,71,88,101,110]. Particularly relevant to our work, [41] uses a reduction from the multi-party tree pointer chasing problem [38] to prove an Ω( log n log log n ) pass lower bound for approximating set cover with m sets and n elements using O(n · poly {log n, log m}) space (this can also be interpreted as a lower bound for the edge-cover problem on hyper-graphs with n vertices and m hyper-edges in the graph streaming model).…”

Set Cover in Sub-linear Time

“…We present two algorithms with sub-linear number of queries. First, we show that the streaming algorithm presented in [17] can be adapted so that it returns an O(α)-approximate cover using O(m(n/k) 1/(α−1) + nk) queries, which could be quadratically smaller than mn. Second, we present a simple algorithm which is tailored to the case when the value of k is large.…”

“…The estimation variant of Vertex Cover under the adjacency-list oracle model has been studied in [30,23,29,33]. Set Cover has been also studied in the sub-linear space context, most notably for the streaming model of computation [32,12,7,3,2,5,18,10,17]. In this model, there are algorithms that compute approximate set covers with only multiplicative errors.…”

“…Set Cover is a well-studied problem with applications in operations research [16], information retrieval and data mining [32], learning theory [19], web host analysis [9], and many others. Recently, this problem and other related coverage problems have gained a lot of attention in the context of massive data sets, e.g., streaming model [32,12,10,17,7,3,24,2,5,18] or map reduce model [22,25,4].…”

Tight Space-Approximation Tradeoff for the Multi-Pass Streaming Set Cover Problem

“…Particularly relevant to our work, Demaine et al [23], have shown an α-approximation algorithm that uses O(α) passes over the stream and needs O(mn Θ(1/ log α) ) space. Recently, Har-Peled et al [32] provide a significant improvement over this algorithm: they developed an α-approximation, O(α)-pass streaming algorithm that requires O(mn Θ(1/α) ) space. They further conjectured that the tradeoff between the number of passes and the space in their algorithm is almost tight: this is supported by a lower bound of Ω(mn 1/2p ) space for p-pass streaming algorithms that compute an exact set cover solution [32].…”