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

Assadi, Sepehr

doi:10.1145/3034786.3056116

Cited by 18 publications

(16 citation statements)

References 71 publications

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“…Let S big,i be a family of sets that contains those sets which have more than 4 · s 2 i many free elements at the beginning of stage i. Then, E [|S big,i |] ≤ n t·s 3 and Pr[|S big,i | > n s 2 ] ≤ 1 st .…”

Section: Approximation Analysis Of Algorithmmentioning

confidence: 99%

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Improved Local Computation Algorithm for Set Cover via Sparsification

Grunau¹,

Mitrović²,

Rubinfeld³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We design a Local Computation Algorithm (LCA) for the set cover problem. Given a set system where each set has size at most s and each element is contained in at most t sets, the algorithm reports whether a given set is in some fixed set cover whose expected size is O(log s) times the minimum fractional set cover value. Our algorithm requires s O(log s) t O(log s·(log log s+log log t)) queries. This result improves upon the application of the reduction of [Parnas and Ron, TCS'07] on the result of [Kuhn et al., SODA'06], which leads to a query complexity of (st) O(log s·log t) .To obtain this result, we design a parallel set cover algorithm that admits an efficient simulation in the LCA model by using a sparsification technique introduced in [Ghaffari and Uitto, SODA'19] for the maximal independent set problem. The parallel algorithm adds a random subset of the sets to the solution in a style similar to the PRAM algorithm of [Berger et al., FOCS'89]. However, our algorithm differs in the way that it never revokes its decisions, which results in a fewer number of adaptive rounds. This requires a novel approximation analysis which might be of independent interest. . This work was done when the author was a graduate student at MIT. √ log t) many queries. 1 T def = log(t) 2 for each pair of (i, k) where i ∈ [log s] and k ∈ [log t] do 3 B i (S) and S i,k are generated in the exact same way as in Algorithm 2 4 for i = 1 to log s do 5 for phase j = 0 to T − 1 do 6 for each set S ∈ S do 7 B i,j (S) ← B i (S) ∩ E i,j // Ei,j denotes the set of free elements at the beginning of the phase 8 if |B i,j (S)| ≥ log 20 s · log 20 t then // test whether S is a bad set 9 add S to S cover . 10 for each klog 20 s · log 20 t · 2 T then // test whether e is a bad element 14 pretend that e is covered. // e will be covered in Line 20. 15 for iteration ℓ = 1 to T do 16 for each set S ∈ S i,j·T +ℓ in parallel do 17 ifd B i (S) ≥ s/2 i then 18 add S to S cover . 19 for each free element in parallel e do // Handling bad elements 20

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Section: Approximation Analysis Of Algorithmmentioning

confidence: 99%

“…By Lemma 14, this only happens with a probability of O 1 (st) log 5 s·log 5 t . As we have at most t · n many sets, E [|S big,i |] ≤ n t·s 3 . Hence, a Markov bound implies that Pr[|S big,i | > n s 2 ] ≤ 1 st .…”

Section: Approximation Analysis Of Algorithmmentioning

confidence: 99%

Improved Local Computation Algorithm for Set Cover via Sparsification

Grunau¹,

Mitrović²,

Rubinfeld³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…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).…”

Section: A Further Related Workmentioning

confidence: 99%

Polynomial pass lower bounds for graph streaming algorithms

Assadi

Chen

Khanna

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a weighted minimum s-t cut in an n-vertex undirected graph requires n 2−o(1) space unless it makes n Ω(1) passes over the stream.To prove our lower bounds, we introduce and analyze a new four-player communication problem that we refer to as the hidden-pointer chasing problem. This is a problem in spirit of the standard pointer chasing problem with the key difference that the pointers in this problem are hidden to players and finding each one of them requires solving another communication problem, namely the set intersection problem. Our lower bounds for graph problems are then obtained by reductions from the hidden-pointer chasing problem.Our hidden-pointer chasing problem appears flexible enough to find other applications and is therefore interesting in its own right. To showcase this, we further present an interesting application of this problem beyond streaming algorithms. Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make n 2−o(1) value queries to the function unless it has a polynomial degree of adaptivity.A vast body of work in graph streaming lower bounds concerns algorithms that make only one or a few passes over the stream. Examples of single-pass lower bounds include the ones for diameter [60], approximate matchings [13,14,63,84], exact minimum/maximum cuts [119], and maximal independent sets [10,46]. Examples of multi-pass lower bounds include the ones for BFS trees [60], perfect matchings [67], shortest path [67], and minimum vertex cover and dominating set [71]. These lower bounds are almost always obtained by considering communication complexity of the problem with limited number of rounds of communication which gives a lower bound on the space complexity of streaming algorithms with proportional number of passes to the limits on rounds of communication (see e.g. [6,66]). The communication lower bounds are then typically proved via reductions from (variants of) the pointer chasing problem [38,105,106] for multi-pass lower bounds and the indexing problem [2,87] and boolean hidden (hyper-)matching problem [61,114] for single-pass lower bounds.In the pointer chasing problem, Alice and Bob are given functions f, g : [n] → [n] and the goal is to compute f (g(· · · f (g(0)))) for k iterations. Computing this function in less than k rounds requires Ω(n/k) communication [118] (see also [52,[105][106][107]). The reductions from pointer chasing to graph streaming lower bounds are based on using vertices of the graph to encode [n] and each edge to encode a pointer [60,67]. Directly using pointer chasing does not imply lower bounds stronger than Ω(n) and hence variants of pointer chasing with multiple pointers such as multi-valued pointer chasing [60,79] and set pointer chasing [67], were considered. Using mul...

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“…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.…”

Section: Related Workmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Set Cover in Sub-linear Time

Indyk¹,

Mahabadi²,

Rubinfeld³

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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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.

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

Cited by 18 publications

References 71 publications

Improved Local Computation Algorithm for Set Cover via Sparsification

Improved Local Computation Algorithm for Set Cover via Sparsification

Polynomial pass lower bounds for graph streaming algorithms

Set Cover in Sub-linear Time

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