Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem

Assadi, Sepehr; Khanna, Sanjeev

doi:10.1137/1.9781611975031.155

Cited by 6 publications

(22 citation statements)

References 59 publications

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“…(1) ∆ TV (Π [10,10] , Π [01, 01] ) = o(ε), (2) ∆ TV (Π [10,10] , Π [11,01] ) = o(ε), (3) ∆ TV (Π [10,10] , Π [01, 11] ) = o(ε), (4) ∆ TV (Π [11,10] , Π [01, 01] ) = o(ε), (5) ∆ TV (Π [11,10] , Π [11,01] ) = o(ε), (6) ∆ TV (Π [11,10] , Π [01, 11] ) = o(ε), (7) ∆ TV (Π [10,11] , Π [01, 01] ) = o(ε), (8) ∆ TV (Π [10,11] , Π [11,01] ) = o(ε), (9) ∆ TV (Π [10,11] , Π [01,11]…”

Section: (By Part (1) and (2) Of The Equation Above)mentioning

confidence: 99%

“…Recall that Π [x 1 x 2 , y 1 y 2 ] denotes the transcript of the protocol conditioned on the input (x 1 , x 2 ) to Alice, and (y 1 , y 2 ) to Bob. (1) h 2 (Π [11,10] , Π [10,10]…”

mentioning

confidence: 99%

“…Define Π [1 * , 10] as the distribution of Π conditioned on the given value for x 1 , y 1 , y 2 (leaving out the assignment for x 2 ). We have, h 2 (Π [10,10] , Π [11,10] ).…”

mentioning

confidence: 99%

“…The distribution of X 2 conditioned on Y 2 = 0, K = 1 in D PI is uniform over {0, 1} (hence the second equality). As such, Π [1 * , 10] = 1 2 · Π [10,10] + Π [11,10] and so we can apply Fact B.10 to obtain the last inequality. As I(X 2 ; Π PI | Y 2 = 0, K = 1) = o(ε 2 ) by Claim 4.10, we obtain the first equation (note that h is symmetric).…”

mentioning

confidence: 99%

“…h 2 (Π [10,10] , Π [01, 01] ) ≤ h(Π [10,10] , Π [11,11] ) + h(Π [11,11] , Π [01, 01] ) 2 (by triangle inequality) ≤ 2 · h 2 (Π [10,10] , Π [11,11] ) + h 2 (Π [11,11] , Π [01, 01] ) (by Cauchy-Schwartz)…”

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confidence: 99%

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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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Section: (By Part (1) and (2) Of The Equation Above)mentioning

confidence: 99%

“…Recall that Π [x 1 x 2 , y 1 y 2 ] denotes the transcript of the protocol conditioned on the input (x 1 , x 2 ) to Alice, and (y 1 , y 2 ) to Bob. (1) h 2 (Π [11,10] , Π [10,10]…”

mentioning

confidence: 99%

“…Define Π [1 * , 10] as the distribution of Π conditioned on the given value for x 1 , y 1 , y 2 (leaving out the assignment for x 2 ). We have, h 2 (Π [10,10] , Π [11,10] ).…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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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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Equivalence classes and conditional hardness in massively parallel computations

Nanongkai

Scquizzato

2022

Distrib. Comput.

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The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical graph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle versus two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., $$\textsf {P}\ne \textsf {NP}$$ P ≠ NP ), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by $$\textsf {MPC}(o(\log N))$$ MPC ( o ( log N ) ) , and the standard space complexity classes $$\textsf {L}$$ L and $$\textsf {NL}$$ NL , and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model. Specifically, our main results are as follows. Lower bounds conditioned on the one cycle versus two cycles conjecture can be instead argued under the $$\textsf {L}\nsubseteq \textsf {MPC}(o(\log N))$$ L ⊈ MPC ( o ( log N ) ) conjecture: these two assumptions are equivalent, and refuting either of them would lead to $$o(\log N)$$ o ( log N ) -round MPC algorithms for a large number of challenging problems, including list ranking, minimum cut, and planarity testing. In fact, we show that these problems and many others require asymptotically the same number of rounds as the seemingly much easier problem of distinguishing between a graph being one cycle or two cycles. Many lower bounds previously argued under the one cycle versus two cycles conjecture can be argued under an even more robust (thus harder to refute) conjecture, namely $$\textsf {NL}\nsubseteq \textsf {MPC}(o(\log N))$$ NL ⊈ MPC ( o ( log N ) ) . Refuting this conjecture would lead to $$o(\log N)$$ o ( log N ) -round MPC algorithms for an even larger set of problems, including all-pairs shortest paths, betweenness centrality, and all aforementioned ones. Lower bounds under this conjecture hold for problems such as perfect matching and network flow.

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Stochastic Submodular Cover with Limited Adaptivity

Agarwal¹,

Assadi²,

Khanna³

2019

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

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In the submodular cover problem, we are given a non-negative monotone submodular function f over a ground set E of items, and the goal is to choose a smallest subset S ⊆ E such that f (S) = Q where Q = f (E). In the stochastic version of the problem, we are given m stochastic items which are different random variables that independently realize to some item in E, and the goal is to find a smallest set of stochastic items whose realization R satisfies f (R) = Q. The problem captures as a special case the stochastic set cover problem and more generally, stochastic covering integer programs.A fully adaptive algorithm for stochastic submodular cover chooses an item to realize and based on its realization, decides which item to realize next. A non-adaptive algorithm on the other hand needs to choose a permutation of items beforehand and realize them one by one in the order specified by this permutation until the function value reaches Q. The cost of the algorithm in both case is the number (or costs) of items realized by the algorithm. It is not difficult to show that even for the coverage function there exist instances where the expected cost of a fully adaptive algorithm and a non-adaptive algorithm are separated by Ω(Q). This strong separation, often referred to as the adaptivity gap, is in sharp contrast to the separations observed in the framework of stochastic packing problems where the performance gap for many natural problem is close to the poly-time approximability of the non-stochastic version of the problem. Motivated by this striking gap between the power of adaptive and non-adaptive algorithms, we consider the following question in this work: does one need full power of adaptivity to obtain a near-optimal solution to stochastic submodular cover? In particular, how does the performance guarantees change when an algorithm interpolates between these two extremes using a few rounds of adaptivity.Towards this end, we define an r-round adaptive algorithm to be an algorithm that chooses a permutation of all available items in each round k ∈ [r], and a threshold τ k , and realizes items in the order specified by the permutation until the function value is at least τ k . The permutation for each round k is chosen adaptively based on the realization in the previous rounds, but the ordering inside each round remains fixed regardless of the realizations seen inside the round. Our main result is that for any integer r, there exists a poly-time r-round adaptive algorithm for stochastic submodular cover whose expected cost isÕ(Q 1/r ) times the expected cost of a fully adaptive algorithm. Prior to our work, such a result was not known even for the case of r = 1 and when f is the coverage function. On the other hand, we show that for any r, there exist instances of the stochastic submodular cover problem where no r-round adaptive algorithm can achieve better than Ω(Q 1/r ) approximation to the expected cost of a fully adaptive algorithm. Our lower bound result holds even for coverage function and for algorithms with unbo...

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Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem

Cited by 6 publications

References 59 publications

Polynomial pass lower bounds for graph streaming algorithms

Polynomial pass lower bounds for graph streaming algorithms

Equivalence classes and conditional hardness in massively parallel computations

Stochastic Submodular Cover with Limited Adaptivity

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