Vertex Ordering Problems in Directed Graph Streams

Chakrabarti, Amit; Ghosh, Prantar; McGregor, Andrew; Vorotnikova, Sofya

doi:10.1137/1.9781611975994.109

Cited by 14 publications

(27 citation statements)

References 21 publications

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“…This problem was studied in the streaming setting by [53,54] and it was shown in [54] that a reduction from BHH implies an Ω( √ n)-space lower bound for better-than-(7/8) approximation of this problem in single-pass streams. We note that some related problems in directed graphs have also been studied recently in [28].…”

Section: Maximum Acyclic Subgraphmentioning

confidence: 97%

“…Let p ≥ 1 be an integer and ε ∈ (0, 1) be a parameter. Any p-pass streaming algorithm that outputs a (1+ε)-approximation to the number of edges in a largest acyclic subgraph of a directed graph with probability at least 2/3 requires ε2 −O(p) •n 1−g(ε,p) space where g is the function in Eq (28).…”

Section: Maximum Acyclic Subgraphmentioning

confidence: 99%

“…Let p ≥ 1 be an integer and ε ∈ (0, 1) be a parameter. Any p-pass streaming algorithm that outputs a (1 + ε)-approximation to the weight of a minimum spanning tree in a graph with maximum weight W with probability at least 2/3 requires εW −1 2 −O(p) • n 1−g(ε/W,p) space where g is the function in Eq (28) (notice that the first argument of g is ε/W and not merely ε).…”

Section: Minimum Spanning Treementioning

confidence: 99%

“…Let p ≥ 1 be an integer and ε ∈ (0, 1) be a parameter. Any p-pass streaming algorithm for ε-property testing of connectivity, bipartiteness, or cycle-freeness in planar graphs, with probability at least 2/3 requires ε • 2 −O(p) • n 1−g(ε,p) space where g is the function in Eq (28).…”

Section: Property Testing Of Connectivity Bipartiteness and Cycle-fre...mentioning

confidence: 99%

“…Graph streaming algorithms process graphs presented as a sequence of edges under the usual constraints of the streaming model, i.e., by making one or a few passes over the input and using a limited memory. There are two main area of research on graph streams: (i) the semi-streaming algorithms that use O(n • polylog(n)) space for n-vertex graphs and target problems on (dense) graphs such as finding MST [42,73], large matchings [3,6,10,45,49,60,72], spanners and shortest paths [15,16,28,39,40,43,52], sparsifiers and minimum cuts [2,4,65,66,78], maximal independent sets [8,33,48], graph coloring [8,19], and the like; and (ii) the o(n)-space streaming algorithms that use polylog(n) space and aim to estimate properties of (sparse) graphs such as max-cut value [21,[62][63][64]67], maximum matching size [9,31,35,41,61,74,75], number of connected components [55], subgraph counting [14,18,24,25,34,<...>…”

Section: Introductionmentioning

confidence: 99%

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Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems

Assadi

Kol

Saxena

et al. 2020

Preprint

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Consider the following gap cycle counting problem in the streaming model: The edges of a 2-regular n-vertex graph G are arriving one-by-one in a stream and we are promised that G is a disjoint union of either k-cycles or 2k-cycles for some small k; the goal is to distinguish between these two cases using a limited memory. Verbin and Yu [SODA 2011] introduced this problem and showed that any single-pass streaming algorithm solving it requires n 1−Ω( 1 /k) space. This result and the proof technique behind it-the Boolean Hidden Hypermatching communication problem-has since been used extensively for proving streaming lower bounds for various problems, including approximating MAX-CUT, matching size, property testing, matrix rank and Schatten norms, streaming unique games and CSPs, and many others.Despite its significance and broad range of applications, the lower bound technique of Verbin and Yu comes with a key weakness that is also inherited by all subsequent results: the Boolean Hidden Hypermatching problem is hard only if there is exactly one round of communication and, in fact, can be solved with logarithmic communication in two rounds. Therefore, all streaming lower bounds derived from this problem only hold for single-pass algorithms. Our goal in this paper is to remedy this state-of-affairs.We prove the first multi-pass lower bound for the gap cycle counting problem: Any p-pass streaming algorithm that can distinguish between disjoint union of k-cycles vs 2k-cycles-or even k-cycles vs one Hamiltonian cycle-requires n 1− 1 /k Ω(1/p) space. This makes progress on multiple open questions in this line of research dating back to the work of Verbin and Yu.

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Section: Maximum Acyclic Subgraphmentioning

confidence: 97%

Section: Maximum Acyclic Subgraphmentioning

confidence: 99%

Section: Minimum Spanning Treementioning

confidence: 99%

Section: Property Testing Of Connectivity Bipartiteness and Cycle-fre...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems

Assadi

Kol

Saxena

et al. 2020

Preprint

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Streaming approximation resistance of every ordering CSP

Singer,

Sudan,

Velusamy

2024

comput. complex.

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An ordering constraint satisfaction problem (OCSP) is defined by a family $$\mathcal F$$ F of predicates mapping permutations on $$\{1,\ldots,k\}$$ { 1 , … , k } to $$\{0,1\}$$ { 0 , 1 } . An instance of ($$\mathcal F$$ F ) on n variables consists of a list of constraints, each consisting of a predicate from $$\mathcal F$$ F applied on k distinct variables. The goal is to find an ordering of the n variables that maximizes the number of constraints for which the induced ordering on the k variables satisfies the predicate. OCSPs capture well-studied problems including ‘maximum acyclic subgraph’ () and “maximum betweenness”. In this work, we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, when an instance is presented as a stream of constraints. We show that for every $$\mathcal F$$ F , ($$\mathcal F$$ F ) is approximation-resistant to o(n)-space streaming algorithms, i.e., algorithms using o(n) space cannot distinguish streams where almost every constraint is satisfiable from streams where no ordering beats the random ordering by a noticeable amount. This space bound is tight up to polylogarithmic factors. In the case of , our result shows that for every $$\epsilon>0$$ ϵ > 0 , is not $$(1/2+\epsilon)$$ ( 1 / 2 + ϵ ) -approximable in o(n) space. The previous best inapproximability result, due to Guruswami & Tao (2019), only ruled out 3/4-approximations in $$o(\sqrt n)$$ o ( n ) space. Our results build on recent works of Chou et al. (2022b, 2024) who provide a tight, linear-space inapproximability theorem for a broad class of “standard” (i.e., non-ordering) constraint satisfaction problems (CSPs) over arbitrary (finite) alphabets. Our results are obtained by building a family of appropriate standard CSPs (one for every alphabet size q) from any given OCSP and applying their theorem to this family of CSPs. To convert the resulting hardness results for standard CSPs back to our OCSP, we show that the hard instances from this earlier theorem have the following “partition expansion” property with high probability: For every partition of the n variables into small blocks, for most of the constraints, all variables are in distinct blocks.

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Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms

Assadi

Raz

2020

2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)

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We give a near-optimal sample-pass trade-off for pure exploration in multi-armed bandits (MABs) via multi-pass streaming algorithms: any streaming algorithm with sublinear memory that uses the optimal sample complexity of O(n/∆ 2 ) requires Ω(log (1/∆)/ log log (1/∆)) passes. Here, n is the number of arms and ∆ is the reward gap between the best and the second-best arms. Our result matches the O(log(1/∆)) pass algorithm of Jin et al. [ICML'21] (up to lower order terms) that only uses O(1) memory and answers an open question posed by Assadi and Wang [STOC'20].

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Vertex Ordering Problems in Directed Graph Streams

Cited by 14 publications

References 21 publications

Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems

Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems

Streaming approximation resistance of every ordering CSP

Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms

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