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

Abstract

We continue the study of the communication complexity of gap cycle counting problems. These problems have been introduced by Verbin and Yu [SODA 2011] and have found numerous applications in proving streaming lower bounds. In the noisy gap cycle counting problem (NGC), there is a small integer k 1 and an n-vertex graph consisted of vertex-disjoint union of either k-cycles or 2k-cycles, plus O(n/k) disjoint paths of length k − 1 in both cases ("noise"). The edges of this graph are partitioned between Alice and Bob whose goal is to decide which case the graph belongs to with minimal communication from Alice to Bob.We study the robust communication complexity-à la Chakrabarti, Cormode, and McGregor [STOC 2008]-of NGC, namely, when edges are partitioned randomly between the players. This is in contrast to all prior work on gap cycle counting problems in adversarial partitions. While NGC can be solved trivially with zero communication when k < log n, we prove that when k is a constant factor larger than log n, the robust (one-way) communication complexity of NGC is Ω(n) bits.As a corollary of this result, we can prove several new graph streaming lower bounds for random order streams. In particular, we show that any streaming algorithm that for every ǫ > 0 estimates the number of connected components of a graph presented in a random order stream to within an ǫ • n additive factor requires 2 Ω(1/ǫ) space, settling a conjecture of Peng and Sohler [SODA 2018]. We further discuss new implications of our lower bounds to other problems such as estimating size of maximum matchings and independent sets on planar graphs, random walks, as well as to stochastic streams.

Graph Streaming Lower Bounds for Parameter Estimation and Property Testing via a Streaming XOR Lemma

“…We present a host of new multi-pass streaming lower bounds that in multiple cases such as property testing of connectivity, achieve optimal lower bounds on the space-pass tradeoffs for the given problems. At the core of our results, similar to [VY11,AKSY20], is a new lower bound for a "gap cycle counting" problem, wherein the goal is to distinguish between graphs consisting of only "short" cycles or only "long" cycles. Our other streaming lower bounds then follow by easy reductions from this problem.…”

“…How well can we perform property testing on G, say, decide whether it is connected or cycle-free versus being far from having these properties? These questions are highly motivated by the growing need in processing massive graphs and have witnessed a flurry of results in recent years: see, e.g., [KK15, KKS15, KKSV17, BDV18, KK19] on maximum cut, [AKL17, EHL + 15, CCE + 16, MV16, MV18, CJMM17, KKS14] on maximum matching size, [BKS02, BOV13, CJ17, MVV16, BC17, BFKP16, KMPV19] on subgraph counting, [GVV17, GT19, CGV20] on CSPs, and [HP16, MMPS17, PS18, CFPS20] on property testing, among others (see also [VY11,CFPS20,AKSY20] for a more detailed discussion of this line of work).…”

“…But until very recently, it was even open if one could solve this problem in O(log n) space and two passes. This question was partially addressed by the first author, Kol, Saxena, and Yu [AKSY20] who proved that any algorithm for this problem requires n Ω(1) space or Ω(log (1/ε)) passes. But this is still far from the only known upper bound of polylog(n) space and O(1/ε) passes obtained via a streaming implementation of the algorithm of [CRT05] (see [PS18]).…”

Noisy Boolean Hidden Matching with Applications

“…Several communication problems inspired by the Boolean Hidden (Hyper)Matching problem have recently been used in the literature to prove tight lower bounds for the single pass or sketching complexity of several graph problems (e.g., [KKSV17,KK19] for the MAX-CUT problem, [KKP18] for subgraph counting, in [GVV17, GT19, CGV20, CGSV21] for general CSPs). The recent work of [AKSY20] gives multipass streaming lower bounds for the space complexity of the aforementioned one-or-many cycles communication problem, which is tightly connected to BHH, extending many of the abovementioned single pass lower bounds to the multipass setting. Although (to the best of our knowledge) property testing for graph isomorphism on streams have not been previously studied, there is an active line of work, e.g.…”

“…Instead, we use our p-Noisy Boolean Hidden Matching communication problem to show a fine-grained lower bound for the maximum acyclic subgraph problem with tradeoffs between approximation guarantee and space. Independently, [AKSY20] showed a lower bound that (1 − ǫ)-approximation requires Ω(n 1−O(ǫ c ) ) space for a fixed constant c > 0 through a reduction from their one-or-many cycles communication problem.…”