## Graph Streaming Lower Bounds for Parameter Estimation and Property Testing

via a Streaming XOR Lemma

Authors:
Sepehr Assadi, Vishvajeet N.

Abstract:
We study space-pass tradeoffs in graph streaming algorithms for parameter estimation and property testing problems such as estimating the size
of maximum matchings and maximum cuts, weight of minimum spanning trees, or testing if a graph is connected or cycle-free versus being far from
these properties. We develop a new lower bound technique that proves that for many problems of interest, including all the above,
obtaining a (1 + ε)-approximation requires either near-linear space or Ω(1/ε) passes, even on highly restricted families of graphs such as
bounded-degree planar graphs. For multiple of these problems, this bound matches those of existing algorithms and is thus (asymptotically) optimal.

Our results considerably strengthen prior lower bounds even for arbitrary graphs: starting from the influential work of [Verbin, Yu; SODA 2011], there has been a plethora of lower bounds for single-pass algorithms for these problems; however, the only multi-pass lower bounds proven very recently in [Assadi, Kol, Saxena, Yu; FOCS 2020] rules out sublinear-space algorithms with exponentially smaller o(log (1/ε)) passes for these problems.

One key ingredient of our proofs is a simple streaming XOR Lemma, a generic hardness amplification result, that we prove: informally speaking, if a p-pass s-space streaming algorithm can only solve a decision problem with advantage δ > 0 over random guessing, then it cannot solve XOR of L independent copies of the problem with advantage much better than δ^L. This result can be of independent interest and useful for other streaming lower bounds as well.

Our results considerably strengthen prior lower bounds even for arbitrary graphs: starting from the influential work of [Verbin, Yu; SODA 2011], there has been a plethora of lower bounds for single-pass algorithms for these problems; however, the only multi-pass lower bounds proven very recently in [Assadi, Kol, Saxena, Yu; FOCS 2020] rules out sublinear-space algorithms with exponentially smaller o(log (1/ε)) passes for these problems.

One key ingredient of our proofs is a simple streaming XOR Lemma, a generic hardness amplification result, that we prove: informally speaking, if a p-pass s-space streaming algorithm can only solve a decision problem with advantage δ > 0 over random guessing, then it cannot solve XOR of L independent copies of the problem with advantage much better than δ^L. This result can be of independent interest and useful for other streaming lower bounds as well.