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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.

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We provide the first separation in the approximation guarantee achievable by truthful and non-truthful combinatorial auctions with polynomial communication. Specifically, we prove that any truthful mechanism guaranteeing a (3 /4 − 1 /240 + ε)approximation for two buyers with XOS valuations over m items requires exp(Ω(ε 2 •m)) communication, whereas a non-truthful algorithm by Dobzinski and Schapira [SODA 2006] and Feige [2009] is already known to achieve a 3 /4-approximation in poly(m) communication.

We study the menu complexity of optimal and approximately-optimal auctions in the context of the "FedEx" problem, a so-called "one-and-a-halfdimensional" setting where a single bidder has both a value and a deadline for receiving an item [FGKK16]. The menu complexity of an auction is equal to the number of distinct (allocation, price) pairs that a bidder might receive [HN13]. We show the following when the bidder has n possible deadlines:• Exponential menu complexity is necessary to be exactly optimal: There exist instances where the optimal mechanism has menu complexity ≥ 2 n − 1. This matches exactly the upper bound provided by • Fully polynomial menu complexity is necessary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative (1 − )-approximation to the optimal revenue with menu complexity O(n 3/2 min{n/ ,ln(v max )} ) = O(n 2 / ), where v max denotes the largest value in the support of integral distributions.• There exist instances where any mechanism guaranteeing a multiplicative (1 − O(1/n 2 ))-approximation to the optimal revenue requires menu complexity Ω(n 2 ).Our main technique is the polygon approximation of concave functions [Rot92], and our results here should be of independent interest. We further show how our techniques can be used to resolve an open question of [DW17] on the menu complexity of optimal auctions for a budget-constrained buyer.