This paper is joint work with Venkatesan Guruswami and Ameya Vellingker, to appear in APPROX 17.
We studied the complexity of estimating the optimum value of a Boolean 2CSP (arity two constraint satisfaction problem) in the single-pass streaming setting, where the algorithm is presented the constraints in an arbitrary order. We give a streaming algorithm to estimate the optimum within a factor approaching 2/5 using logarithmic space, with high probability. This beats the trivial factor 1/4 estimate obtained by simply outputting 1/4 th of the total number of constraints.
The inspiration for our work is a lower bound of Kapralov, Khanna, and Sudan (SODA '15) who showed that a similar trivial estimate (of factor 1/2) is the best one can do for Max CUT. This lower bound implies that beating a factor 1/2 for Max 2CSP, in particular, to distinguish between the case when the optimum is m/2 versus when it is at most (1/4+ \eps) m, where m is the total number of constraints, requires polynomial space. We complement this hardness result by showing that one can distinguish between the case in which the optimum exceeds (1/2 + \eps)m and the case in which it is close to m/4.
We also prove that estimating the size of the maximum acyclic subgraph of a graph, when its edges are presented in a single-pass stream, within a factor better than 7/8 requires polynomial space.
In this talk, I'll present the main results of the paper. The proofs are quite simple and would not require any previous knowledge