Open Problems:29

Suggested by	Krzysztof Onak
Source	Kanpur 2009
Short link	https://sublinear.info/29

A large number of streaming papers consider graph problems. Typically, the input stream is an arbitrarily-ordered sequence of edges. For many problems, one can show that solving the problem, even approximately, requires $\Omega(n)$ bits of space. For instance, one can relatively easily prove that finding a constant-factor approximation to the maximum matching problem requires $\Omega(n)$ bits of space. Therefore, in many cases, the desired space complexity is of the form $\tilde O(n)$. Despite this relaxation, it is plausible that for some popular problems, there are barriers that cannot be overcome by (approximate) algorithms that use $n^{1+o(1)}$ space and a small number of passes.

For example, let $M(G)$ be the maximum matching size in the input graph $G$. McGregor [McGregor-05] shows that there is an algorithm that finds a matching of size $(1-\epsilon) \cdot M(G)$ in a number of passes that is a function of only $\epsilon$. It is plausible that for any constant $k$, there is no $k$-pass $\tilde O(n)$-space algorithm that finds a matching of size greater than $(1-\epsilon_k) \cdot M(G)$ times the optimum, where $\epsilon_k$ is a positive constant. In particular, to the best of my knowledge, no one-pass $\tilde O(n)$-space algorithm that finds a $(1-\epsilon)$-approximation for any constant $\epsilon \in (0,1/2)$ is known. Can one prove lower bounds as suggested above? The question generalizes to other problems. For instance, the best known $\tilde O(n)$-space algorithms for simulating random walks require a large number of passes (see [DasSarmaGP-08] and Rina Panigrahy's question). Can one prove for these problems that a small number of passes requires $n^{1+\Omega(1)}$ space?

To the best of my knowledge, the only problem for which this kind of lower bound is known is approximating graph distances. Feigenbaum et al. [FeigenbaumKMSZ-08] show that obtaining a $t$-approximation for the distance between two nodes in a single pass requires $\Omega(n^{1+1/t})$ space.