# Problem 29: Strong Lower Bounds for Graph Problems

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, computing the BFS tree and computing the diameter are the only problems for which an $n^{1+\Omega(1)}$ lower bound for more than one pass is known [FeigenbaumKMSZ-08].

## Update[edit]

Guruswami and Onak [GuruswamiO-13] showed that the following problems require roughly $n^{1+\Omega(1/p)}$ bits of space in $p$ passes: testing if there is a perfect matching, checking if $v$ and $w$ are at distance at most $2(p+1)$, and checking if there is a directed path from $v$ to $w$, where $v$ and $w$ are known to the algorithm in advance.