Problem 63: Submodular Matching Maximization

Suggested by	Amit Chakrabarti
Source	Bertinoro 2014
Short link	https://sublinear.info/63

Let $G = (V, E)$ be a graph. Fix a monotone submodular function $f : 2^E \rightarrow \mathbb{R}$. A maximum submodular matching $M$ is a subset of $E$ that forms a matching and maximizes $f(E)$. Suppose the graph edges are streaming. It is known that we cannot compute a maximum weight matching in one pass and $n \operatorname{polylog}(n)$ space to a better approximation than $\frac{e}{e-1}$. Can we show a stronger lower bound for maximum submodular matchings? A conjecture is that it will be hard to get a better than 2-approximation in one pass with the same space constraints.

A related question (due to Deeparnab Chakrabarty): Is there an instance-wise gap between MWMs and MSMs in the stream setting, for some choice of submodular $f$ and with the MWM instance being derived by evaluating $f$ at singleton sets?