Problem 63: Submodular Matching Maximization

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Suggested by Amit Chakrabarti
Source Bertinoro 2014
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Let $G = (V, E)$ be a graph. Fix a monotone submodular function $f : 2^E \rightarrow \mathbb{R}$. A matching $M \subseteq E$ is called a maximum submodular matching (MSM) with respect to $f$ if it maximizes $f(E)$. This generalizes maximum weight matching (MWM). Suppose the graph edges are streaming and we are allowed only one pass. It is known that using $O(n\log n)$ space we can approximate MWM within a factor of $4+\epsilon$ [CrouchS-14] and MSM (for any $f$) within $7.75$ [ChakrabartiK-14]. It is also known that we cannot approximate MWM to a factor better than $\frac{e}{e-1}$ using $n \operatorname{polylog}(n)$ space [Kapralov-12].

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?