Editing Open Problems:63
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{{Header | {{Header | ||
+ | |title=Submodular Matching Maximization | ||
|source=bertinoro14 | |source=bertinoro14 | ||
|who=Amit Chakrabarti | |who=Amit Chakrabarti | ||
}} | }} | ||
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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 $4+\epsilon$ {{cite|CrouchS-14}} and MSM (for any $f$) within $7.75$ {{cite|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 {{cite|Kapralov-12}}. | 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 $4+\epsilon$ {{cite|CrouchS-14}} and MSM (for any $f$) within $7.75$ {{cite|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 {{cite|Kapralov-12}}. | ||