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| {{Header | | {{Header |

| + | |title=Title of the problem |

| |source=baltimore16 | | |source=baltimore16 |

| |who=Robert Krauthgamer | | |who=Robert Krauthgamer |

| }} | | }} |

− | ~~Suppose we want to design a data structure that stores, for a given edge-weighted (undirected) graph $G = (V,E_G,w_G)$, the values of the minimum $st$-cuts for all $s,t \in V$. ~~
| + | The open problem will appear here. |

− | ~~A naive method is to construct a table containing the value for each pair, requiring $O(|V|^2)$ space (machine words).~~
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− | ~~Alternatively, one may construct a Gomory–Hu tree {{cite|GomoryH-61}}.~~
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− | ~~This is a tree $T = (V,E_T,w_T)$, in which the minimum $st$-cut values are equal to those in $G$.~~
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− | ~~Since $T$ is a tree, it requires only $O(|V|)$ space.~~
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− | ~~Thus for this ~~problem~~, a very space-efficient data structure (perhaps even the best one) is itself a graph $G'$, and it encodes the desired values in a natural manner, just compute the same function (min $st$-cut) on $G'$. ~~
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− | ~~But is this the case for all such functions on graphs, or is there a (natural) case where a potentially complicated data structure outperforms a graphical encoding? The question applies both to exact and approximate computations of the function.~~
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− | ~~'''Some relevant examples:'''~~
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− | ~~* A randomized data structure for $(1+\varepsilon)$-approximating any cut value (or Rayleigh quotient) in $G$, which is more space-efficient than the known graphical representation, can be found in {{cite|AndoniCKQWZ-16}}.~~
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− | ~~* For a given graph $G$ with $k$ terminals, the exact values of all the minimum cuts between subsets of terminals can be stored simply as a list of $2^k$ numbers. The best graphical representation known is of size roughly $2^{2^k}$ {{cite|HagerupKNR-98|KhanR-14}}. ~~
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− | ~~* A deterministic data structure for $(1+\varepsilon)$-approximating any multicommodity flow on a set of $k$ given terminals in $G$. There is no known graphical representation for this, whose size depends on $k$ and $\varepsilon$, but not on $G$ {{cite|AndoniGK-14}}~~.
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