Problem 74: Succinct Representation for Functions on Graphs

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Suggested by Robert Krauthgamer
Source Baltimore 2016
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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 $s,t \in V$. A naive method is to construct a table containing the value for each pair, requiring $O(|V|^2)$ space (machine words). Alternatively, one may construct a Gomory–Hu tree [GomoryH-61]. This is a tree $T = (V,E_T,w_T)$, in which the minimum $st$-cut values are equal to those in $G$. Since $T$ is a tree, it requires only $O(|V|)$ space.

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 $G'$. 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.

Some relevant examples:

  • 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 [AndoniCKQWZ-16].
  • 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 $2^{2^k}$ [HagerupKNR-98,KhanR-14].
  • 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 $G$ [AndoniGK-14].