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This question is motivated by the Leighton-Rao relaxation for SPARSEST-CUT.

Suppose one wants to find a non-trivial semimetric over an $n$-point space that minimizes a certain linear functional. More formally:

  • minimize $\sum_{ij = 1}^n c_{ij} d_{ij}$
  • where $d$ is a semimetric over $\{1, 2, \ldots, n\}$
  • and $\sum_{ij = 1}^n d_{ij} = 1$

Of course it is a linear program that can be solved in polynomial time using ellipsoid method. But what I would like to know is if there are any combinatorial algorithms for this optimization problem. I suspect that the answer is 'no' since the problem looks pretty general to be solved combinatorially, but who knows...

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Yes: the dual is a multicommodity flow problem which can be solved using the Garg-Koenemann algorithm http://dx.doi.org/10.1137/S0097539704446232.

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    $\begingroup$ I can not download the article, but what I don't like is a word "approximate" in the abstract. $\endgroup$ – ilyaraz Jul 12 '11 at 6:23
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    $\begingroup$ Google for the title: it's in citeseer. All combinatorial algorithms of this style give a $1 +\epsilon$ approximation in time that depends on $\epsilon$ as $\frac{1}{\epsilon^2}$. The usual argument is that if your goal is to compute the $O(\log n)$ approximation, then the $1+\epsilon$ factor is acceptable in comparison. I believe if you want $\log \frac{1}{\epsilon}$ dependence, then your only choice is a "real" LP algorithm. BTW, your LP can be written with $O(n^3)$ constraints, so you can use interior point methods. $\endgroup$ – Sasho Nikolov Jul 12 '11 at 7:38

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