TR09-125 Authors: David Steurer, Nisheeth Vishnoi

Publication: 24th November 2009 18:08

Downloads: 1923

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In this paper we demonstrate a close connection between UniqueGames and

MultiCut.

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In MultiCut, one is given a ``network graph'' and a ``demand

graph'' on the same vertex set and the goal is to remove as few

edges from the network graph as possible such that every two

vertices connected by a demand edge are separated.

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On the other hand, UniqueGames is a certain family of constraint

satisfaction problems.

In one direction, we show that, at least with respect to current

algorithmic techniques, MultiCut is not harder than UniqueGames.

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Specifically, we can adapt most known algorithms for UniqueGames to

work for MultiCut and obtain new approximation guarantees for

MultiCut that depend on the maximum degree of the demand graph.

This degree plays the same role as the alphabet size plays in

approximation guarantees for UniqueGames.

In the other direction, we show that MultiCut is not easier than

UniqueGames ($\Gamma$-max-$2$-lin to be precise).

We exhibit a reduction from UniqueGames to MultiCut so that the fraction of

edges in the optimal multicut is up to a factor of $2$ equal to the

fraction of constraint violated by the optimal assignment for the

UniqueGames instance.

In contrast to the vast majority of Unique Games reductions whose

analysis relies on Fourier analysis and isoperimetric inequalities,

this reduction is simple and the analysis is elementary.

Further, the maximum degree of the demand graph in the instance

produced by the reduction is less than the size of the alphabet size

in the Unique Games instance.

Our results rely on a simple but previously unknown characterization

of multicut in terms of $\ell_1$ metrics.