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Electronic Colloquium on Computational Complexity

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TR00-021 | 19th April 2000 00:00

Improved Approximation of MAX-CUT on Graphs of Bounded Degree



We analyze the addition of a simple local improvement step to various known
randomized approximation algorithms.
Let $\alpha \simeq 0.87856$ denote the best approximation ratio currently
known for the Max Cut problem on general graphs~\cite{GW95}.
We consider a semidefinite relaxation of the Max Cut problem,
round it using the random hyperplane rounding technique of (\cite{GW95}),
and then add a local improvement step.
We show that for graphs of degree at most $\Delta$, our algorithm
achieves an approximation ratio of at least $\alpha + \epsilon$, where
$\epsilon > 0$ is a constant that depends only on $\Delta$.
In particular, using computer assisted analysis, we show that
for graphs of maximal degree $3$, our algorithm obtains an approximation
ratio of at least $0.921$, and for 3-regular graphs, the approximation ratio
is at least $0.924$. We note that for the semidefinite relaxation of Max Cut
used in~\cite{GW95}, the integrality gap is at least $1/0.884$, even for
2-regular graphs.

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