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A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs
Kolla and Tulsiani [KT07,Kolla11} and Arora, Barak and Steurer [ABS10]
introduced the technique of subspace enumeration, which gives approximation
algorithms for graph problems such as unique games and small set expansion; the
running time of such algorithms is exponential in the threshold-rank of the
graph.
Guruswami and Sinop [GS11,GS12], and Barak, Raghavendra, and Steurer [BRS11]
developed an alternative approach to the design of approximation algorithms for
graphs of bounded threshold-rank, based on semidefinite programming relaxations
in the Lassere hierarchy and on novel rounding techniques. These algorithms are
faster than the ones based on subspace enumeration and work on a broad class of
problems.
In this paper we develop a third approach to the design of such algorithms.
We show, constructively, that graphs of bounded threshold-rank satisfy a weak
Szemeredi regularity lemma analogous to the one proved by Frieze and Kannan
[FK99] for dense graphs. The existence of efficient approximation algorithms is
then a consequence of the regularity lemma, as shown by Frieze and Kannan.
Applying our method to the Max Cut problem, we devise an algorithm that is
faster than all previous algorithms, and is easier to describe and analyze
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