Lecture 7 Approximation algorithms for MAXCUT and MAXQP
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where B = d 4 I− A 2 and Ei is a matrix whose i th diagonal entry is 1 and the others are 0. From now on let’s assume G is d-regular (i.e. each vertex has degree exactly d). Then the adjacency matrix A has at most d non-zero entries, each equal to 1/2, in every row, so ‖A‖ ≤ d/2 and 0 B d 2 I. Note that if α is the optimal value for the SDP we have |E| 2 = nd 4 ≤ α ≤ |E| = nd 2 . The first inequality follows since there is always a cut of size |E|/2 (a random cut will cut half the edges), and the second follows from the bound on the norm of B.
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