An efficient sparse regularity concept

Let A be a 0/1 matrix of size m × n, and let p be the density of A (i.e., the number of ones divided by m · n). We show that A can be approximated in the cut norm within ε ·mnp by a sum of cut matrices (of rank 1), where the number of summands is independent of the sizem ·n of A, provided that A satisfies a certain boundedness condition. This decomposition can be computed in polynomial time. This result extends the work of Frieze and Kannan [16] to sparse matrices. As an application, we obtain efficient 1 − ε approximation algorithms for “bounded” instances of MAX CSP problems.