Positive semidefinite relaxation and approximation algorithm for triple patterning lithography

Abstract

Triple patterning lithography (TPL) is one of the major techniques for 14 nm technology node and beyond. This paper discusses TPL layout decomposition which maximizes objective value representing decomposition quality. We introduce a maximization problem of the weighted sum of resolved conflicts and unused stitch candidates. We propose a polynomial time (7/9)-approximation algorithm based on positive semidefinite relaxation and randomized rounding procedure. Our algorithm returns a decomposition such that the expectation of the corresponding objective value is at least (7/9) times the optimal value even in the worst case problem instance. To our knowledge, the result is the first approximation algorithm with a constant approximation ratio for TPL.