Layout Decomposition via Boolean Satisfiability

Abstract

Multiple patterning lithography (MPL) has been introduced in the integrated circuits manufacturing industry to enhance feature density as the technology node advances. A crucial step of MPL is assigning layout features to different masks, namely layout decomposition. Exact algorithms like integer linear programming (ILP) can solve layout decomposition to optimality but lack scalability for dense patterns. Relaxation algorithms (e.g., linear programming and semi-definite programming) and heuristics (e.g., exact cover) are capable of handling large cases at the cost of inferior solution quality. These methods rely on different mathematical solvers and expert-designed heuristics to offer a balance between solution quality and computational efficiency. In this article, we propose a unified layout decomposition framework comprising three algorithms: 1) satisfiability (SAT)-exact; 2) SAT-bilevel; and 3) SAT-fast, all leveraging the capabilities of Boolean SAT solvers. The SAT-exact ensures optimality, but with faster convergence than ILP, SAT-bilevel addresses the decomposition as a bilevel optimization problem for rapid near-optimal solutions, and SAT-fast handles very large layouts in an incremental manner. Experimental results demonstrate our framework’s superiority over existing state-of-the-art methods in terms of solution quality and runtime.