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.

Integer linear programming (ILP) models a wide range of practical combinatorial optimization problems and significantly impacts industry and management sectors. This work proposes new characterizations of ILP with the concept of boundary solutions. Motivated by the new characterizations, we develop a new local search algorithm Local-ILP, which is efficient for solving general ILP validated on a large heterogeneous problem dataset. We propose a new local search framework that switches between three modes, namely Search, Improve, and Restore modes. Two new operators are proposed, namely the tight move and the lift move operators, which are associated with appropriate scoring functions. Different modes apply different operators to realize different search strategies and the algorithm switches between three modes according to the current search state. Putting these together, we develop a local search ILP solver called Local-ILP. Experiments conducted on the MIPLIB dataset show the effectiveness of our algorithm in solving large-scale hard ILP problems. In the aspect of finding a good feasible solution quickly, Local-ILP is competitive and complementary to the state-of-the-art commercial solver Gurobi and significantly outperforms the state-of-the-art non-commercial solver SCIP. Moreover, our algorithm establishes new records for 6 MIPLIB open instances. The theoretical analysis of our algorithm is also presented, which shows our algorithm could avoid visiting unnecessary regions.

The Guaranteed Delivery (GD) advertising is a crucial component of the online advertising industry, and the allocation of inventory in GD advertising is an important procedure that influences directly the ability of the publisher to fulfill the requirements and increase its revenues. Nowadays, as the requirements of advertisers become more and more diverse and fine-grained, the focus ratio requirement, which states that the portion of allocated impressions of a designated contract on focus media among all possible media should be greater than another contract, often appears in business scenarios. However, taking these requirements into account brings hardness for the GD advertising inventory allocation as the focus ratio requirements involve non-convex multilinear constraints. Existing methods which rely on the convex properties are not suitable for processing this problem, while mathematical programming or constraint-based heuristic solvers are unable to produce high-quality solutions within the time limit. Therefore, we propose a local search framework to address this challenge. It incorporates four new operators designed for handling multilinear constraints and a two-mode algorithmic architecture. Experimental results demonstrate that our algorithm is able to compute high-quality allocations with better business metrics compared to the state-of-the-art mathematical programming or constraint based heuristic solvers. Moreover, our algorithm is able to handle the general multilinear constraints and we hope it could be used to solve other problems in GD advertising with similar requirements.

Abstract. Modular decomposition focuses on repeatedly identifying a module [Formula: see text] (a collection of vertices that shares exactly the same neighborhood outside of [Formula: see text]) and collapsing it into a single vertex. This notion of exactitude of neighborhood is very strict, especially when dealing with real-world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an [Formula: see text]- module, a relaxation that maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal [Formula: see text]-modules can be computed in polynomial time, and we generalize series and parallel operation between graphs. This leads to [Formula: see text]-cographs which have interesting properties. We study how to generalize Gallai’s theorem corresponding to the case for [Formula: see text], but unfortunately we give evidence that computing such a decomposition tree can be difficult.

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 lacks scalability for very dense patterns. Approximation algorithms (e.g., linear programming, semi-definite programming) and heuristics (e.g., Exact-Cover) are capable of handling large cases but can only get inferior solutions. In this paper, we propose a new exact algorithm that tackles layout decomposition by solving a series of boolean satisfiability instances. Our algorithm can preserve optimality and achieve more than 4× speedup compared to ILP. In addition, we provide an approximation algorithm by reformulating the layout decomposition to a bilevel optimization problem. Experiments show that our approximation algorithm can attain higher solution quality compared to SDP and heuristics within faster convergence.