Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

Tractable Optimization Problems through Hypergraph-Based Structural Restrictions

Several variants of the Constraint Satisfaction Problem have been proposed and investigated in the literature for modelling those scenarios where solutions are associated with some given costs. Within these frameworks computing an optimal solution is an NP-hard problem in general; yet, when restricted over classes of instances whose constraint interactions can be modelled via (nearly-)acyclic graphs, this problem is known to be solvable in polynomial time. In this paper, larger classes of tractable instances are singled out, by discussing solution approaches based on exploiting hypergraph acyclicity and, more generally, structural decomposition methods, such as (hyper)tree decompositions

Multilevel Solvers for Unstructured Surface Meshes

Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner

Algorithms for DFM in electronic design automation

As the dimension of features in integrated circuits (IC) keeps shrinking to fulfill Moore’s law, the manufacturing process has no choice but confronting the limit of physics at the expense of design flexibility. On the other hand, IC designs inevitably becomes more complex to meet the increasing demand of computational power. To close this gap, design for manufacturing (DFM) becomes the key to enable an easy and low-cost IC fabrication. Therefore, efficient electronic design automation (EDA) algorithms must be developed for DFM to address the design constraints and help the designers to better facilitate the manufacture process. As the core of manufacturing ICs, conventional lithography systems (193i) reach their limit for the 22 nm technology node and beyond. Consequently, several advanced lithography techniques are proposed, such as multiple patterning lithography (MPL), extreme ultra-violet lithography (EUV), electron beam (E-beam), and block copolymer directed self-assembly (DSA); however, DFM algorithms are essential for them to achieve better printability of a design. In this dissertation, we focus on analyzing the compatibility of designs and various advanced lithography techniques, and develop efficient algorithms to enable the manufacturing. We first explore E-Beam, one of the promising candidates for IC fabrication beyond the 10 nm technology node. To address its low throughput issue, the character projection technique has been proposed, and its stencil planning can be optimized with an awareness of overlapping characters. 2D stencil planning is proved NP-Hard. With the assumption of standard cells, the 2D problem can be partitioned into 1D row ordering subproblems; however, it is also considered hard, and no efficient optimal solution has been provided so far. We propose a polynomial time optimal algorithm to solve the 1D row ordering problem, which serves as the major subroutine for the entire stencil planning problem. Technical proofs and experimental results verify that our algorithm is efficient and indeed optimal. As the most popular and practical lithography technique, MPL utilizes multiple exposures to print a single layout and thus allows placement of features within the minimum distance. Therefore, a feasible decomposition of the layout is a must to adopt MPL, and it is usually formulated as a graph k-coloring problem, which is computationally difficult for k > 2. We study the k-colorability of rectangular and diagonal grid graphs as induced subgraphs of a rectangular or diagonal grid respectively, since it has direct applications in printing contact/via layouts. It remains an open question on how hard it is to color grid graphs due to their regularity and sparsity. In this dissertation, we conduct a complete analysis of the k-coloring problems on rectangular and diagonal grid graphs, and particularly the NP-completeness of 3-coloring on a diagonal grid graph is proved. In practice, we propose an exact 3-coloring algorithm for those graphs and conduct experiments to verify its performance and effectiveness. Besides, we also develop an efficient algorithm for model based MPL, because it is more expensive but accurate than the rule based decomposition. As one of the alternative lithography techniques, block copolymer directed self-assembly (DSA) is studied. It has emerged as a low-cost, high- throughput option in the pursuit of alternatives to traditional optical lithography. However, issues of defectivity have hampered DSA’s viability for large-scale patterning. Recent studies have shown the copolymer fill level to be a crucial factor in defectivity, as template overfill can result in malformed DSA structures and poor LCDU after etching. For this reason, the use of sub-DSA resolution assist features (SDRAFs) as a method of evening out template density has been demonstrated. In this dissertation, we propose an algorithm to place SDRAFs in random logic contact/via layouts. By adopting this SDRAF placement scheme, we can significantly improve the density unevenness and the resources used are also optimized. We also apply our knowledge in coloring grid graphs to the problem of group-and-coloring in DSA-MPL hybrid lithography. We derive a solution to group-3-coloring and prove the NP-completeness of grouping-2-coloring