Center for Discrete Mathematics and Theoretical Computer Science

About: Center for Discrete Mathematics and Theoretical Computer Science is a facility organization based out in Piscataway, New Jersey, United States. It is known for research contribution in the topics: Local search (optimization) & Optimization problem. The organization has 140 authors who have published 175 publications receiving 2345 citations.

Discrete Relaxation Method for Triple Patterning Lithography Layout Decomposition

Abstract: In this paper, we consider the triple patterning lithography layout decomposition problem. To address the problem, a discrete relaxation theory is built. For designing a discrete relaxation based decomposition framework, we propose a surface projection method for identifying native conflicts in a layout, and then constructing the conflict graph. Guided by the theory, the conflict graph is reduced to small size subgraphs by vertex removals, which is a discrete relaxation. Furthermore, by ignoring stitch insertions and assigning weights to features, the layout decomposition problem on the small subgraphs is further relaxed to a 0-1 program, which is solved by the Branch-and-Bound method. To obtain a feasible solution of the original problem, legalization methods are introduced to legalize a relaxation solution. At the legalization stage, we prior utilize one-stitch insertion to eliminate conflicts, and use a backtrack coloring algorithm to obtain a better solution. We test our decomposition approach on the ISCAS-85 & 89 benchmarks. Comparisons of experimental results show that our approach finds solutions of some benchmarks better than those by the state-of-the-art decomposers. Especially, according to our discrete relaxation theory, some optimal decompositions are obtained.

An Efficient Memetic Algorithm for theMax-Bisection Problem

Abstract: The max-bisection problem consists in partitioning the vertices of a weighted undirected graph into two equally sized subsets so as to maximize the sum of the weights of crossing edges. It is an NP-hard combinatorial optimization problem that arises in many applications. In this paper, we present a memetic algorithm for the max-bisection problem, which integrates a new fast local search procedure, a crossover operator, and a pool updating strategy. These strategies achieve a balance between intensification and diversification. Extensive experiments were performed on a number of benchmark instances with 800 to 10,000 vertices from the literature. The proposed memetic algorithm improved the best known solutions for all benchmark instances tested in this paper. The improvement in terms of cut value over the CirCut by Burer et al. ranging from 0.02 to 4.15 percent, and the average time of our proposed memetic algorithm is much lower than that of CirCut. It shows that the proposed memetic algorithm can find high quality solutions in an acceptable running time.

Max-k-Cut by the Discrete Dynamic Convexized Method

Abstract: In this paper, we propose a “multistart-type” algorithm for solving the max-k-cut problem. Central to our algorithm is an auxiliary function we propose. We formulate the max-k-cut problem as an explicit mathematical form, which allows us to use an easy implementable local search. The construction of the auxiliary function requires a local maximizer of the max-k-cut problem. If the best local maximizer obtained is used in the construction of the auxiliary function, then the local maximization of the auxiliary function leads to a better maximizer of the max-k-cut problem. This proves to be a good strategy to escape from the current local optima and to search a broader solution space. Indeed, we have shown, both numerically and theoretically, that the maximization of the auxiliary function by the local search method can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. Computational results on many test instances with different sizes and densities show that the proposed algorithm is efficient and stable to find approximate global solutions for the max-k-cut problems. Although we have presented results for k ≥ 2, the robustness of our algorithm is shown for k = 2 by comparisons with a number of recent methods. A number of theoretical results are also presented, which justify the design of our algorithm.