Solving Irregular Strip Packing Problems With Free Rotations Using Separation Lines

Solving nesting problems or irregular strip packing problems is to position polygons in a fixed width and unlimited length strip, obeying polygon integrity containment constraints and non-overlapping constraints, in order to minimize the used length of the strip. To ensure non-overlapping, we used separation lines. A straight line is a separation line if given two polygons, all vertices of one of the polygons are on one side of the line or on the line, and all vertices of the other polygon are on the other side of the line or on the line. Since we are considering free rotations of the polygons and separation lines, the mathematical model of the studied problem is nonlinear. Therefore, we use the nonlinear programming solver IPOPT (an algorithm of interior points type), which is part of COIN-OR. Computational tests were run using established benchmark instances and the results were compared with the ones obtained with other methodologies in the literature that use free rotation

Approximating the Permanent of a Matrix with Deep Rejection Sampling

Computing the permanent of a matrix is a famous #P-hard problem with a wide range of applications. The fastest known exact algorithms for the problem require an exponential number of operations, and all known fully polynomial randomized approximation schemes are rather complicated to implement and have impractical time complexities. The most promising recent advancements on approximating the permanent are based on rejection sampling and upper bounds for the permanent. In this thesis, we improve the current state of the art by developing the deep rejection sampling method, which combines an exact algorithm with the rejection sampling method. The algorithm precomputes a dynamic programming table that tightens the initial upper bound used by the rejection sampling method. In a sense, the table is used to jump-start the sampling process. We give a high probability upper bound for the time complexity of the deep rejection sampling method for random (0, 1)-matrices in which each entry is 1 with probability p. For matrices with p < 1/5, our high probability bound is stronger than in previous work. In addition to that, we empirically observe that our algorithm outperforms earlier rejection sampling methods by testing it with different parameters against other algorithms on multiple classes of matrices. The improvements in sampling times are especially notable in cases in which the ratios of the permanental upper bounds and the exact value of the permanent are huge