Hard and Easy Instances of L-Tromino Tilings

We study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle and an Aztec diamond has an L-tromino tiling. Then, we study tilings of arbitrary regions where only $180^\circ$ rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region does not contains certain so-called "forbidden polyominoes" as sub-regions, then there exists a polynomial time algorithm for deciding a tiling.Comment: Full extended version of LNCS 11355:82-95 (WALCOM 2019

Hard and easy instances of L-Tromino tilings

In this work we study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle has an L-tromino tiling, and hence also an Aztec diamond; if an Aztec rectangle has an unknown number of defects or holes, however, the problem of deciding a tiling is NP-complete. Then, we study tilings of arbitrary regions where only 180∘ rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region does not contain so-called “forbidden polyominoes” as subregions, then there exists a polynomial time algorithm for deciding a tiling

Essays on Integer Programming in Military and Power Management Applications

This dissertation presents three essays on important problems motivated by military and power management applications. The array antenna design problem deals with optimal arrangements of substructures called subarrays. The considered class of the stochastic assignment problem addresses uncertainty of assignment weights over time. The well-studied deterministic counterpart of the problem has many applications including some classes of the weapon-target assignment. The speed scaling problem is of minimizing energy consumption of parallel processors in a data warehouse environment. We study each problem to discover its underlying structure and formulate tailored mathematical models. Exact, approximate, and heuristic solution approaches employing advanced optimization techniques are proposed. They are validated through simulations and their superiority is demonstrated through extensive computational experiments. Novelty of the developed methods and their methodological contribution to the field of Operations Research is discussed through out the dissertation

Reversible Image Watermarking Using Modified Quadratic Difference Expansion and Hybrid Optimization Technique

With increasing copyright violation cases, watermarking of digital images is a very popular solution for securing online media content. Since some sensitive applications require image recovery after watermark extraction, reversible watermarking is widely preferred. This article introduces a Modified Quadratic Difference Expansion (MQDE) and fractal encryption-based reversible watermarking for securing the copyrights of images. First, fractal encryption is applied to watermarks using Tromino's L-shaped theorem to improve security. In addition, Cuckoo Search-Grey Wolf Optimization (CSGWO) is enforced on the cover image to optimize block allocation for inserting an encrypted watermark such that it greatly increases its invisibility. While the developed MQDE technique helps to improve coverage and visual quality, the novel data-driven distortion control unit ensures optimal performance. The suggested approach provides the highest level of protection when retrieving the secret image and original cover image without losing the essential information, apart from improving transparency and capacity without much tradeoff. The simulation results of this approach are superior to existing methods in terms of embedding capacity. With an average PSNR of 67 dB, the method shows good imperceptibility in comparison to other schemes

Expert Cognition During Proof Construction Using the Principle of Mathematical Induction

The purpose of this study was to identify and analyze the observable cognitive processes of experts in mathematics while they work on proof-construction activities using the Principle of Mathematical Induction (PMI). Graduate student participants in the study worked on ``nonstandard mathematical induction problems that did not involve algebraic identities or finite sums. This study identified some of the problem solving-strategies used by the participants during a Cognitive Task Analysis (Feldon, 2007) as well as epistemological obstacles they encountered while working with PMI. After the Cognitive Task Analysis, the graduate students participated in two semi-structured interviews. These interviews explored graduate students\u27 beliefs about proofs and proof techniques and situates their use of PMI within the contexts of these beliefs. Two primary theoretical frameworks were used to analyze participant cognition and the qualitative data collected. First, the study used Action, Process, Object, Schema (APOS) Theory (Asiala et al., 1996) to to study and analyze the participants\u27 conceptual understanding of the technique of mathematical induction and to test a preliminary genetic decomposition adapted from previous studies on PMI (Dubinsky \& Lewin 1996, 1999; Garcia-Martinez & Parraguez, 2017). Second, an Expert Knowledge Framework (Bransford, Brown, & Cocking, 1999; Shepherd & Sande, 2014) was used to classify the participants\u27 responses to the semi-structured interview questions according to several characteristics of expertise. The study identified several results which (1) give insight to the mental constructions used by mathematical experts when solving problem involving PMI; (2) offer some implications for improving the instruction of PMI in introductory proofs classrooms; and (3) provide results that allow for future comparison between expert and novice mathematical learners

Towards Developing Efficient Area Coverage Approach in Domestic Robots

東京電機大学201