Stochastic Local Search and the Lovasz Local Lemma

Stochastic Local Search and the Lovasz Local LemmabyFotios IliopoulosDoctor of Philosophy in Computer ScienceUniversity of California, BerkeleyProfessor Alistair Sinclair, ChairThis thesis studies randomized local search algorithms for finding solutions of constraint satisfaction problems inspired by and extending the Lovasz Local Lemma (LLL). The LLL is a powerful probabilistic tool for establishing the existence of objects satisfying certain properties (constraints). As a probability statement it asserts that, given a family of “bad” events, if each bad event is individually not very likely and independent of all but a small number of other bad events, then the probability of avoiding all bad events is strictly positive. In a celebrated breakthrough, Moser and Tardos made the LLL constructive for any product probability measure over explicitly presented variables. Specifically, they proved that whenever the LLL condition holds, their Resample algorithm, which repeatedly selects any occurring bad event and resamples all its variables according to the measure, quickly converges to an object with desired properties. In this dissertation we present a framework that extends the work of Moser and Tardos and can be used to analyze arbitrary, possibly complex, focused local search algorithms, i.e., search algorithms whose process for addressing violated constraints, while local, is more sophisticated than obliviously resampling their variables independently of the current configuration. We give several applications of this framework, notably a new vertex coloring algorithm for graphs with sparse vertex neighborhoods that uses a number of colors that matches the algorithmic barrier for random graphs, and polynomial time algorithms for the celebrated (non-constructive) results of Kahn for the Goldberg-Seymour and List-Edge-Coloring Conjectures.Finally, we introduce a generalization of Kolmogorov’s notion of commutative algorithms, cast as matrix commutativity, and show that their output distribution approximates the so-called “LLL-distribution”, i.e., the distribution obtained by conditioning on avoiding all bad events. This fact allows us to consider questions such as the number of possible distinct final states and the probability that certain portions of the state space are visited by a local search algorithm, extending existing results for the Moser-Tardos algorithm to commutative algorithms

Cancer rate of Bethesda category II thyroid nodules

Aim Thyroid nodules are very common and may be found in more than 50% of the population. Fine-needle aspiration cytology (FNAC) of thyroid nodules is a very useful diagnostic tool with high sensitivity and predictive value for diagnosis. The Bethesda System for Reporting Thyroid Cytopathology (BSRTC) uses six categories for thyroid cytology reporting (I-nondiagnostic, IIbenign, III-atypia of undetermined significance (AUS)/ follicular lesion of undetermined significance (FLUS), IV-follicular neoplasm/suspicious for follicular neoplasm (SFN), V-suspicious for malignancy, and VI-malignant. Our objective was to determine the malignancy rate in Bethesda II nodules. Methods From June 2010 to May 2020 a retrospective analysis was performed among 1166 patients who underwent thyroid surgery for benign thyroid diseases in our institution. Thyroid cytopathological slides and Ultrasound (US) reports were reviewed and classified according to the BSRTC. Data collected included age, gender, cytological features, and histological type of thyroid cancer. Results During the study period, 44.77% (522/1166) of patients with an FNA categorized as Bethesda II underwent thyroid surgery. Incidental malignancy was found in 1.53% (8/522) cases of Bethesda II. The most common malignant tumour type was papillary thyroid carcinoma. Conclusion The current study demonstrates that incidental thyroid carcinoma can be diagnosed after thyroidectomy even in patients with an FNA categorized as Bethesda II

Stochastic Local Search and the Lovasz Local Lemma

Stochastic Local Search and the Lovasz Local LemmabyFotios IliopoulosDoctor of Philosophy in Computer ScienceUniversity of California, BerkeleyProfessor Alistair Sinclair, ChairThis thesis studies randomized local search algorithms for finding solutions of constraint satisfaction problems inspired by and extending the Lovasz Local Lemma (LLL). The LLL is a powerful probabilistic tool for establishing the existence of objects satisfying certain properties (constraints). As a probability statement it asserts that, given a family of “bad” events, if each bad event is individually not very likely and independent of all but a small number of other bad events, then the probability of avoiding all bad events is strictly positive. In a celebrated breakthrough, Moser and Tardos made the LLL constructive for any product probability measure over explicitly presented variables. Specifically, they proved that whenever the LLL condition holds, their Resample algorithm, which repeatedly selects any occurring bad event and resamples all its variables according to the measure, quickly converges to an object with desired properties. In this dissertation we present a framework that extends the work of Moser and Tardos and can be used to analyze arbitrary, possibly complex, focused local search algorithms, i.e., search algorithms whose process for addressing violated constraints, while local, is more sophisticated than obliviously resampling their variables independently of the current configuration. We give several applications of this framework, notably a new vertex coloring algorithm for graphs with sparse vertex neighborhoods that uses a number of colors that matches the algorithmic barrier for random graphs, and polynomial time algorithms for the celebrated (non-constructive) results of Kahn for the Goldberg-Seymour and List-Edge-Coloring Conjectures.Finally, we introduce a generalization of Kolmogorov’s notion of commutative algorithms, cast as matrix commutativity, and show that their output distribution approximates the so-called “LLL-distribution”, i.e., the distribution obtained by conditioning on avoiding all bad events. This fact allows us to consider questions such as the number of possible distinct final states and the probability that certain portions of the state space are visited by a local search algorithm, extending existing results for the Moser-Tardos algorithm to commutative algorithms