This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set. The book starts with a broad introduction to the field and progresses to advanced results. Contents include: definition of Turing machines and basic time and space complexity classes, probabilistic algorithms, interactive proofs, cryptography, quantum computation, lower bounds for concrete computational models (decision trees, communication complexity, constant depth, algebraic and monotone circuits, proof complexity), average-case complexity and hardness amplification, derandomization and pseudorandom constructions, and the PCP theorem.

Treatment with a once-daily, single-tablet regimen of ledipasvir and sofosbuvir resulted in high rates of sustained virologic response among patients with HCV genotype 1 infection who had not had a sustained virologic response to prior interferon-based treatment. (Funded by Gilead Sciences; ION-2 ClinicalTrials.gov number, NCT01768286.).

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Background The Extension for Community Healthcare Outcomes (ECHO) model was developed to improve access to care for complex health problems such as hepatitis C virus (HCV) infection for underserved populations. Using videoconferencing technology, ECHO trains primary care providers to treat complex diseases. Methods A prospective cohort study compared treatment of HCV at the University of New Mexico (UNM) HCV clinic to treatment by primary care clinicians at 21 ECHO sites in rural areas and prisons in New Mexico. A total of 407 treatment naive patients with chronic HCV were enrolled. The primary end point was a sustained viral response (SVR). Results The rate of SVR was 57.5% (84/146) for patients treated at UNM and 58.2% (152 /261) at ECHO sites (P=0.89); difference between SVR rates 0.7% (95% CI -9.2%, 10.7%). In genotype 1 infection the SVR rate was 45.8% (38 /83) at UNM and 49.7% (73 /147) at ECHO sites (P=0.57). Serious adverse events occurred in 13.7% of the UNM HCV clinic cohort and 6.9% of the ECHO cohort. Conclusions This study demonstrates that the ECHO model is an effective way to treat HCV in underserved communities. Implementation of this model would allow other states and nations to treat more patients with HCV.

We give a O( √ log n)-approximation algorithm for the sparsest cut, edge expansion, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in d , whose proof makes essential use of a phenomenon called measure concentration.We also describe an interesting and natural "approximate certificate" for a graph's expansion, which involves embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.