This chapter is about optimization problems, where the goal is to output not just any solution, but an optimal solution, which maximizes or minimizes some objective function. The chapter covers approximation algorithms (which output near-optimal solutions), hardness of approximation, and the connection between hardness of approximation and probabilistically checkable proofs via the profound PCP theorem.

A 48-year-old male is brought in for emergency evaluation from the local correctional facility. He complains of chest pain, shortness of breath, and a cough. He is tachycardic, tachypneic, hypoxic, febrile, and borderline hypotensive. He is alert but diaphoretic and leaning forward in the gurney, with increased work of breathing. As the treating physician, you learn that he has AIDS with a CD4 count of 94. Your quick exam reveals a rapid heart rate, bilateral rales, and white plaques in the posterior oropharynx. As you generate your differential diagnosis, you simultaneously begin to address the patient’s abnormal vital signs. You start a 1 L bolus of normal saline, place the patient on a nonrebreather face mask at 15 L per minute, and treat his fever with acetaminophen. Because of the patient’s fever, a pulmonary infection is highest on your differential. Given the patient’s compromised immune system and incarcerated status, you realize the need to do more than treat bacterial pneumonia. While starting antibiotics, what are your next moves?