Raising NP Lower Bounds to Parallel NP Lower Bounds

A decade ago, a beautiful paper by Wagner developed a ``toolkit'' that in certain cases allows one to prove problems hard for parallel access to NP. However, the problems his toolkit applies to most directly are not overly natural. During the past year, problems that previously were known only to be NP-hard or coNP-hard have been shown to be hard even for the class of sets solvable via parallel access to NP. Many of these problems are longstanding and extremely natural, such as the Minimum Equivalent Expression problem (which was the original motivation for creating the polynomial hierarchy), the problem of determining the winner in the election system introduced by Lewis Carroll in 1876, and the problem of determining on which inputs heuristic algorithms perform well. In the present article, we survey this recent progress in raising lower bounds.