Approximation algorithms

Increasing global competition, rapidly changing markets, and greater consumer awareness have altered the way in which corporations do business. To become more efficient, many industries have sought to model some operational aspects by gigantic optimization problems. It is not atypical to encounter models that capture 106 separate “yes” or “no” decisions to be made. Although one could, in principle, try all 2106 possible solutions to find the optimal one, such a method would be impractically slow. Unfortunately, for most of these models, no algorithms are known that find optimal solutions with reasonable computation times. Typically, industry must rely on solutions of unguaranteed quality that are constructed in an ad hoc manner. Fortunately, for some of these models there are good approximation algorithms: algorithms that produce solutions quickly that are provably close to optimal. Over the past 6 years, there has been a sequence of major breakthroughs in our understanding of the design of approximation algorithms and of limits to obtaining such performance guarantees; this area has been one of the most flourishing areas of discrete mathematics and theoretical computer science.

The acceleration and collimation of jets.

I will discuss several issues related to the acceleration, collimation, and propagation of jets from active galactic nuclei. Hydromagnetic stresses provide the best bet for both accelerating relativistic flows and providing a certain amount of initial collimation. However, there are limits to how much "self-collimation" can be achieved without the help of an external pressurized medium. Moreover, existing models, which postulate highly organized poloidal flux near the base of the flow, are probably unrealistic. Instead, a large fraction of the magnetic energy may reside in highly disorganized "chaotic" fields. Such a field can also accelerate the flow to relativistic speeds, in some cases with greater efficiency than highly organized fields, but at the expense of self-collimation. The observational interpretation of jet physics is still hampered by a dearth of unambiguous diagnostics. Propagating disturbances in flows, such as the oblique shocks that may constitute the kiloparsec-scale "knots" in the M87 jet, may provide a wide range of untapped diagnostics for jet properties.