We develop an idempotent version of probabilistic potential theory. The goal is to describe the set of max-plus harmonic functions, which give the stationary solutions of deterministic optimal control problems with additive reward. The analogue of the Martin compactification is seen to be a generalisation of the compactification of metric spaces using (generalised) Busemann functions. We define...

Two noninteger parameters are defined for MAX statistics, which are maxima of d simpler test statistics. The first parameter, dMAX, is the fractional number of tests, representing the equivalent numbers of independent tests in MAX. If the d tests are dependent, dMAX < d. The second parameter is the fractional degrees of freedom k of the chi-square distribution χk that fits the MAX null distribu...

We have seen several randomized approximation algorithms before. Recall that for the MAXCUT and MAX-3SAT problems we proved that choosing any solution uniformly at random gives a constant factor approximation in the expectation. For MAX-CUT the approximation factor was 2. For MAX-3SAT, a random assignment of truth values to a variable satisfies a given clause with probability 78 . In the case o...