In this lecture we discuss a few applications of the results on symmetric computation presented in this course. First, we discuss an application to proving unconditional lower bounds for certain algorithmic methods used to solve constraint satisfaction problems. Second, we discuss an application to hardness of approximation. These hardness results establish that certain NP-hard problems cannot be approximated beyond some threshold of accuracy in polynomial-time so long as P is not equal to NP. We discuss recent work showing that for certain NP-hard problems inapproximability in FPC can be established without reference to any condition on P and NP.

We finally discuss other notions of symmetric computation which, like FPC, can express only problems in P but can also express problems known to inexpressible in FPC. In particular, we discuss choiceless polynomial time and fixed-point logic with rank.

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.