The class Max (r,2)-CSP consists of constraint satisfaction problems with at most two r-valued variables per clause. For instances with n variables and m binary clauses, we present an O(T19m/100) -time algorithm. It is the fastest algorithm for most problems in the class (including Max Cut and Max 2-Sat), and in combination with "Generalized CSPs" introduced in a companion paper, also allows counting, sampling, and the solution of problems like Max Bisection that escape the usual CSP framework. Linear programming is key to the design as well as the analysis of the algorithm.