An instance of Max 2-CSP consists of a graph G and a set of score functions. A potential solution is an assignment of values (colours) from {0,...,r-1} (often r = 2) to the vertices of the graph. The score of the assignment is the sum of the values of the score functions (applied to the assignment), which are of three kinds (i) there is a single constant function, (ii) for each vertex, there is a function whose value depends (only) on the colour of the vertex, and (iii) for each edge, there is a function whose value depends (only) on the colours of the two endpoints of the edge. (In general, distinct vertices and distinct edges have different functions.)
An optimum solution is one which achieves the maximum possible total score.