## The characteristics of the benchmark instances used

The characteristics of the benchmark instances used during the experiments are summarised in Table 1. For graph colouring, 19 benchmark instances are used in which the number of colours, vertices, edges as well as edge densities vary. The instances in the upper half of the table are from the COLOR02 website,2 which was initially compiled for a competition. Myciel graphs are based on Mycielski THZ1 and are considered to be difficult to solve and the colouring number increases in problem size. A queenn.n graph is a graph on n2n2 nodes, each represents a square on an n by n chessboard. If two squares are in the same row, column, or diagonal, then their corresponding nodes on the graph are considered to be connected. The objective is to place n sets of n queens on the board so bronchitis no two queens of the same set can capture one another, which could be achieved only if the graph has a colouring number n. In addition to these data sets, problem instances in the bottom half of the table are from the well known DIMACS 3 challenge suite. For examination timetabling, we used a subset from the Toronto benchmark suite referred to as Toronto a instances ( Qu et al., 2009). Table 1 also shows the range of k values used during the experiments for each instance.