"... In PAGE 17: ... Edge weights were generated by using a parameter k 2 f1; 2; 3; 4; 5g. Our computational results for these instances are listed below in Table1 (positive edge weights) and Table 2 (mixed, i:e:, positive and negative edge weights). The column \gap % quot; displays the relative di erence between the optimal solution and the upper bound computed at depth 0.... ..."

"... In PAGE 5: ...1 Four Color Raster Signature (4CRS) The 4CRS (Zimbrao and Souza, 1998) is used for representing polygons, and it is a small bit-map using four colors. Each color represents an intersection type between the object and the cell ( Table1 ). In Figure 1, an ... ..."

"... In PAGE 24: ... For these instances, previous work includes linear programming re- laxations and corresponding order-based heuristics by Cavalcante, de Souza, Savelsbergh, Wang, and Wolsey [11], a tabu search algorithm by Cavalcante and de Souza [10, 11] (a refined and extended im- plementation of which has been proposed recently in [9]), as well as a constraint propagation based branch-and-bound algorithm by Heipcke [33]. Before we turn to the combined algorithm, in Table7 we compare the weak linear programming relaxation (14), (2), (6), (4), and (15) to results obtained with the Lagrangian approach. Note that the computation times for solving the strong linear programming relaxation (14), (2), (3), (4), and (15) are prohibitive large for the larger of these benchmark instances which is the reason to only consider the weak formulation.... In PAGE 24: ...avalcante et al. [11]. Moreover, for this experiment we did not consider additional inequalities of the type (22). In Table7 , the columns #jobs and #tasks show the number of job and tasks of an instance, respec- tively. CP is the length of a longest (critical) path in the project network, and the next columns show the respective lower bound values obtained with the weak linear program (14), (2), (6), (4), and (15) and the Lagrangian relaxation.... In PAGE 25: ... The computation times for medium size instances (40-80 jobs) were up to 2,000 seconds, for large size instances (more than 80 jobs) the computation times varied between 2,000 and 14,000 seconds, on a Sun Sparc 1000. Results for the LP-based ordering heuristics from [11] are not available for all of these instances, which is due to the fact that it took too much time to solve the corresponding LP-relaxations for the large size instances with more than 80 jobs (see also Table7 ). For the remaining instances, the results shown in Table 8 have been obtained using different LP-relaxations, based upon formulations in x- as well as z-variables, additionally using different improvement heuristics.... ..."