We compare our approach to the MTZ-based formulation of Santos et al. [43] (variant (B) and (C) for even and odd diameters, respectively) and the MCF formulation of Gouveia and Magnanti [59] (HopDMCF and Enh-HopMCF, respectively). The same test instances as in these previous works are used, which have been created the following way: The integral edge costs for random instances have been chosen from the interval [1,100] using a uniform distribution. For Euclidean instances, the integer coordinates of the nodes have been computed randomly within a 100×100 square grid, whereas for the edge weights just the integer parts of the Euclidean distances have been utilized. From these complete graphs, Gouveia and Magnanti only used the|E|cheapest edges for their tests. To ensure that their sparse instance graphs have feasible solutions, Santos et al. built them in a special way by first computing a minimum spanning tree with a diameter bound ofD= 2 (a star), and afterwards augmenting this tree with the least cost edges until reaching the desired number of edges for the instance. Our experiments were performed on a PentiumR4 2.8 GHz system with 2 GB of RAM using Linux 2.4.21 and CPLEX 8.1 under default parameters as MIP solver.
Table 5.1 lists CPU times for finding optimal solutions and proving optimality on complete and sparse instances for our reimplementation of Santos et al. and our ILP formulation. Variants marked by ‘+’ apply connection cuts, whereas those marked by ‘∗’ use cycle elimination cuts. Enclosed in parentheses we list the percentage gap between the optimal solution (opt) and the LP-relaxation (lb) at the first node within the Branch&Cut tree after performing all separations of our own and the standard cuts of CPLEX: gap= (opt−lb)/opt·100%. In case there are two values listed the first one depicts the gap at the very beginning only depending on the model used and so invariant in terms of applying cuts.
In general, no approach dominates any other. However, it can be observed that the ILP formulation performs better on tighter constrained instances with rather small diameters, whereas the model of Santos et al. becomes faster than the ILP for looser diameter constraints.
Applying our cuts to the different formulations lead to ambivalent results, except on sparse instances where connection cuts show significantly better performance. It also turns out that the ILP model in general benefits more from cycle elimination cuts, in particular on bigger instances with larger diameters.
Concerning the listed LP-relaxation gaps we further remark that in addition to our connection and cycle elimination cuts, CPLEX sometimes adds additional general-purpose cuts, which further reduce the LP-bound. In a few cases, this leads to the
5.4Computational
|V| |E| D Santos Santos+ Santos∗ Santos+∗ ILP ILP+ ILP∗ ILP+∗
15 105 4 4.7 (37.4/23.9) 9.8 (22.4) 18.88 (21.0) 29.9 (20.1) 0.7(36.4/22.0) 0.9 (25.1) 1.1 (19.7) 1.7 (20.1) 15 105 5 22.8 (33.8/28.9) 36.9 (25.5) 44.1 (20.8) 78.7 (18.2) 3.0(33.6/29.2) 3.2 (25.3) 5.8 (19.2) 7.0 (18.1) 15 105 6 21.1 (31.5/19.7) 18.6 (16.4) 52.6 (16.9) 48.6 (13.7) 8.1(38.9/32.8) 24.1 (25.0) 15.6 (16.3) 33.6 (13.7) 15 105 7 44.8 (28.1/23.7) 29.3 (18.0) 38.8 (13.9) 26.9 (10.6) 26.4 (33.6/28.3) 37.2 (28.3) 20.5 (14.0) 20.0 (10.6) 15 105 9 18.4 (24.8/21.1) 16.5 (15.1) 15.8 (10.4) 6.2 (6.6) 150.6 (32.5/26.8) 47.0 (21.1) 20.7 (10.3) 10.7 (6.6) 15 105 10 1.5 (24.8/12.1) 1.0 (8.3) 5.1 (9.2) 2.4 (5.3) 65.8 (36.1/29.1) 43.3 (20.6) 15.8 (9.2) 4.3 (5.3) 20 190 4 562.9 (29.7/24.4) 1,063.6 (21.6) 1,232.5 (24.1) 4,315.1 (21.4) 2.5(28.8/25.0) 3.3 (21.3) 3.7 (22.3) 5.0 (20.2) 20 190 5 436.7 (25.2/20.9) 662.5 (20.3) 572.9 (17.4) 1,260.6 (17.3) 8.1(27.4/17.8) 10.2 (17.8) 9.3 (15.2) 12.8 (15.6) 20 190 6 577.0 (20.3/13.8) 489.3 (13.3) 455.2 (12.5) 531.9 (12.1) 95.0 (28.2/21.7) 210.5 (19.7) 396.4 (12.5) 382.8 (12.5) 20 190 7 8.1 (13.4/9.7) 5.1 (7.7) 7.8 (5.9) 10.2 (5.1) 10.0 (19.3/11.3) 12.9 (8.3) 4.5 (5.0) 7.2 (5.0) 20 190 9 241.9 (22.2/18.5) 105.7 (14.4) 244.3 (11.2) 73.4 (8.9) 1,209.1 (30.0/24.4) 923.7 (18.6) 194.6 (11.0) 66.7 (8.8) 20 190 10 64.6 (21.9/15.4) 41.9 (10.6) 205.1 (11.2) 29.7 (8.1) 13,755.5 (34.0/26.1) 13,972.1 (18.4) 226.4 (11.3) 101.0 (8.1) 25 300 4 15,203.7 (30.3/26.6) >20,000.0 (25.9) >20,000.0 (26.1) >20,000.0 (25.1) 12.0 (31.2/23.2) 14.2 (22.9) 16.5 (22.2) 20.3 (21.7) 25 300 5 >20,000.0 (32.4/28.7) >20,000.0 (25.1) >20,000.0 (25.5) >20,000.0 (22.6) 76.3 (33.5/27.4) 64.3 (23.1) 102.7 (23.7) 127.3 (20.9) 25 300 6 1,282.5 (18.6/12.4) 826.7 (11.4) 1,241.7 (10.9) 1,151.3 (10.1) 26.4 (28.4/17.0) 166.7 (13.9) 75.8 (10.6) 146.1 (10.1) 25 300 7 11,521.3 (18.7/15.9) >20,000.0 (14.0) >20,000.0 (13.3) 17,014.1 (11.4) 770.5(26.5/17.6) 2,719.3 (14.9) 1,090.3 (13.0) 989.6 (11.6) 25 300 9 >20,000.0 (22.7/18.4) 246.0 (8.0) 13,677.6 (15.5) 429.7 (5.3) >20,000.0 (31.3/22.7) 2160.1 (11.9) 4,143.3 (15.4) 295.4 (5.2) 25 300 10 278.2 (10/8.6) 254.8 (5.9) 401.4 (7.0) 327.0 (5.3) 3,666.1 (24.0/12.6) 2,904.2 (9.2) 1,127.9 (7.2) 404.9 (5.3) 20 50 4 1.0 (32.9/19.1) 1.0 (16.5) 4.7 (18.8) 3.8 (16.4) 0.2(29.0/13.9) 0.2 (13.4) 0.3 (13.8) 0.4 (13.2) 20 50 5 4.6 (60.7/57.8) 9.0 (52.5) 15.6 (54.6) 23.7 (51.2) 1.0(62.1/58.8) 2.5 (53.6) 3.5 (53.8) 4.9 (50.6) 20 50 6 34.6 (28.9/20.8) 0.8 (9.5) 45.6 (15.2) 1.5 (8.8) 8.7 (35.1/27.2) 5.1 (11.6) 19.9 (15.2) 6.0 (8.8) 20 50 7 13.3 (26.1/22.4) 0.8 (8.4) 13.3 (20.8) 1.2 (7.9) 1.2 (27.4/25.5) 1.7 (7.8) 2.5 (19.5) 3.0 (7.4) 20 50 9 76.2 (24.3/19.5) 0.7 (7.7) 108.5 (14.5) 0.8 (4.6) 42.5 (31.8/24.9) 2.8 (10.4) 25.6 (14.4) 3.7 (4.6) 20 50 10 98.5 (29.5/21.7) 0.2 (3.9) 187.7 (16.8) 0.2 (2.2) 505.3 (40.4/32.8) 1.8 (7.4) 79.1 (16.7) 1.3 (2.2) 40 100 4 43.7 (41.9/27.7) 82.4 (23.5) 516.5 (24.6) 2,352.1 (23.0) 1.9(39.6/20.4) 1.9 (19.4) 8.5 (19.0) 10.0 (18.5) 40 100 5 471.0 (65.4/56.7) 291.7 (52.9) 1,646.5 (56.1) 893.2 (52.3) 6.4(62.9/47.5) 6.8 (46.9) 13.1 (46.9) 18.2 (46.7) 40 100 6 1,991.6 (29.9/25.7) 50.9 (4.6) 13,719.8 (23.3) 100.2 (4.1) 182.9 (44.8/30.2) 13.2 (6.4) 449.6 (23.3) 37.4 (4.6) 40 100 7 >20,000.0 (36.6/21.1) 459.4 (14.7) 3,731.7 (20.4) 10,224.2 (13.4) 212.4(45.5/34.5) 4,463.2 (15.2) 525.2 (20.9) 9455.3 (13.2)
Table 5.2: CPU times (in seconds) and LP-relaxation gaps on incomplete instances from Gouveia and Magnanti [59] (gaps for G&M at the beginning of the first Branch&Cut node without CPLEX generated cuts).
|V| |E| D G&M ILP ILP+ ILP∗ ILP+∗
Random 20 100 4 0.5 (0.0) 0.9 (35.8/23.9) 0.9 (23.9) 1.5 (24.3) 1.6 (23.4) 20 100 5 6.3 (0.0) 2.7 (35.5/27.0) 2.4 (26.9) 2.4 (25.6) 3.2 (25.6) 20 100 6 5.8 (1.0) 2.9 (34.7/21.0) 7.2 (20.6) 5.9 (16.3) 7.9 (18.0) 20 100 7 94.0 (1.5) 10.6 (31.6/17.1) 5.7 (16.2) 2.6 (13.1) 15.8 (13.2) 20 100 8 1.3 (0.0) 4.4 (28.1/10.0) 4.3 (9.4) 3.4 (5.8) 5.7 (5.8) Random 30 200 4 0.8 (0.0) 3.5 (39.8/30.1) 6.1 (29.7) 6.3 (29.8) 8.7 (29.5) 30 200 5 58.6 (0.0) 377.8 (48.1/41.9) 283.8 (41.9) 428.1 (42.1) 482.4 (41.6) 30 200 6 2.9 (0.0) 5.7 (28.4/19.4) 20.4 (14.9) 11.5 (14.6) 39.6 (13.5) 30 200 7 529.4 (0.0) 112.4 (21.3/16.6) 261.1 (16.3) 53.9 (14.3) 100.8 (14.2) 30 200 8 2.3 (0.0) 10.6 (18.3/14.3) 13.9 (5.4) 3.67 (8.3) 4.9 (2.2) Euclidian 20 100 4 0.1 (0.0) 1.1 (20.3/17.5) 1.4 (17.4) 2.3 (17.2) 2.2 (17.1) 20 100 5 5.3 (0.0) 1.7 (16.8/13.2) 2.0 (13.0) 2.5 (12.8) 3.5 (12.6) 20 100 6 3.1 (0.2) 7.3 (16.3/10.4) 9.5 (9.9) 15.5 (8.0) 21.2 (8.5) 20 100 7 49.5 (0) 10.0 (14.1/8.3) 24.3 (8.2) 11.2 (7.0) 31.3 (7.6) 20 100 8 1.1 (0.0) 10.7 (14.2/10.0) 12.9 (7.7) 18.9 (6.8) 31.2 (5.2) Euclidian 30 200 4 130.8 (1.7) 148.6 (35.2/25.8) 84.9 (26.0) 59.5 (26.3) 107.0 (25.3) 30 200 5 25.1 (0.1) 36.1 (33.9/25.7) 42.2 (26.1) 65.3 (24.1) 61.5 (25.6) 30 200 6 1,381.9 (0.8) 348.0 (28.8/19.4) 4,022.7 (17.6) 7,224.2 (16.9) 17,144.3 (15.0) 30 200 7 6,912.1 (1.2) 1,339.7 (23.5/17.5) 3,713.8 (16.3) 1,014.4 (13.6) 4,205.4 (13.4) 30 200 8 1,111.0 (0.8) 2,864.7 (22.6/16.2) 9,298.8 (12.6) 2,430.6 (11.5) >20,000.0 (9.4)
effect that the LP-bound of a variant where only one type of our cuts is used is slightly smaller than when applying both of them.
Table 5.2 shows running times and LP-bounds for the ILP variants and Gouveia and Magnanti’s (G&M) approach. The times listed for G&M are adopted from [59]
and scaled by a factor of 1/8 to account for different hardware. This factor has been determined by considering the widely used floating point benchmarks published at http://www.spec.org. Nevertheless, we remark that this scaling is only a rough estimation and running time comparisons should be taken with care.
Concerning the LP-relaxation gaps it can easily be seen that the ILP model cannot compete with the flow formulations of G&M. When looking at the computation times no single variant dominates any other. The different ILP approaches outperform G&M on several occasions. However, no real pattern can be observed for conditions under which a certain method performs best; speed-up factors are in general not as large as those of the first series of experiments (Table 5.1).
In this context we want to point out that the above results are not sufficient to draw more general conclusions on expected running times for specific classes of
instances. During our tests we experienced highly varying computation times for instances randomly generated all the same way. For example, when running ILP+
on 10 different complete Euclidean graphs with 20 randomly distributed nodes, CPU times ranged from 51 to 54,400 seconds. A similar behavior was observed for the model of Santos et al. and the flow formulations as well.
In addition to the presented results we also made first experiments on instances with 40 nodes and more. As expected due to the observed LP-relaxation gaps the size of the Branch&Cut tree grows fast and so does the computation time. Without further improvements the ILP results will not be competitive to state-of-the-art flow formulations.
We also tried to reduce the initialization overhead always involved when calculating connection or cycle elimination cuts. A number of two to three cuts generated at once (i.e., cuts added to the model before a new LP is solved) has proven to be a good choice.