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1. Vehicle Routing Application of MWIS
Given an undirected graph where is its set of nodes and its set of edges, a subset of nodes is an independent set if the elements of are pairwise nonadjacent in . If is the weight of node , the weight of independent set is . In the maximum weight independent set (MWIS) problem we seek an independent set such that for all independent sets in . This optimization problem is NP-hard (Garey and Johnson, 1979)
and it is often solved using heuristic algorithms.
We provide a collection of instances of an MWIS problem that appeared as subproblems in algorithms solving real-life long-haul vehicle routing problems at Amazon. Our goal is to enhance the set of benchmark instances available to algorithm researchers working on MWIS. Our instances differ from other publicly available instances and the new collection includes some large instances.
To gain intuition into the application, consider a stochastic heuristic for the problem. This heuristic produces different solutions for different pseudo-random generator seeds. Each solution consists of a set of routes. We want to recombine routes from multiple solutions to obtain a better solution.
Each route consists of a driver and a set of loads assigned to the driver. A subset of routes is feasible if no two routes in the subset share a driver or a load. Each route has a weight. The objective function is the sum of route weights. The problem is to find a feasible solution of the maximum total weight.
To state this problem as MWIS, we build a conflict graph as follows. Nodes of the graph correspond to routes and weights correspond to route weights. We connect two nodes by an edge if the corresponding routes have a conflict, i.e., they share a driver or a load.
| Instance | Initial Sol. | LP bound | Filename | Mbytes | ||
|---|---|---|---|---|---|---|
| MT-D-01 | 979 | 3 841 | 228 874 404 | 238 166 485 | MT-D-01.tar.gz | 0.03 |
| MT-D-200 | 10 880 | 547 529 | 286 750 411 | 287 228 467 | MT-D-200.tar.gz | 1.77 |
| MT-D-FN | 10 880 | 645 026 | 290 723 959 | 290 881 566 | MT-D-FN.tar.gz | 2.07 |
| MT-W-01 | 1 006 | 3 140 | 299 132 358 | 312 121 568 | MT-W-01.tar.gz | 0.03 |
| MT-W-200 | 12 320 | 554 288 | 383 620 215 | 384 099 118 | MT-W-200.tar.gz | 1.86 |
| MT-W-FN | 12 320 | 593 328 | 390 596 383 | 390 869 891 | MT-W-FN.tar.gz | 1.97 |
| MR-D-01 | 14 058 | 60 738 | 1 664 446 852 | 1 695 332 636 | MR-D-01.tar.gz | 0.48 |
| MR-D-03 | 21 499 | 168 504 | 1 739 544 141 | 1 763 685 757 | MR-D-03.tar.gz | 0.97 |
| MR-D-05 | 27 621 | 295 700 | 1 775 123 794 | 1 796 703 313 | MR-D-05.tar.gz | 1.35 |
| MR-D-FN | 30 467 | 367 408 | 1 794 070 793 | 1 809 854 459 | MR-D-FN.tar.gz | 1.75 |
| MR-W-FN | 15 639 | 267 908 | 5 386 472 651 | 5 386 842 781 | MR-W-FN.tar.gz | 1.18 |
| MW-D-01 | 3 988 | 19 522 | 465 730 126 | 477 563 775 | MW-D-01.tar.gz | 0.14 |
| MW-D-20 | 20 054 | 718 152 | 522 485 254 | 531 510 712 | MW-D-20.tar.gz | 2.50 |
| MW-D-40 | 33 563 | 2 169 909 | 533 938 531 | 543 396 252 | MW-D-40.tar.gz | 7.20 |
| MW-D-FN | 47 504 | 4 577 834 | 542 182 073 | 549 872 520 | MW-D-FN.tar.gz | 15.17 |
| MW-W-01 | 3 079 | 48 386 | 1 268 370 807 | 1 270 311 626 | MW-W-01.tar.gz | 0.21 |
| MW-W-05 | 10 790 | 789 733 | 1 328 552 109 | 1 334 413 294 | MW-W-05.tar.gz | 2.49 |
| MW-W-10 | 18 023 | 2 257 068 | 1 342 415 152 | 1 360 791 627 | MW-W-10.tar.gz | 6.76 |
| MW-W-FN | 22 316 | 3 495 108 | 1 350 675 180 | 1 373 020 454 | MW-W-FN.tar.gz | 10.41 |
| CW-T-C-1 | 266 403 | 162 263 516 | 1 298 968 | 1 353 493 | CW-T-C-1.tar.gz | 547.73 |
| CW-T-C-2 | 194 413 | 125 379 039 | 933 792 | 957 291 | CW-T-C-2.tar.gz | 417.49 |
| CW-T-D-4 | 83 091 | 43 680 759 | 457 715 | 463 672 | CW-T-D-4.tar.gz | 140.88 |
| CW-T-D-6 | 83 758 | 44 702 150 | 457 605 | 463 946 | CW-T-D-6.tar.gz | 143.95 |
| CR-T-C-1 | 602 472 | 216 862 225 | 4 605 156 | 4 801 515 | CR-T-C-1.tar.gz | 746.32 |
| CR-T-C-2 | 652 497 | 240 045 639 | 4 844 852 | 5 032 895 | CR-T-C-2.tar.gz | 828.21 |
| CR-T-D-4 | 651 861 | 245 316 530 | 4 789 561 | 4 977 981 | CR-T-D-4.tar.gz | 845.85 |
| CR-T-D-6 | 381 380 | 128 658 070 | 2 953 177 | 3 056 284 | CR-T-D-6.tar.gz | 441.42 |
| CR-T-D-7 | 163 809 | 49 945 719 | 1 451 562 | 1 469 259 | CR-T-D-7.tar.gz | 168.95 |
| CW-S-L-1 | 411 950 | 316 124 758 | 1 622 723 | 1 677 563 | CW-S-L-1.tar.gz | 1 071.34 |
| CW-S-L-2 | 443 404 | 350 841 894 | 1 692 255 | 1 759 158 | CW-S-L-2.tar.gz | 1 192.32 |
| CW-S-L-4 | 430 379 | 340 297 828 | 1 709 043 | 1 778 589 | CW-S-L-4.tar.gz | 1 156.28 |
| CW-S-L-6 | 267 698 | 191 469 063 | 1 159 946 | 1 192 899 | CW-S-L-6.tar.gz | 644.49 |
| CW-S-L-7 | 127 871 | 89 873 520 | 589 723 | 599 271 | CW-S-L-7.tar.gz | 294.53 |
| CR-S-L-1 | 863 368 | 368 431 905 | 5 548 904 | 5 768 579 | CR-S-L-1.tar.gz | 1 271.78 |
| CR-S-L-2 | 880 974 | 380 666 488 | 5 617 351 | 5 867 579 | CR-S-L-2.tar.gz | 1 314.11 |
| CR-S-L-4 | 881 910 | 383 405 545 | 5 629 351 | 5 869 439 | CR-S-L-4.tar.gz | 1 323.34 |
| CR-S-L-6 | 578 244 | 245 739 404 | 3 841 538 | 3 990 563 | CR-S-L-6.tar.gz | 845.81 |
| CR-S-L-7 | 270 067 | 108 503 583 | 1 969 254 | 2 041 822 | CR-S-L-7.tar.gz | 370.47 |
List of VR instances in the library. For each of the 38 instances, the table lists the instance name, the number of nodes and edges in the conflict graph, the total weight of a starting solution, the linear programming (LP) upper bound, the compressed tar files of the directory with the files that define the instance, and the size (in Mbytes) of the compressed tar file.
Our application has additional information that one can (optionally) use in an algorithm. First, we have a good initial solution, the best of the solutions we combine. We provide initial solutions for our instances. One can use this solution to possibly warm-start a MWIS algorithm.
Second, we have information about many cliques in the conflict graph. For a fixed load (or driver), nodes corresponding to the routes containing the load (driver) form a clique: every pair of such nodes is connected. This allows us to use the well-known clique integer linear programming (ILP) formulation of the problem:
| subject to | |||
where are, respectively, the sets of 2-clique, 3-clique and -clique inequalities. In general, for cliques of size , we have the set of -clique inequalities
One can solve a linear programming (LP) relaxation of the problem, which assigns each node a value in the closed real interval . Note that the objective function of the LP relaxation provides an upper bound on the corresponding MWIS solution value. We provide both the cliques and the relaxed LP solutions with our instances.
Table 1 lists the instances we provide and includes the graph size, the initial solution value, and the relaxed LP bound.
2. Input Graph Format
We place each instance in a separate directory containing several files with instance name, graph edge set, node weights, clique information, and relaxed LP solution values. Directory names correspond to the instance names. Next we describe the file formats.
For an undirected, node-weighted graph with nodes, edges and integral node IDs from , we give the following files:
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instance_name.txt – Name of the instance.
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conflict_graph.txt – Edges of . The file has a total of lines. The first line gives the numbers of nodes and edges: “”. Each of the lines describes an edge as “”.
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node_weights.txt – Node weights. The file has a total of lines, each describing the weight of node as “”. The weights are integers.
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solution.txt – Initial solution for warm start. It contains one line per node in the initial solution, giving its node index: if a node in the solution, the file contains a line with “” in it.
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cliques.txt – Set of cliques in . For each clique , the file contains one line as “”.
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lploads.txt – Solution for the relaxed LP problem for the MWIS problem on the clique graph, where each node has a relaxed LP value . The file has lines, each with the LP value of a node as “”, where is a floating point number.
The files conflict_graph.txt and node_weights.txt are needed by any MWIS algorithm. The other files are optional.
Note that some of our graphs are large, with the compressed tar file being over 1 Gbyte in size. 32-bit integers are insufficient to represent the total weight of a solution. An implementation needs to use 64-bit integers or doubles to represent the weight of these independent sets.
3. Links to Instances
The full set of 38 instances are downloaded at
as gzipped tar files.
4. Concluding Remarks
In this paper we introduce a set of large-scale maximum weight independent set instances arising in a real-world vehicle routing application. Our goal in making these instances available to other researchers is that progress can be made in the field. Other researchers can try their existing MWIS solvers on these instances and can be motivated to develop new solvers for them.
References
- Computers and intractability: a guide to the theory of np-completeness. W.H. Freeman and Company, San Francisco. Cited by: §1.