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 NPhard (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 reallife longhaul 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 pseudorandom 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  

MTD01  979  3 841  228 874 404  238 166 485  MTD01.tar.gz  0.03 
MTD200  10 880  547 529  286 750 411  287 228 467  MTD200.tar.gz  1.77 
MTDFN  10 880  645 026  290 723 959  290 881 566  MTDFN.tar.gz  2.07 
MTW01  1 006  3 140  299 132 358  312 121 568  MTW01.tar.gz  0.03 
MTW200  12 320  554 288  383 620 215  384 099 118  MTW200.tar.gz  1.86 
MTWFN  12 320  593 328  390 596 383  390 869 891  MTWFN.tar.gz  1.97 
MRD01  14 058  60 738  1 664 446 852  1 695 332 636  MRD01.tar.gz  0.48 
MRD03  21 499  168 504  1 739 544 141  1 763 685 757  MRD03.tar.gz  0.97 
MRD05  27 621  295 700  1 775 123 794  1 796 703 313  MRD05.tar.gz  1.35 
MRDFN  30 467  367 408  1 794 070 793  1 809 854 459  MRDFN.tar.gz  1.75 
MRWFN  15 639  267 908  5 386 472 651  5 386 842 781  MRWFN.tar.gz  1.18 
MWD01  3 988  19 522  465 730 126  477 563 775  MWD01.tar.gz  0.14 
MWD20  20 054  718 152  522 485 254  531 510 712  MWD20.tar.gz  2.50 
MWD40  33 563  2 169 909  533 938 531  543 396 252  MWD40.tar.gz  7.20 
MWDFN  47 504  4 577 834  542 182 073  549 872 520  MWDFN.tar.gz  15.17 
MWW01  3 079  48 386  1 268 370 807  1 270 311 626  MWW01.tar.gz  0.21 
MWW05  10 790  789 733  1 328 552 109  1 334 413 294  MWW05.tar.gz  2.49 
MWW10  18 023  2 257 068  1 342 415 152  1 360 791 627  MWW10.tar.gz  6.76 
MWWFN  22 316  3 495 108  1 350 675 180  1 373 020 454  MWWFN.tar.gz  10.41 
CWTC1  266 403  162 263 516  1 298 968  1 353 493  CWTC1.tar.gz  547.73 
CWTC2  194 413  125 379 039  933 792  957 291  CWTC2.tar.gz  417.49 
CWTD4  83 091  43 680 759  457 715  463 672  CWTD4.tar.gz  140.88 
CWTD6  83 758  44 702 150  457 605  463 946  CWTD6.tar.gz  143.95 
CRTC1  602 472  216 862 225  4 605 156  4 801 515  CRTC1.tar.gz  746.32 
CRTC2  652 497  240 045 639  4 844 852  5 032 895  CRTC2.tar.gz  828.21 
CRTD4  651 861  245 316 530  4 789 561  4 977 981  CRTD4.tar.gz  845.85 
CRTD6  381 380  128 658 070  2 953 177  3 056 284  CRTD6.tar.gz  441.42 
CRTD7  163 809  49 945 719  1 451 562  1 469 259  CRTD7.tar.gz  168.95 
CWSL1  411 950  316 124 758  1 622 723  1 677 563  CWSL1.tar.gz  1 071.34 
CWSL2  443 404  350 841 894  1 692 255  1 759 158  CWSL2.tar.gz  1 192.32 
CWSL4  430 379  340 297 828  1 709 043  1 778 589  CWSL4.tar.gz  1 156.28 
CWSL6  267 698  191 469 063  1 159 946  1 192 899  CWSL6.tar.gz  644.49 
CWSL7  127 871  89 873 520  589 723  599 271  CWSL7.tar.gz  294.53 
CRSL1  863 368  368 431 905  5 548 904  5 768 579  CRSL1.tar.gz  1 271.78 
CRSL2  880 974  380 666 488  5 617 351  5 867 579  CRSL2.tar.gz  1 314.11 
CRSL4  881 910  383 405 545  5 629 351  5 869 439  CRSL4.tar.gz  1 323.34 
CRSL6  578 244  245 739 404  3 841 538  3 990 563  CRSL6.tar.gz  845.81 
CRSL7  270 067  108 503 583  1 969 254  2 041 822  CRSL7.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 warmstart 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 wellknown clique integer linear programming (ILP) formulation of the problem:
subject to  
where are, respectively, the sets of 2clique, 3clique 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, nodeweighted graph with nodes, edges and integral node IDs from , we give the following files:

instance_name.txt – Name of the instance.

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 “”.

node_weights.txt – Node weights. The file has a total of lines, each describing the weight of node as “”. The weights are integers.

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.

cliques.txt – Set of cliques in . For each clique , the file contains one line as “”.

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. 32bit integers are insufficient to represent the total weight of a solution. An implementation needs to use 64bit 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 largescale maximum weight independent set instances arising in a realworld 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 npcompleteness. W.H. Freeman and Company, San Francisco. Cited by: §1.
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