This document contains supplementary information on the empirical experiments conducted for the article
Kevin K.H. Cheung, Ambros Gleixner, and Daniel E. Steffy: Verifying Integer Programming Results, ZIB-Report 16-58, November 2016,
urn:nbn:de:0297-zib-61044.
We created a modified version of the exact rational MIP solver SCIP described by Cook et al. 2013 and used it to compute certificates for several MIP instances from the literature. Exact SCIP uses a hybrid of floating-point and exact rational arithmetic to efficiently compute exact solutions using a pure branch-and-bound algorithm. In our experiments, the rational MIP solver uses CPLEX 12.6.0.0 as its underlying floating-point LP solver and a patched version of QSopt_ex 2.5.10 as its underlying exact LP solver. The exact MIP solver supports several methods for computing valid dual bounds. Our certificate printing functionality is currently supported by the project-and-shift method (for dual feasible LP relaxations) and the exact LP method (for dual feasible LP relaxations and Farkas proofs). For details on these methods see Cook et al. 2013. Future plans are to include certificate printing functionality in all dual bounding methods and release this in subsequent versions of exact SCIP. Our development version is currently available from the authors by request.
We ran both exact SCIP with default settings and our modified SCIP version with a time limit of 3,600 seconds each and a
memory limit of 10GB on the certificate file size. On the certifcate files generated by our modified version of SCIP,
we called viprttn -t to tighten the certificates and viprchk to verify
their correctness. Note that, although we cannot verify the precise optimal value of the MIP instance when SCIP hits
the time or memory limit, we produce and verify a consistent certificate file for the primal and dual bounds available
at this point of the solution process.
All experiments were conducted on a cluster of Intel(R) Xeon(R) E5-2660 v3 CPUs running at 2.60GHz with 30GB RAM. Jobs were run exclusively to ensure accurate time measurement.
As a test bed, we considered the easy and numerically difficult (below referred to as hard) test sets from Cook
et al. 2013. These test sets consist of 107 publically available
instances from well-known libraries including MIPLIB 2003, MIPLIB 3, MIPLIB 2010, COR@L, and Hans Mittelmann. We had to
exclude only one instance (nw04) because viprttn ran out of memory.
Each certificate file could be verified as correct: viprchk did not exhibit any case in which exact SCIP pruned an
optimal solution or incorrectly accepted an infeasible solution in exact rational arithmetic. Aggregate statistics of
the computational performance are given in ZIB-Report 16-58 and the
following table:
| Test set | inst | SCIP solved | SCIP time (s) | SCIP+C solved | SCIP+C time (s) | ttn time (s) | chk time (s) | raw size (MB) | ttn size (MB) | gz size (MB) |
|---|---|---|---|---|---|---|---|---|---|---|
| easy-all | 56 | 53 | 62.0 | 39 | 180.8 | 25.8 | 28.9 | 214 | 72 | 22 |
| easy-solved | 39 | 39 | 23.2 | 39 | 48.0 | 9.6 | 13.4 | 77 | 34 | 10 |
| easy-memout | 5 | 4 | 600.6 | 0 | 1769.4 | 377.5 | 169.8 | 10286 | 513 | 159 |
| easy-timeout | 12 | 10 | 357.4 | 0 | 3600.0 | 83.7 | 97.5 | 1151 | 368 | 108 |
| hard-all | 50 | 23 | 725.1 | 14 | 976.6 | 31.2 | 15.1 | 372 | 38 | 11 |
| hard-solved | 13 | 13 | 22.9 | 13 | 40.8 | 7.1 | 6.3 | 49 | 15 | 5 |
| hard-memout | 10 | 0 | 3600.0 | 0 | 1833.9 | 275.7 | 53.8 | 10269 | 146 | 39 |
| hard-timeout | 27 | 10 | 1811.1 | 1 | 3255.1 | 20.7 | 11.9 | 286 | 35 | 9 |
Here, the SCIP columns report on the performance of the exact version of SCIP, using its default dual bounding strategy. The SCIP+C columns state the performance for the version of exact SCIP that generates certificates during the solution process. Since certificate printing is not supported for all dual bounding methods it uses only a subset of the dual bounding methods, contributing to its slower average speed. In an independent experiment, we estimated the average overhead for collecting dual LP solutions and file input and output to be about 7% over these test sets, which is included in the time reported for SCIP+C.
Columns inst and solved refer to the number of instances in the subset and the number of instances solved to
optimality or proven infeasibility by SCIP and SCIP+C, respectively. The last five columns report time for tightening
and checking certificate files and the sizes of the certifcate files: before tightening, after tightening, and after
running them through gzip compression.
The rows labeled all contain all instances of the easy and hard set, respectively. The solved rows comprise the instances solved by both SCIP versions. The memout rows give averages over the instances where the certificate produced by SCIP-C hit the file size limit of 10GB. The timeout row contains the remaining instances. Average running times and certificate file sizes were computed as shifted geometric means with a shift of 10 seconds and 1MB, respectively.
Finally, in the following tables, we report instance-wise results on the time, nodes, and and memory required to produce
and verify certificates. Note that file sizes are stated in kB in these tables. Additionally, the final column states
the range of primal and dual bound verified for the instance, or infeasible for the three unsatisfiable MIPs
alu10_5, alu16_2, and opti_157_0. The instance names link to gzipped certificates for the proven primal and dual
bounds stated in the last column.
Results for 56 numerically easy MIPs from Cook et al. 2013
| Instance | SCIP nodes | SCIP time (s) | SCIP+C nodes | SCIP+C time (s) | ttn time (s) | chk time (s) | raw size (kB) | tightened size (kB) | gzipped size (kB) | verified range [dualbound, primalbound] |
|---|---|---|---|---|---|---|---|---|---|---|
| 30:70:4_5:0_95:100 | 549 | 188.3 | 321 | 116.7 | 0.5 | 0.4 | 8488 | 3888 | 500 | [3, 3] |
| acc-0 | 63 | 10.7 | 63 | 11.1 | 0.1 | 0.1 | 464 | 476 | 68 | [0, 0] |
| acc-1 | 50 | 16.6 | 48 | 8.2 | 0.1 | 0.1 | 640 | 664 | 88 | [0, 0] |
| acc-2 | 65 | 28.1 | 80 | 15.9 | 0.1 | 0.1 | 824 | 744 | 96 | [0, 0] |
| air03 | 11 | 1.1 | 21 | 2.1 | 0.3 | 1.5 | 4552 | 3672 | 1128 | [340160, 340160] |
| air05 | 653 | 26.2 | 26964 | 3600.0 | 333.7 | 426.1 | 3293068 | 880364 | 316200 | [7217371784160637/274877906944, 26536] |
| bc1 | 44528 | 3600.0 | 4667 | 3600.0 | 7.1 | 81.5 | 1127016 | 468912 | 127264 | [7677622021646343/4503599627370496, 509779105561676040950537410251567595449186579772827333490018700226416986215080747642965436704233842514333106675453237057/152703338324330694722974844929002695674318892373641886227384705977426250447047855421387906252680844296971834035564360269] |
| bell3a | 65134 | 51.4 | 57522 | 60.3 | 20.0 | 21.0 | 535820 | 262324 | 25460 | [219607579/250, 219607579/250] |
| bell5 | 413280 | 222.3 | 41798 | 33.3 | 10.1 | 7.6 | 301620 | 94384 | 12088 | [28020020286/3125, 28020020286/3125] |
| bienst1 | 12730 | 35.5 | 13308 | 53.2 | 11.5 | 14.8 | 396564 | 223788 | 35936 | [187/4, 187/4] |
| bienst2 | 130318 | 494.6 | 130286 | 648.0 | 183.7 | 173.2 | 4579632 | 2586104 | 499740 | [273/5, 273/5] |
| blend2 | 8827 | 14.6 | 5023 | 14.4 | 3.3 | 3.6 | 174748 | 58192 | 18272 | [1519797/200000, 1519797/200000] |
| dano3_3 | 47 | 254.2 | 15 | 3600.0 | 0.4 | 1.5 | 77136 | 17296 | 4536 | [5068857164776087/8796093022208, 6589775704487905050110770026286969379461979446598769419744790255440005736228788540991415781158356452754204670278380019058141209966434704266446177356135997272060256249092620464129512648317495443417360287/11432318575327552324477631004975495533432641145022022129256961419612508472812576776037186454949311211259399712218543935603219466111928952029438441229110711213217229923602353376452873487155258705448000] |
| dano3_4 | 27 | 189.2 | 180 | 3600.0 | 4.0 | 29.5 | 900060 | 390344 | 103960 | [316893624371107/549755813888, 1148657247713192226440847917316426522651303694883397150229092774351699319089184144898848926431555267350949163345318733454397895738349771615655166740670704384599773207385027081234662909968064627/1991842246937951258182079729901639848023338281929864053027065669347921319647701735968702723794379481044749138761996504856056032972580515466309503291105966248265239629331492728275071099653250] |
| dano3_5 | 323 | 3231.6 | 13 | 3600.0 | 0.4 | 1.5 | 67132 | 16880 | 4432 | [5068857164776087/8796093022208, inf) |
| dcmulti | 2192 | 6.0 | 2115 | 7.5 | 1.9 | 2.4 | 63360 | 23052 | 6480 | [188182, 188182] |
| egout | 8832 | 4.0 | 8337 | 8.6 | 2.0 | 1.8 | 65532 | 19944 | 3808 | [5681007/10000, 5681007/10000] |
| eilD76 | 14341 | 95.5 | 75714 | 3600.0 | 267.2 | 425.7 | 4852640 | 1538060 | 496964 | [441702135556829/549755813888, 456246431/500000] |
| enigma | 566 | 0.2 | 566 | 0.4 | 0.1 | 0.1 | 1008 | 12 | 4 | [0, 0] |
| flugpl | 4302 | 0.5 | 4204 | 1.0 | 0.2 | 0.1 | 3376 | 676 | 168 | [1201500, 1201500] |
| gen | 1005 | 2.4 | 491 | 3.8 | 0.7 | 0.6 | 32336 | 6400 | 1100 | [56156681359/500000, 56156681359/500000] |
| gesa3 | 5633 | 54.0 | 6050 | 70.0 | 15.7 | 15.2 | 1260436 | 240188 | 52716 | [125881983070952346961091799641922753/4497223795921220170000000000, 125881983070952346961091799641922753/4497223795921220170000000000] |
| gesa3_o | 5765 | 51.8 | 5615 | 62.4 | 14.2 | 14.0 | 1162740 | 226528 | 45328 | [125881983070952346961091799641922753/4497223795921220170000000000, 125881983070952346961091799641922753/4497223795921220170000000000] |
| irp | 18730 | 164.5 | 11520 | 3600.0 | 716.2 | 633.6 | 5103920 | 2143688 | 714112 | [3337159806685465/274877906944, 2432196571/200000] |
| khb05250 | 2816 | 0.8 | 2816 | 8.2 | 3.6 | 5.9 | 34516 | 15748 | 4084 | [106940226, 106940226] |
| l152lav | 1140 | 3.7 | 1140 | 19.5 | 3.1 | 9.0 | 155552 | 55724 | 15552 | [4722, 4722] |
| lseu | 35116 | 3.0 | 33224 | 20.6 | 5.1 | 8.6 | 156168 | 64624 | 19124 | [1120, 1120] |
| markshare1_1 | 22633808 | 3600.0 | 6099519 | 2306.1 | 433.4 | 0.1 | 10597168 | 8 | 4 | [0, 8387/398953428] |
| markshare4_0 | 1297895 | 137.1 | 1297895 | 334.1 | 47.2 | 127.0 | 1489740 | 1377988 | 424928 | [1, 1] |
| mas284 | 29342 | 10.1 | 43371 | 841.8 | 15.0 | 418.8 | 2949580 | 1217628 | 332396 | [457028618411/5000000, 457028618411/5000000] |
| mas76 | 730148 | 98.9 | 374461 | 2017.3 | 132.1 | 368.8 | 10510396 | 1991808 | 558652 | [5448098594849289/137438953472, 10320245343/250000] |
| misc03 | 810 | 0.5 | 810 | 1.5 | 0.2 | 0.4 | 5924 | 2684 | 724 | [3360, 3360] |
| misc07 | 25144 | 16.8 | 26080 | 98.8 | 13.9 | 26.6 | 361292 | 175844 | 47236 | [2810, 2810] |
| mod008 | 14743 | 1.9 | 10422 | 30.0 | 4.9 | 14.6 | 235704 | 114780 | 35848 | [307, 307] |
| mod010 | 231 | 1.9 | 90045 | 1755.4 | 399.9 | 813.4 | 2427720 | 1275876 | 460824 | [6548, 6548] |
| mod011 | 33077 | 3600.0 | 26892 | 3600.0 | 497.8 | 360.5 | 8442864 | 2483232 | 702640 | [-3838009375069719/67108864, -81482099889824797479/1705625000000] |
| neos11 | 16704 | 971.5 | 17429 | 1157.6 | 35.0 | 31.8 | 2468120 | 542452 | 83992 | [9, 9] |
| neos21 | 10538 | 76.3 | 188737 | 3600.0 | 131.5 | 73.1 | 903064 | 336512 | 94948 | [6582478788394115/1125899906842624, 7] |
| neos5 | 2332860 | 1004.2 | 1864282 | 3054.7 | 420.5 | 678.8 | 10548348 | 2615112 | 616192 | [8303511812463569/562949953421312, 15] |
| neos8 | 1180 | 233.5 | 2875 | 3600.0 | 4.2 | 1.8 | 52608 | 18952 | 2540 | [-7521/2, inf) |
| neos897005 | 12 | 139.2 | 92 | 214.8 | 2.6 | 1.0 | 28876 | 7916 | 1676 | [14, 14] |
| nug08 | 15 | 14.8 | 13 | 14.3 | 0.1 | 0.2 | 1660 | 1084 | 284 | [214, 214] |
| p0033 | 1376 | 0.1 | 1381 | 0.4 | 0.1 | 0.1 | 1332 | 412 | 68 | [3089, 3089] |
| p0201 | 379 | 0.5 | 394 | 1.0 | 0.2 | 0.5 | 5172 | 2496 | 560 | [7615, 7615] |
| pk1 | 721285 | 998.8 | 763306 | 1321.5 | 194.9 | 281.8 | 8105196 | 1833072 | 490880 | [11, 11] |
| qap10 | 9 | 49.6 | 248 | 855.1 | 1.1 | 2.4 | 34396 | 12936 | 4044 | [340, 340] |
| qnet1_o | 1531 | 2.8 | 1640 | 18.3 | 3.6 | 4.2 | 160072 | 51976 | 15252 | [16029692681/1000000, 16029692681/1000000] |
| ran13x13 | 797429 | 1312.8 | 517808 | 1076.9 | 560.2 | 263.6 | 10501992 | 2635972 | 701488 | [7063355329680749/2199023255552, 3311] |
| rentacar | 61 | 12.3 | 171 | 28.4 | 0.5 | 0.7 | 16128 | 3632 | 1108 | [61302410414087064221219/2019398922240000, 61302410414087064221219/2019398922240000] |
| rgn | 5161 | 11.8 | 4181 | 15.2 | 1.3 | 1.7 | 45076 | 16312 | 2964 | [2054999981/25000000, 2054999981/25000000] |
| stein27 | 4345 | 0.6 | 4345 | 2.0 | 0.3 | 0.5 | 7892 | 3424 | 540 | [18, 18] |
| stein45 | 50206 | 13.9 | 50206 | 49.9 | 9.0 | 14.3 | 259776 | 128268 | 17484 | [30, 30] |
| swath1 | 15737 | 352.2 | 46957 | 3600.0 | 497.6 | 379.5 | 6376408 | 1984748 | 666204 | [6203932979709361/17592186044416, 7685346609/20000000] |
| swath2 | 20269 | 525.3 | 46508 | 3600.0 | 489.1 | 280.8 | 6306004 | 1471424 | 494116 | [6176554855184325/17592186044416, 7815448301/20000000] |
| vpm1 | 307794 | 48.3 | 307794 | 583.9 | 298.3 | 277.3 | 6132956 | 2997324 | 227164 | [20, 20] |
| vpm2 | 780423 | 150.9 | 466524 | 1130.6 | 554.7 | 250.7 | 10505052 | 2913832 | 477432 | [7437176122954717/562949953421312, 55/4] |
Results for 50 numerically difficult MIPs from Cook et al. 2013
| Instance | SCIP nodes | SCIP time (s) | SCIP+C nodes | SCIP+C time (s) | tightening time (s) | checking time (s) | raw size (kB) | tightened size (kB) | gzipped size (kB) | verified range [dualbound, primalbound] |
|---|---|---|---|---|---|---|---|---|---|---|
| alu10_1 | 1630142 | 3600.0 | 884351 | 2682.4 | 852.2 | 119.6 | 10499492 | 894736 | 65152 | [2544271988888683/2199023255552, inf) |
| alu10_5 | 3661 | 2.1 | 4075 | 13.4 | 2.1 | 0.2 | 111884 | 1376 | 128 | infeasible |
| alu10_7 | 5951049 | 3600.0 | 697972 | 2279.9 | 797.6 | 69.7 | 10495864 | 648644 | 76204 | [7703588447558093/70368744177664, inf) |
| alu10_8 | 5626153 | 3600.0 | 347382 | 1333.9 | 384.3 | 40.0 | 10492720 | 351032 | 53268 | [1894667407139053/17592186044416, inf) |
| alu10_9 | 6182364 | 3600.0 | 430245 | 1489.7 | 444.4 | 50.7 | 10495812 | 542072 | 89236 | [7600158590118817/70368744177664, inf) |
| alu16_1 | 2227840 | 3600.0 | 712994 | 2613.6 | 584.3 | 34.1 | 10494324 | 284984 | 50596 | [91442538121529/2147483648, inf) |
| alu16_2 | 63 | 3.3 | 63 | 0.6 | 0.1 | 0.1 | 1224 | 200 | 24 | infeasible |
| alu16_5 | 419052 | 3600.0 | 362067 | 3600.0 | 369.8 | 29.7 | 7155180 | 25860 | 3864 | [6155442237676161/16777216, inf) |
| alu16_7 | 106094 | 3600.0 | 447490 | 1734.0 | 380.6 | 0.1 | 10498452 | 364 | 52 | [439803675788291/8796093022208, inf) |
| alu16_8 | 5293381 | 3600.0 | 217446 | 807.5 | 265.8 | 0.1 | 10490860 | 304 | 44 | [-496378337539802944, inf) |
| alu16_9 | 1333351 | 3600.0 | 403935 | 2561.6 | 454.7 | 0.1 | 10493788 | 304 | 44 | [-293196311638809/536870912, inf) |
| bernd2 | 2242 | 3600.0 | 26 | 3600.0 | 0.5 | 2.5 | 179824 | 45972 | 14156 | [7086424560532419/137438953472, inf) |
| cnr_dual_mip1 | 41355 | 3064.8 | 6231 | 3600.0 | 190.2 | 123.1 | 2589424 | 417812 | 82028 | [7917105942922069/134217728, 29918065150113917/500000000] |
| cnr_heur_mip1 | 22708 | 1551.4 | 147 | 3600.0 | 2.4 | 2.4 | 76344 | 19372 | 4304 | [7897528697417871/134217728, inf) |
| dfn6_load | 19645 | 3600.0 | 11721 | 3600.0 | 3.2 | 4.4 | 432628 | 82288 | 22708 | [7625009310040395/2251799813685248, inf) |
| dfn6f_cost | 47163 | 3600.0 | 145 | 3600.0 | 0.1 | 0.1 | 9304 | 1264 | 176 | [800, inf) |
| dfn6fp_cost | 50273 | 3600.0 | 181 | 3600.0 | 0.4 | 0.4 | 25672 | 4704 | 1256 | [7036972962610833/8796093022208, inf) |
| dfn6fp_load | 21406 | 3600.0 | 90 | 3600.0 | 0.3 | 1.3 | 83056 | 23220 | 9240 | [1788417086098823/281474976710656, inf) |
| ilp_sh5 | 12021 | 3600.0 | 440 | 2512.5 | 21.8 | 902.4 | 10614908 | 9206896 | 3156672 | [5783875749308023/4398046511104, inf) |
| ilp_sh6 | 10515 | 3600.0 | 401 | 1496.9 | 20.7 | 846.2 | 10582972 | 8601020 | 3022588 | [5849140831198403/4398046511104, inf) |
| neos-1053591 | 243537 | 3030.0 | 222824 | 2525.0 | 420.6 | 162.4 | 10489696 | 2259976 | 338972 | [-4027416974360919/1099511627776, -4578643/1250] |
| neos-1062641 | 806 | 62.2 | 123 | 8.0 | 0.1 | 0.1 | 472 | 348 | 64 | [0, 0] |
| neos-1367061 | 876 | 3600.0 | 1331 | 3600.0 | 55.9 | 1.3 | 1088060 | 27636 | 2432 | [8407520959488351/268435456, inf) |
| neos-1603965 | 52 | 3600.0 | 39 | 3600.0 | 0.5 | 0.6 | 12988 | 8312 | 848 | [5194598256532745/8388608, inf) |
| neos-522351 | 31472 | 229.3 | 29914 | 281.5 | 110.7 | 54.1 | 3598564 | 757064 | 106952 | [4472769279/250000, 4472769279/250000] |
| neos-619167 | 41573 | 3600.0 | 270 | 3600.0 | 0.4 | 0.2 | 33724 | 2980 | 328 | [3342063234358541/2251799813685248, inf) |
| neos-799716 | 662 | 3600.0 | 2705 | 3600.0 | 2.6 | 0.5 | 41484 | 6636 | 1116 | [5296414793476493/1073741824, 348369865482/70625] |
| neos-839838 | 53218 | 2722.5 | 14551 | 3600.0 | 196.4 | 29.1 | 3871272 | 252008 | 85996 | [3630582153038007/268435456, 216154648669/2000] |
| neumaiershcherbina | 5 | 0.1 | 5 | 0.1 | 0.1 | 0.1 | 12 | 8 | 4 | [-2, -2] |
| normalized-aim-200-1_6-yes1-3 | 13271 | 26.1 | 13795 | 73.8 | 6.7 | 1.3 | 325452 | 10992 | 1528 | [200, 200] |
| npmv07 | 20 | 3600.0 | 2 | 3600.0 | 2.0 | 5.5 | 66420 | 63376 | 14288 | [3434403169950365/32768, inf) |
| ns1629327 | 40193 | 1491.5 | 5091 | 3600.0 | 20.2 | 10.4 | 973328 | 71656 | 21848 | [-1582671328730437/140737488355328, inf) |
| ns1770598 | 9001 | 38.9 | 4841 | 593.2 | 23.9 | 321.2 | 10499104 | 3113440 | 756040 | [3241814004977687/137438953472, 89834353929974953382923610590747242935159173444391256802327665676316204815539763065760508542311794737181077558299227732351929735234406318219580335312630112135112834127664065830445059526404979965208963482226734722452432847263457002478183396711660658623847481278452527348213655658438044433468661993638518105568090351661835885769730044066113913158513874849135968248655381190429834311692704341489870113679068570471340793946694720426548997505133020895986294235469578597302314870180978311152054113868383892871889291430282083384972182392282800955416024898881798840342404506991103026319584359201785114855735141968909416466402913845247410157956085184276012449116579667484276143004298513396703306238042628011167294091326596944161325831198051010532298825922441801019577096143193949495154777436424621213630190657907558014619753764602511056621703405246227737829971801164951581182490885308624847792961577888932676258996667732517892019761818717196788874031766211756355294319100587231867124885716329615143636553299490438185336467581755709922430092975524618744393431202707607875364476611938937301923730572577071153942267024578221/3194161461050852704307502106391729430472763027318339131230473113410199213995688417624274320334180040072604686198494803919807307749535485168880498458771658080453522816508393610766056099212203609043478063738566604237910046315125165002373275256325064584772511348671803693358961156758335269966847078607611945124053145392280607367957523780425969306541828046604915192932150958146414814614413300571309009293932930914092300083963547165652785481808562307502710207175132794694920323971328968051919139441766101658693433983540758019074009411782956820685080520790119285502873753055200335575839504129464533903054076074508282119762345100929212285323833752916857712903989368798284908561003651862813379956142883569002813457643252841356061740351050779954765860039064193111701481710707026489021488761153826695974721208846687378979946563889508432192891950006366834486731973726433522234484203441294758974332258312511663359407549704983196821429242584727458778058579001134641216832353890745953002987110739163093245516357022930118771766825092401115417441838322217268594393508608026273582431962716864188177976509106072401881700000000 |
| ns1859355 | 193802 | 285.3 | 21074 | 3600.0 | 1.0 | 0.1 | 29132 | 1436 | 192 | [0, inf) |
| ns1866531 | 168 | 11.1 | 183 | 10.5 | 0.1 | 0.1 | 2564 | 740 | 152 | [10, 10] |
| ns1900685 | 34498 | 6.5 | 29816 | 31.8 | 8.6 | 5.3 | 191872 | 48920 | 7540 | [3453, 3453] |
| ns1925218 | 319880 | 3600.0 | 128811 | 3600.0 | 238.1 | 0.4 | 7023868 | 4244 | 688 | [6931675893999861/34359738368, inf) |
| ns2017839 | 2188502 | 3600.0 | 1 | 3600.0 | 0.6 | 1.6 | 19964 | 20784 | 3244 | [4929640932321283/64, inf) |
| ns2080781 | 72617 | 3600.0 | 27922 | 3600.0 | 6.4 | 0.1 | 393360 | 172 | 20 | [0, inf) |
| opti_157_0 | 119 | 2.9 | 3 | 1.2 | 0.2 | 0.2 | 2704 | 2780 | 440 | infeasible |
| p4 | 1 | 184.8 | 938 | 3600.0 | 66.7 | 18.0 | 1193552 | 73520 | 16308 | [4014352503518715/8, inf) |
| prodplan1 | 1 | 3600.0 | 2 | 3600.0 | 2.7 | 7.4 | 96728 | 91208 | 16412 | [-7325416861334513/134217728, inf) |
| prodplan2 | 4 | 2.5 | 27 | 1401.1 | 0.9 | 0.7 | 28984 | 9532 | 1624 | [-185497557187228867655290921686042318951/774845425502612687685652190000000, -185497557187228867655290921686042318951/774845425502612687685652190000000] |
| ran14x18 | 2188502 | 3600.0 | 4620 | 3600.0 | 3.5 | 3.8 | 264592 | 35316 | 11064 | [1907094538583883/549755813888, 4864] |
| sp98ir | 24636 | 298.4 | 28091 | 3600.0 | 77.9 | 61.5 | 1033008 | 164252 | 59884 | [7318013340697871/33554432, 225571452] |
| tkat3K | 119250 | 266.9 | 6118 | 60.1 | 12.5 | 14.3 | 574632 | 217744 | 52440 | [47728181/10, 47728181/10] |
| tkat3T | 8074 | 29.6 | 7984 | 115.2 | 20.8 | 36.4 | 1087572 | 547192 | 143372 | [22259567/4, 22259567/4] |
| tkat3TV | 15533 | 73.2 | 6370 | 104.0 | 18.7 | 28.3 | 891956 | 417988 | 109240 | [167767973/20, 167767973/20] |
| tkatTV5 | 2432005 | 3600.0 | 40038 | 2036.3 | 148.5 | 156.3 | 1287316 | 526508 | 181960 | [1124705769/40, 1124705769/40] |
| x01 | 1 | 58.6 | 10 | 3600.0 | 1.5 | 2.3 | 35196 | 19936 | 4580 | [5522129193649787/262144, inf) |