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Computer Science

Generating Difficult SAT Instances by Preventing Triangles

Profile image of Steven PrestwichSteven Prestwich

2019, arXiv (Cornell University)

Abstract

When creating benchmarks for SAT solvers, we need SAT instances that are easy to build but hard to solve. A recent development in the search for such methods has led to the Balanced SAT algorithm, which can create k-SAT instances with m clauses of high difficulty, for arbitrary k and m. In this paper we introduce the No-Triangle SAT algorithm, a SAT instance generator based on the cluster coefficient graph statistic. We empirically compare the two algorithms by fixing the arity and the number of variables, but varying the number of clauses. The hardest instances that we find are produced by No-Triangle SAT. Furthermore, difficult instances from No-Triangle SAT have a different number of clauses than difficult instances from Balanced SAT, potentially allowing a combination of the two methods to find hard SAT instances for a larger array of parameters.

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  14. 100 100 43 24919 1896 84 10546 129 2.03 20400 2.05 0.137 0.150 710 100 100 41 29599 2107 86 10939 132 2.02 20321 2.05 0.140 0.153 720 100 100 47 40389 3364 86 11231 141 2.01 20240 2.04 0.141 0.153 730 100 100 47 54922 6300 88 11499 136 2.01 20157 2.03 0.140 0.153 740 100 100 66 74831 5181 94 11908 146 2.00 20088 2.02 0.140 0.155 750 100 100 42 104146 7522 99 12103 148 1.99 20400 3.99 0.141 0.155 760 100 100 40 121366 7526 99 12376 144 1.98 19930 2.01 0.143 0.156 770 100 100 36 189630 19232 104 12951 155 1.97 20061 3.96 0.143 0.157 780 100 100 87 515266 41407 100 13284 158 1.97 20008 3.96 0.146 0.158 790 100 100 185 366716 28057 107 13580 164 1.96 19737 1.98 0.145 0.161 800 100 98 105 704235 82068 108 13855 167 1.96 19678 1.98 0.147 0.159 810 100 94 196 1171447 74977 113 14221 168 1.95 19616 1.97 0.149 0.160 820 100 90 127 1678682 93907 128 14662 180 1.95 19565 1.97 0.147 0.160 830 100 85 81 2167970 92260 127 14899 179 1.94 19710 3.93 0.150 0.161 840 100 71 153 2279300 125413 128 15302 186 1.93 19457 1.96 0.148 0.162 850 100 48 335 2768423 81628 129 15731 203 1.93 19608 3.92 0.151 0.162 860 100 38 459 2818349 100712 132 15943 193 1.93 19346 1.94 0.152 0.166 870 100 29 704 2820201 104830 139 16379 195 1.92 19300 1.94 0.152 0.163 880 100 25 1208 2220243 72721 135 16804 192 1.92 19262 1.94 0.154 0.166 890 100 17 871 2386582 63759 147 17221 204 1.91 19215 1.93 0.155 0.167 900 100 3 5678 2063986 47589 143 17480 205 1.91 19175 1.93 0.154 0.167 910 100 4 6457 1924045 56304 135 17780 214 1.90 19126 1.92 0.156 0.168 920 100 1 6089 1640938 65832 150 18185 214 1.90 19091 1.92 0.158 0.168 930 100 1 3585 1501617 41479 151 18603 211 1.90 19039 1.92 0.159 0.169 940 100 0 5379 1406421 37810 151 18979 228 1.89 19010 1.91 0.159 0.171 950 100 0 4606 1209729 26448 165 19170 222 1.89 18980 1.91 0.161 0.172 960 100 0 3501 1216098 35770 171 19737 232 1.89 18925 1.90 0.162 0.173 970 100 0 3019 1056247 36095 175 20166 230 1.88 18904 1.90 0.162 0.174 980 100 0 4140 920851 15625 174 20613 239 1.88 18885 1.90 0.164 0.174 990 100 0 3081 861910 21373 180 21280 242 1.88 18839 1.89 0.164 0.175 1000 100 0 3721 780571 15328 182 21436 247 1.87 18807 1.89 0.163 0.177 Table 4. Full results of the Random SAT algorithm for N=225. 675 25 25 58 2056 114 65 1977 93 2.18 5661 4.16 0.124 0.135 685 25 25 59 2195 172 69 2030 103 2.17 5438 2.18 0.127 0.136 695 25 25 65 2725 495 66 2123 106 2.16 5610 4.14 0.126 0.134 705 25 25 63 3117 394 60 2129 104 2.15 5585 4.13 0.126 0.137 715 25 25 52 2884 340 79 2247 112 2.13 5562 4.12 0.127 0.135 725 25 25 66 3318 582 70 2269 108 2.13 5338 2.14 0.127 0.137 735 25 25 57 5029 579 77 2311 108 2.12 5512 4.11 0.128 0.136 745 25 25 54 4302 663 77 2382 115 2.10 5292 2.13 0.129 0.138 755 25 25 47 4072 559 74 2460 115 2.10 5274 2.12 0.130 0.137 765 25 25 57 4857 452 78 2516 125 2.09 5247 2.11 0.127 0.138 775 25 25 54 5673 812 80 2578 121 2.08 5222 2.10 0.127 0.138 785 25 25 62 5953 863 92 2729 130 2.07 5207 2.10 0.131 0.139 795 25 25 142 11769 1099 84 2706 137 2.06 5185 2.08 0.130 0.137 805 25 25 52 17170 2303 88 2839 129 2.06 5168 2.07 0.132 0.138 815 25 25 75 18034 2358 87 2870 132 2.05 5149 2.07 0.131 0.142 825 25 25 103 32529 15195 88 2896 140 2.04 5129 2.06 0.130 0.140 835 25 25 61 44702 14001 99 2939 146 2.04 5109 2.05 0.135 0.141 845 25 25 64 34479 6863 108 3108 146 2.03 5095 2.05 0.133 0.143 855 25 25 174 51101 13828 104 3108 146 2.02 5076 2.04 0.134 0.143 865 25 25 127 105208 30884 114 3275 149 2.02 5054 2.03 0.135 0.141 875 25 25 235 125632 35415 108 3340 150 2.01 5048 2.03 0.134 0.143 885 25 25 102 137376 26304 120 3437 153 2.00 5027 2.02 0.135 0.145 895 25 24 249 411540 177642 100 3513 162 2.00 5014 2.02 0.137 0.145 905 25 25 820 432532 186807 119 3520 158 1.99 4996 2.01 0.139 0.144 915 25 25 1949 617802 137764 120 3612 162 1.98 4984 2.00 0.136 0.146 925 25 21 1521 1636535 209420 118 3655 164 1.98 4969 1.99 0.139 0.146 935 25 20 3247 1391203 170795 129 3705 175 1.97 4951 1.99 0.139 0.147 945 25 19 3128 1291132 166016 129 3759 178 1.97 4945 1.98 0.140 0.148 955 25 13 3834 2271068 245304 138 3917 176 1.97 4933 1.98 0.140 0.151 965 25 11 1616 1918131 148155 126 4010 178 1.96 4919 1.98 0.142 0.151 975 25 7 603 1856267 138588 137 4112 199 1.96 4906 1.97 0.143 0.148 985 25 3 10011 2096434 192684 144 4143 189 1.95 4892 1.97 0.142 0.151 995 25 7 8391 2047771 142322 150 4212 202 1.95 4881 1.96 0.143 0.152 1005 25 2 5771 1511449 157374 151 4313 197 1.94 4873 1.96 0.145 0.153 1015 25 1 20027 1506162 108728 156 4403 192 1.94 4860 1.95 0.145 0.154 1025 25 1 28137 1398087 106039 153 4476 197 1.93 4852 1.95 0.144 0.154 1035 25 1 20890 1268619 113910 159 4502 204 1.93 4842 1.94 0.143 0.153 1045 25 0 15987 1074700 87185 158 4568 211 1.92 4827 1.94 0.147 0.155 1055 25 0 9698 813706 63433 167 4806 229 1.92 4826 1.94 0.147 0.155 1065 25 0 12595 831258 75407 166 4903 221 1.92 4814 1.93 0.148 0.156 1075 25 0 14172 715952 80468 158 4879 217 1.91 4805 1.93 0.148 0.158 1085 25 0 10164 622022 85008 172 5055 222 1.91 4800 1.93 0.151 0.156 1095 25 0 9596 575054 36154 186 5152 234 1.91 4792 1.92 0.149 0.158 1105 25 0 5925 515638 40923 178 5298 239 1.91 4782 1.92 0.149 0.159 1115 25 0 8207 460458 34261 188 5272 241 1.90 4773 1.92 0.150 0.163 1125 25 0 7249 486975 44923 189 5448 249 1.90 4765 1.91 0.153 0.161 Table 5. Full results of the Balanced SAT algorithm for N=175.
  15. 2709697 229764 0 60 3 1.97 19705 1.98 0.136 0.148 585 100 100 735 10110047 596479 0 69 3 1.95 19604 1.97 0.133 0.148 595 100 97 640 27515142 2944991 0 73 3 1.95 19511 1.96 0.137 0.150 605 100 83 505 49015355 2761113 0 73 4 1.94 19431 1.95 0.141 0.151 615 100 51 1524 73100116 1636515 0 82 3 1.93 19362 1.94 0.140 0.152 625 100 28 2123 77268614 1265079 0 79 3 1.92 19291 1.93 0.141 0.154 635 100 11 4368 78283199 1095609 0 78 3 1.92 19201 1.93 0.141 0.154 645 100 2 407930 68378942 991447 0 78 3 1.91 19116 1.92 0.144 0.154 655 100 3 109013 49640591 675064 0 77 4 1.90 19082 1.91 0.145 0.155 665 100 0 322458 42706631 579850 0 91 3 1.90 19000 1.90 0.148 0.156 675 100 0 283517 37265554 484737 0 77 3 1.89 18942 1.90 0.147 0.160 685 100 0 263031 33057198 425587 0 94 4 1.89 18900 1.89 0.148 0.160 695 100 0 234532 30268101 373855 0 102 4 1.88 18824 1.89 0.151 0.159 705 100 0 197172 26637122 330557 0 125 4 1.88 18800 1.88 0.151 0.163 715 100 0 155588 23258828 294058 0 131 4 1.87 18743 1.88 0.154 0.164 725 100 0 118915 20551706 267380 0 109 4 1.87 18700 1.87 0.154 0.167 735 100 0 91289 18750455 243652 0 138 5 1.86 18696 1.87 0.156 0.169 745 100 0 88356 15771666 219329 0 109 4 1.86 18600 1.86 0.159 0.167 755 100 0 70045 14493008 213955 0 125 4 1.86 18600 1.86 0.159 0.169 765 100 0 68095 11546446 187105 0 121 3 1.85 18555 1.86 0.160 0.172 775 100 0 50243 8782812 139058 0 125 4 1.85 18500 1.85 0.162 0.172 785 100 0 39062 7598549 141292 0 122 5 1.85 18500 1.85 0.162 0.172 795 100 0 37432 6058963 106575 0 138 5 1.84 18491 1.85 0.167 0.174 805 100 0 28661 5200819 103522 0 136 4 1.84 18400 1.84 0.166 0.176 815 100 0 26623 4464272 83275 0 120 5 1.84 18400 1.84 0.168 0.179 825 100 0 19140 3588886 60989 0 167 6 1.84 18400 1.84 0.171 0.178 835 100 0 15780 2956467 60339 0 130 5 1.83 18331 1.84 0.172 0.180 845 100 0 14869 2566463 43399 0 142 5 1.83 18300 1.83 0.171 0.181 855 100 0 13580 2271585 41845 0 169 5 1.83 18300 1.83 0.176 0.183 865 100 0 12249 2011425 35015 0 144 5 1.83 18300 1.83 0.174 0.186 875 100 0 11322 1834440 29685 0 202 7 1.82 18232 1.83 0.178 0.187 Table 6. Full results of the Balanced SAT algorithm for N=200.
  16. 735 25 25 1903 3374264 1167695 0 24 4 2.07 5182 2.08 0.116 0.124 745 25 25 21436 9952167 2311298 0 24 4 2.06 5153 2.07 0.116 0.127 755 25 25 31045 33682937 7455734 0 32 4 2.05 5135 2.06 0.119 0.124 765 25 25 47021 50270191 20427104 0 30 3 2.04 5112 2.05 0.118 0.127 775 25 24 147821 262667419 65268387 0 33 4 2.03 5096 2.04 0.119 0.128 785 25 19 53865 437422958 49715216 0 23 3 2.02 5071 2.03 0.119 0.129 .132 885 25 0 2974408 147699827 9228776 0 40 4 1.96 4900 1.96 0.126 0.133 895 25 0 3025087 128306110 8337797 0 34 3 1.95 4879 1.96 0.126 0.134 905 25 0 1895433 94539106 5306052 0 43 5 1.94 4872 1.95 0.127 0.133 915 25 0 1681623 72389634 4248709 0 47 5 1.94 4856 1.95 0.129 0.137 925 25 0 1356969 51835486 3698778 0 38 4 1.93 4849 1.94 0.129 0.136 935 25 0 1062796 40665858 3185711 0 39 4 1.93 4826 1.94 0.130 0.137 945 25 0 983563 32042746 1704670 1 58 5 1.93 4825 1.93 0.129 0.138 955 25 0 816725 28099763 1689906 0 46 6 1.92 4807 1.93 0.131 0.137 965 25 0 791516 26757741 1500555 0 43 5 1.92 4800 1.92 0.133 0.139 975 25 0 711557 24236465 1266714 0 45 5 1.91 4794 1.92 0.132 0.140 985 25 0 563889 19713949 1002265 0 52 5 1.91 4775 1.91 0.134 0.140 995 25 0 479524 16308456 830998 0 62 5 1.91 4775 1.91 0.134 0.141 1005 25 0 454633 14960311 815451 0 60 4 1.90 4762 1.91 0.135 0.141 1015 25 0 412768 13475652 673255 0 52 5 1.90 4750 1.90 0.133 0.143 1025 25 0 394527 12194125 554821 0 42 4 1.90 4750 1.90 0.137 0.143 1035 25 0 339646 10969882 510657 0 39 4 1.89 4742 1.90 0.138 0.144 1045 25 0 338860 10427428 498608 0 63 7 1.89 4725 1.89 0.138 0.145 1055 25 0 283251 9401119 493375 0 62 6 1.89 4725 1.89 0.139 0.146 1065 25 0 271950 8891702 446883 0 62 6 1.88 4724 1.89 0.141 0.148 1075 25 0 285650 8330238 401701 0 69 6 1.88 4705 1.89 0.141 0.148 1085 25 0 258551 7828755 379711 0 57 6 1.88 4700 1.88 0.141 0.148 1095 25 0 200311 7219457 339582 0 55 6 1.88 4700 1.88 0.143 0.148 1105 25 0 219805 6912573 337282 0 53 6 1.87 4699 1.88 0.144 0.150 1115 25 0 176719 6500832 306719 1 69 6 1.87 4678 1.88 0.146 0.151 1125 25 0 202431 6321020 318091 0 74 7 1.87 4675 1.87 0.146 0.152 Table 8. Full results of the No-Triangle SAT algorithm for N=175. .91 19192 1.92 0.057 5815 0.060 615 100 100 64 276891 13203 0 83 3 1.91 19100 1.91 0.056 5786 0.059 625 100 100 45 388124 23571 0 94 3 1.90 19023 1.91 0.056 5766 0.059 635 100 100 47 795201 39282 0 102 4 1.89 18999 1.90 0.056 5800 0.059 645 100 100 105 1072264 64565 0 117 5 1.89 18908 1.90 0.058 5891 0.060 655 100 100 225 1776136 107172 0 125 4 1.88 18895 1.89 0.057 5873 0.061 665 100 100 320 3252788 232143 0 129 4 1.88 18800 1.88 0.057 5869 0.060 675 100 100 409 6499525 404988 0 107 5 1.87 18799 1.88 0.058 5877 0.060 685 100 100 1134 13975071 2984858 0 121 4 1.87 18703 1.88 0.058 5906 0.060 695 100 92 851 29184354 2433899 0 118 4 1.87 18700 1.87 0.058 5982 0.062 705 100 73 958 51389353 1913059 0 150 6 1.86 18686 1.87 0.059 6103 0.062 715 100 53 665 61205027 1165890 0 174 5 1.86 18600 1.86 0.060 6129 0.063 725 100 36 4078 61913402 1064983 0 177 5 1.86 18600 1.86 0.060 6141 0.063 735 100 14 5328 66645732 1010250 0 169 5 1.85 18573 1.86 0.060 6180 0.064 745 100 12 148635 58297842 799695 0 163 5 1.85 18500 1.85 0.061 6253 0.064 755 100 3 154118 53801432 689309 0 158 5 1.85 18500 1.85 0.062 6394 0.066 765 100 1 128819 46673669 578716 0 157 5 1.85 18500 1.85 0.063 6495 0.066 775 100 1 341182 42206118 508810 0 154 5 1.84 18408 1.85 0.064 6537 0.068 785 100 0 317253 38202177 469590 0 189 6 1.84 18400 1.84 0.064 6586 0.068 795 100 1 303081 34972710 407504 0 171 6 1.84 18400 1.84 0.065 6671 0.068 805 100 0 260478 32099887 385377 0 163 5 1.84 18400 1.84 0.066 6795 0.070 815 100 0 240068 29659596 345337 0 162 6 1.83 18389 1.84 0.068 6964 0.071 825 100 0 187611 26867156 306272 0 170 8 1.83 18300 1.83 0.069 7073 0.073 835 100 0 170978 24218808 293286 0 187 5 1.83 18300 1.83 0.070 7147 0.073 845 100 0 170620 22976290 265154 0 175 4 1.83 18300 1.83 0.070 7234 0.074 855 100 0 117120 20380808 255330 0 216 6 1.83 18300 1.83 0.071 7344 0.076 865 100 0 119202 18756778 236289 0 181 5 1.82 18203 1.83 0.073 7475 0.076 875 100 0 78535 16117067 222456 0 246 8 1.82 18200 1.82 0.076 7726 0.079 Table 9. Full results of the No-Triangle SAT algorithm for N=200.
  17. 100 0 440363 53499751 727897 0 177 6 1.85 18500 1.85 0.059 0.061 970 100 0 382836 49044965 655341 0 144 5 1.85 18500 1.85 0.059 0.062 980 100 0 380332 45898581 574910 0 172 6 1.85 18500 1.85 0.060 0.062 990 100 0 344509 41741061 537047 0 175 5 1.85 18500 1.85 0.061 0.063 1000 100 0 325419 38779506 466097 0 235 7 1.84 18485 1.85 0.062 0.065 Table 10. Full results of the No-Triangle SAT algorithm for N=225. 675 25 25 68 10099 2231 0 30 3 2.10 5252 2.11 0.060 1504 0.061 685 25 25 86 9247 1101 0 24 3 2.08 5224 2.09 0.058 1473 0.059 695 25 25 74 13100 2818 0 39 3 2.07 5197 2.08 0.057 1448 0.058 705 25 25 57 17163 2175 0 34 5 2.06 5173 2.07 0.057 1427 0.058 715 25 25 67 22706 3682 0 30 4 2.05 5149 2.06 0.056 1406 0.057 725 25 25 94 41106 6085 0 37 4 2.04 5123 2.05 0.055 1389 0.056 735 25 25 545 77983 12472 0 26 3 2.03 5096 2.04 0.054 1375 0.056 745 25 25 226 70447 12598 0 33 4 2.02 5073 2.03 0.054 1359 0.055 755 25 25 1005 156997 21906 0 39 4 2.02 5050 2.02 0.054 1358 0.055 765 25 25 664 194346 27684 0 51 6 2.01 5026 2.02 0.053 1341 0.054 775 25 25 325 399061 182180 0 40 4 2.00 5004 2.01 0.052 1322 0.054 785 25 25 516 526834 104346 0 32 3 1.99 4986 2.00 0.051 1305 0.053 795 25 25 367 811282 179971 0 41 3 1.98 4973 1.99 0.051 1291 0.052 805 25 25 1948 889473 129537 0 34 4 1.98 4950 1.98 0.051 1282 0.052 815 25 25 1830 2743849 511557 0 42 3 1.97 4930 1.98 0.050 1279 0.052 825 25 25 1012 4418564 668641 0 43 4 1.96 4923 1.97 0.051 1289 0.052 835 25 25 6416 7745404 1182193 0 57 5 1.96 4900 1.96 0.050 1277 0.052 845 25 25 251756 20618063 3119407 0 35 4 1.95 4887 1.96 0.050 1266 0.051 855 25 25 28762 42204876 14326970 0 45 7 1.95 4875 1.95 0.050 1259 0.051 865 25 25 7148 65319255 18582405 0 42 4 1.94 4857 1.95 0.049 1248 0.051 875 25 22 156150 306679993 63315167 0 38 4 1.94 4850 1.94 0.049 1247 0.051 885 25 13 1591849 788038314 60397295 0 42 4 1.93 4830 1.94 0.049 1243 0.050 895 25 11 17476 772210331 52629470 0 46 5 1.93 4825 1.93 0.049 1244 0.050 905 25 3 1343366 858005875 44622644 0 54 5 1.92 4812 1.93 0.050 1258 0.051 915 25 4 604105 718516087 38708440 0 54 6 1.92 4800 1.92 0.049 1248 0.051 925 25 0 21929255 748475514 34726145 0 62 9 1.91 4799 1.92 0.049 1247 0.051 935 25 0 24184043 704517010 31124675 0 49 4 1.91 4777 1.92 0.049 1241 0.050 945 25 0 22659603 638177233 30156699 0 56 4 1.91 4775 1.91 0.048 1236 0.050 955 25 0 16249420 609166985 28810905 0 50 7 1.90 4769 1.91 0.049 1240 0.051 965 25 1 4962911 511347497 24479783 0 46 4 1.90 4750 1.90 0.049 1246 0.051 975 25 0 12643543 507406618 24573862 0 54 6 1.90 4750 1.90 0.050 1267 0.051 985 25 0 10500301 410234852 21394493 0 54 6 1.89 4748 1.90 0.050 1267 0.051 995 25 0 5730065 318041287 18685310 0 70 5 1.89 4726 1.90 0.050 1265 0.051 1005 25 0 6884868 283880075 15595774 0 52 4 1.89 4725 1.89 0.050 1265 0.051 1015 25 0 6235113 250031251 14612456 0 65 6 1.89 4725 1.89 0.050 1267 0.051 1025 25 0 5019300 196195331 11284038 0 70 7 1.88 4715 1.89 0.050 1267 0.052 1035 25 0 2866480 163642207 9445875 0 67 5 1.88 4700 1.88 0.050 1270 0.052 1045 25 0 2581415 126487598 7529615 0 66 6 1.88 4700 1.88 0.051 1288 0.052 1055 25 0 2578222 100366781 6906600 0 85 8 1.88 4700 1.88 0.051 1308 0.053 1065 25 0 2043869 73964238 4423565 0 73 6 1.88 4700 1.88 0.052 1313 0.053 1075 25 0 1348411 60677064 3895537 0 66 5 1.87 4675 1.87 0.052 1314 0.054 1085 25 0 1342830 55097337 5192919 0 75 7 1.87 4675 1.87 0.051 1320 0.054 1095 25 0 1120126 43772330 3239436 0 64 6 1.87 4675 1.87 0.052 1324 0.054 1105 25 0 979929 39216020 2521350 0 76 7 1.87 4675 1.87 0.052 1326 0.054 1115 25 0 946247 32348483 1819514 0 70 6 1.87 4675 1.87 0.053 1343 0.054 1125 25 0 803117 26630337 1349768 0 94 7 1.86 4657 1.87 0.054 1375 0.056

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