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https://github.com/pgdr/cc-degeneracy
https://github.com/pgdr/cc-degeneracy
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- Host: GitHub
- URL: https://github.com/pgdr/cc-degeneracy
- Owner: pgdr
- Created: 2023-05-02T08:24:36.000Z (over 1 year ago)
- Default Branch: master
- Last Pushed: 2023-05-16T09:50:07.000Z (over 1 year ago)
- Last Synced: 2023-09-18T16:51:41.937Z (about 1 year ago)
- Language: Python
- Size: 6.11 MB
- Stars: 0
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# FPT in P for c-closure, parameterized by degeneracy
We compute the c-closure of a graph parameterized by its degeneracy.
If the degeneracy is smaller than the c-closure, we compute the
c-closure, and output a witness in P-parameterized time $O(d^3 \cdot n)$.
In the case where the c-closure is smaller than the degeneracy, we fall
back to a quadratic time algorithm, $O(d \cdot n^2)$.
However, our analyses show that in general, degeneracy is in fact
smaller than the c-closure.
In fact, the best known algorithm for computing the c-closure is in
$O(c\cdot n^\omega)$ time, where $\omega$ is the matrix multiplication
constant. This means that if $d \in O(n^{0.37})$, then this algorithm
is faster than the best known algorithm. In the case where degeneracy
is smaller than the c-closure, as we see below it is in most cases, the
$O(d^3 \cdot n)$ is much faster than $O(c \cdot n^\omega)$ for all our
dataset.
## Some experiments
We ran our algorithm on all the datasets from
[_network-corpus_](https://github.com/microgravitas/network-corpus/), the results are
below. We note that only 6 of 216 graphs used more than 5 seconds, while
195 graphs were solved in less than a second (in Python).
Only 2 of 216 graphs used more than 30 seconds
* `dogster_friendships.txt.gz` (48 sec) ($n=426821, m=8546581, d=249, c=32804$)
* `livemocha` (53 sec) ($n=104104, m=2193083, d=92, c=1129$)
In this dataset, 9 / 216 networks have $c < d$, whereas in 207, we have $c \geq d$.
This means that $207/216 \approx 0.95833... > 95.8\%$ are solved completely by _Case 1_.
### Analysis
| name | n | m | deg | c-c | time (ms) | smaller |
|------------------------------------------|---------|---------|-------|-------|-----------|-----------|
| `iscas89-s27` | 9 | 8 | 1 | 2 | 0.2 | d |
| `wafa-padgett` | 15 | 27 | 3 | 4 | 0.2 | d |
| `BioGrid-Human-Immunodeficiency-Virus-2` | 19 | 15 | 1 | 2 | 0.2 | d |
| `wafa-hightech` | 22 | 159 | 12 | 17 | 0.7 | d |
| `wafa-ceos` | 26 | 93 | 5 | 7 | 0.3 | d |
| `BioGrid-Dictyostelium-Discoideum-Ax4` | 27 | 20 | 1 | 2 | 0.2 | d |
| `seventh-graders` | 29 | 376 | 20 | 18 | 3.6 | c |
| `karate` | 34 | 78 | 4 | 7 | 0.3 | d |
| `windsurfers` | 43 | 336 | 11 | 17 | 3.8 | d |
| `BioGrid-Glycine-Max` | 44 | 39 | 2 | 3 | 0.2 | d |
| `wafa-eies` | 45 | 652 | 24 | 31 | 8.8 | d |
| `dutch-textiles` | 48 | 134 | 7 | 12 | 0.3 | d |
| `bergen` | 53 | 272 | 9 | 13 | 0.9 | d |
| `iscas89-s208.1` | 61 | 67 | 2 | 3 | 0.3 | d |
| `dolphins` | 62 | 159 | 4 | 5 | 0.5 | d |
| `BioGrid-Emericella-Nidulans-Fgsc-A4` | 64 | 62 | 2 | 3 | 0.3 | d |
| `train_bombing` | 64 | 243 | 10 | 9 | 1.4 | c |
| `pollination-tenerife` | 68 | 129 | 4 | 11 | 0.4 | d |
| `BioGrid-Cricetulus-Griseus` | 69 | 57 | 1 | 2 | 0.3 | d |
| `Noordin-terror-relation` | 70 | 251 | 11 | 11 | 1 | d |
| `mg_watchmen` | 76 | 201 | 7 | 6 | 0.7 | c |
| `lesmiserables` | 77 | 254 | 9 | 9 | 0.8 | d |
| `mg_godfatherII` | 78 | 219 | 8 | 9 | 0.6 | d |
| `iscas89-s298` | 92 | 131 | 2 | 6 | 0.4 | d |
| `mg_forrestgump` | 94 | 271 | 8 | 4 | 1.7 | c |
| `win95pts` | 99 | 112 | 2 | 4 | 0.3 | d |
| `iscas89-s344` | 100 | 122 | 2 | 5 | 0.3 | d |
| `iscas89-s641` | 100 | 144 | 3 | 7 | 0.4 | d |
| `movies` | 101 | 192 | 3 | 6 | 0.5 | d |
| `iscas89-s349` | 102 | 127 | 2 | 5 | 0.4 | d |
| `polbooks` | 105 | 441 | 6 | 16 | 1 | d |
| `mg_casino` | 109 | 326 | 9 | 8 | 0.9 | c |
| `word_adjacencies` | 112 | 425 | 6 | 14 | 1 | d |
| `iscas89-s386` | 114 | 200 | 3 | 13 | 0.5 | d |
| `hypertext_2009` | 114 | 2196 | 28 | 54 | 16.8 | d |
| `StackOverflow-tags` | 115 | 245 | 6 | 7 | 0.6 | d |
| `football` | 115 | 613 | 8 | 10 | 5 | d |
| `iscas89-s382` | 116 | 168 | 2 | 5 | 0.5 | d |
| `BioGrid-Human-Herpesvirus-5` | 121 | 107 | 1 | 2 | 0.4 | d |
| `iscas89-s400` | 121 | 182 | 2 | 5 | 0.6 | d |
| `Noordin-terror-loc` | 127 | 190 | 3 | 7 | 0.5 | d |
| `iscas89-s420.1` | 129 | 145 | 2 | 3 | 0.4 | d |
| `Noordin-terror-orgas` | 129 | 181 | 3 | 11 | 0.5 | d |
| `iscas89-s444` | 134 | 206 | 2 | 5 | 0.5 | d |
| `BioGrid-Hepatitus-C-Virus` | 136 | 134 | 1 | 2 | 0.4 | d |
| `iscas89-s713` | 137 | 180 | 3 | 7 | 0.5 | d |
| `capitalist` | 139 | 1071 | 19 | 35 | 11 | d |
| `american_revolution` | 141 | 160 | 3 | 10 | 0.4 | d |
| `foodweb-otago` | 141 | 832 | 14 | 36 | 3.2 | d |
| `BioGrid-Canis-Familiaris` | 143 | 125 | 2 | 3 | 0.4 | d |
| `iscas89-s526n` | 159 | 268 | 3 | 9 | 0.6 | d |
| `iscas89-s526` | 160 | 270 | 3 | 9 | 0.6 | d |
| `iscas89-s510` | 172 | 251 | 2 | 3 | 0.6 | d |
| `BioGrid-Human-Papillomavirus-16` | 173 | 186 | 2 | 18 | 0.5 | d |
| `BioGrid-Human-Herpesvirus-1` | 178 | 208 | 3 | 11 | 0.5 | d |
| `CoW-interstate` | 182 | 319 | 4 | 13 | 0.7 | d |
| `jazz` | 199 | 2742 | 29 | 42 | 17.5 | d |
| `mousebrain` | 213 | 16089 | 111 | 179 | 37.5 | d |
| `residence_hall` | 218 | 1839 | 11 | 18 | 18.6 | d |
| `airlines` | 235 | 1297 | 13 | 43 | 8 | d |
| `sp_data_school_day_2` | 238 | 5539 | 33 | 54 | 36.2 | d |
| `iscas89-s820` | 239 | 480 | 3 | 16 | 1.1 | d |
| `rhesusbrain` | 242 | 3054 | 19 | 38 | 23.4 | d |
| `iscas89-s832` | 245 | 498 | 3 | 19 | 1.2 | d |
| `BioGrid-Danio-Rerio` | 261 | 266 | 3 | 4 | 0.6 | d |
| `iscas89-s838.1` | 265 | 301 | 2 | 3 | 0.7 | d |
| `haggle` | 275 | 2124 | 39 | 41 | 6.6 | d |
| `celegans` | 297 | 2148 | 10 | 41 | 8.6 | d |
| `BioGrid-Human-Herpesvirus-4` | 323 | 326 | 2 | 6 | 0.7 | d |
| `hex` | 331 | 930 | 3 | 3 | 27.2 | d |
| `iscas89-s953` | 332 | 454 | 2 | 4 | 1 | d |
| `autobahn` | 374 | 478 | 2 | 3 | 1 | d |
| `photoviz_dynamic` | 376 | 610 | 4 | 13 | 1.1 | d |
| `iscas89-s1196` | 377 | 537 | 2 | 3 | 1.2 | d |
| `ia-infect-dublin` | 411 | 2765 | 17 | 23 | 16.9 | d |
| `infectious` | 411 | 2765 | 17 | 23 | 17.5 | d |
| `BioGrid-Gallus-Gallus` | 413 | 436 | 4 | 22 | 1 | d |
| `iscas89-s1238` | 416 | 625 | 2 | 3 | 1.3 | d |
| `iscas89-s1423` | 423 | 554 | 2 | 5 | 1.1 | d |
| `ecoli-transcript` | 424 | 578 | 3 | 11 | 1.1 | d |
| `muenchen-bahn` | 447 | 578 | 2 | 2 | 11.7 | d |
| `BioGrid-Bos-Taurus` | 454 | 424 | 3 | 7 | 0.9 | d |
| `iscas89-s1488` | 463 | 779 | 3 | 11 | 2 | d |
| `iscas89-s1494` | 473 | 796 | 3 | 11 | 1.7 | d |
| `pigs` | 492 | 592 | 2 | 7 | 1.1 | d |
| `foodweb-caribbean` | 492 | 3313 | 13 | 155 | 10.8 | d |
| `ratbrain` | 503 | 23030 | 67 | 127 | 37.9 | d |
| `BioGrid-Human-Herpesvirus-8` | 716 | 691 | 3 | 6 | 1.4 | d |
| `codeminer` | 724 | 1015 | 4 | 7 | 1.8 | d |
| `pollination-daphni` | 797 | 2933 | 9 | 45 | 15.3 | d |
| `cpan-authors` | 839 | 2112 | 9 | 52 | 11.1 | d |
| `columbia-mobility` | 864 | 4147 | 9 | 12 | 17.9 | d |
| `columbia-social` | 864 | 7724 | 18 | 54 | 25.5 | d |
| `unicode_languages` | 868 | 1255 | 4 | 19 | 3.7 | d |
| `soc-wiki-Vote` | 890 | 2914 | 9 | 18 | 15.2 | d |
| `link-pedigree` | 898 | 1125 | 2 | 14 | 3.1 | d |
| `Opsahl-forum` | 899 | 7036 | 14 | 38 | 32.4 | d |
| `pollination-uk` | 984 | 16712 | 35 | 202 | 23.9 | d |
| `EU-email-core` | 986 | 16064 | 34 | 162 | 29.1 | d |
| `roget-thesaurus` | 1010 | 3648 | 6 | 8 | 21.2 | d |
| `bn-mouse_retina_1` | 1076 | 90811 | 121 | 403 | 869.1 | d |
| `BioGrid-Candida-Albicans-Sc5314` | 1121 | 1609 | 9 | 64 | 5 | d |
| `ia-email-univ` | 1134 | 5451 | 11 | 19 | 28.7 | d |
| `BioGrid-Human-Immunodeficiency-Virus-1` | 1138 | 1319 | 3 | 43 | 5 | d |
| `euroroad` | 1174 | 1417 | 2 | 4 | 4.1 | d |
| `BioGrid-Far-Western` | 1199 | 1089 | 3 | 11 | 4.2 | d |
| `polblogs` | 1224 | 16715 | 36 | 130 | 49.4 | d |
| `BioGrid-Escherichia-Coli-K12-Mg1655` | 1273 | 1889 | 5 | 13 | 7.1 | d |
| `web-google` | 1299 | 2773 | 17 | 18 | 17.4 | d |
| `munin` | 1324 | 1397 | 3 | 29 | 4.2 | d |
| `iscas89-s5378` | 1411 | 1639 | 3 | 5 | 5.3 | d |
| `diseasome` | 1419 | 2738 | 11 | 9 | 16.1 | c |
| `BioGrid-Dosage-Growth-Defect` | 1447 | 2193 | 5 | 70 | 7 | d |
| `netscience` | 1461 | 2742 | 19 | 6 | 28.4 | c |
| `chicago` | 1468 | 1298 | 1 | 2 | 6.5 | d |
| `pollination-carlinville` | 1500 | 15255 | 18 | 67 | 45.3 | d |
| `bitcoin-otc-negative` | 1606 | 3259 | 16 | 39 | 15 | d |
| `BioGrid-Fret` | 1700 | 2395 | 19 | 38 | 17.4 | d |
| `BioGrid-Dosage-Lethality` | 1776 | 2289 | 4 | 33 | 6.9 | d |
| `bn-fly-drosophila_medulla_1` | 1781 | 8911 | 18 | 42 | 39.8 | d |
| `BioGrid-Affinity-Capture-Luminescence` | 1840 | 2312 | 6 | 38 | 5.5 | d |
| `DNC-emails` | 1866 | 4384 | 17 | 74 | 19.7 | d |
| `wikipedia-norm` | 1881 | 15372 | 22 | 138 | 47.6 | d |
| `exnet-water` | 1893 | 2416 | 2 | 3 | 7.1 | d |
| `Opsahl-socnet` | 1899 | 13838 | 20 | 112 | 44 | d |
| `Y2H_union` | 1966 | 2705 | 4 | 30 | 7 | d |
| `iscas89-s9234` | 1985 | 2370 | 4 | 10 | 7.1 | d |
| `NZ_legal` | 2141 | 15739 | 25 | 129 | 44.1 | d |
| `BioGrid-Co-Crystal-Structure` | 2291 | 2021 | 5 | 6 | 6.6 | d |
| `Yeast` | 2362 | 7182 | 10 | 22 | 27.3 | d |
| `soc-hamsterster` | 2426 | 16630 | 24 | 77 | 39.8 | d |
| `iscas89-s13207` | 2492 | 3406 | 4 | 31 | 8.4 | d |
| `moreno_health` | 2539 | 10455 | 7 | 17 | 20.5 | d |
| `minnesota` | 2642 | 3303 | 2 | 3 | 8.3 | d |
| `ODLIS` | 2900 | 16377 | 12 | 86 | 40.8 | d |
| `openflights` | 2940 | 15677 | 28 | 111 | 33.1 | d |
| `iscas89-s15850` | 3247 | 4004 | 4 | 16 | 9.1 | d |
| `BioGrid-Dosage-Rescue` | 3380 | 6444 | 7 | 35 | 15.4 | d |
| `BioGrid-Co-Localization` | 3543 | 4452 | 6 | 25 | 13.7 | d |
| `twittercrawl` | 3656 | 154824 | 142 | 507 | 641.6 | d |
| `BioGrid-Escherichia-Coli-K12-W3110` | 4063 | 181620 | 156 | 442 | 3634.4 | d |
| `boards_gender_1m` | 4134 | 19993 | 25 | 20 | 34.3 | c |
| `boards_gender_2m` | 4220 | 5598 | 4 | 21 | 12.2 | d |
| `web-EPA` | 4271 | 8909 | 6 | 64 | 16 | d |
| `BioGrid-Co-Purification` | 4326 | 5970 | 12 | 71 | 19.4 | d |
| `ingredients` | 4372 | 591463 | 462 | 836 | 3424.3 | d |
| `advogato` | 5155 | 39285 | 25 | 216 | 51 | d |
| `soc-advogato` | 5167 | 39432 | 25 | 218 | 51.7 | d |
| `ca-GrQc` | 5242 | 14484 | 43 | 43 | 39.2 | d |
| `bitcoin-otc-positive` | 5573 | 18591 | 20 | 92 | 43.9 | d |
| `JUNG-javax` | 6120 | 50290 | 65 | 619 | 38.9 | d |
| `web-california` | 6175 | 15969 | 11 | 69 | 23.5 | d |
| `reactome` | 6328 | 147547 | 176 | 446 | 101.6 | d |
| `BioGrid-Caenorhabditis-Elegans` | 6394 | 23646 | 64 | 82 | 42.4 | d |
| `JDK_dependency` | 6435 | 53658 | 65 | 707 | 41.8 | d |
| `as20000102` | 6474 | 12572 | 12 | 43 | 26.7 | d |
| `zewail` | 6651 | 54182 | 18 | 99 | 48.7 | d |
| `ia-reality` | 6810 | 7680 | 5 | 19 | 8.9 | d |
| `wiki-vote` | 7115 | 100762 | 53 | 441 | 126.1 | d |
| `eva-corporate` | 7253 | 6711 | 3 | 7 | 9 | d |
| `chess` | 7301 | 55899 | 29 | 64 | 259.8 | d |
| `lederberg` | 8324 | 41532 | 15 | 175 | 51.1 | d |
| `BioGrid-Biochemical-Activity` | 8620 | 17746 | 11 | 182 | 33.1 | d |
| `iscas89-s38584` | 9193 | 12573 | 4 | 13 | 11.5 | d |
| `BioGrid-Drosophila-Melanogaster` | 9330 | 60556 | 83 | 115 | 73.8 | d |
| `iscas89-s38417` | 9500 | 10635 | 4 | 12 | 14.2 | d |
| `movielens_1m` | 9746 | 1000209 | 221 | 2356 | 12513.7 | d |
| `BioGrid-Arabidopsis-Thaliana-Columbia` | 10417 | 47916 | 26 | 385 | 37 | d |
| `p2p-Gnutella04` | 10876 | 39994 | 7 | 29 | 32.1 | d |
| `BioGrid-Co-Fractionation` | 11017 | 56354 | 83 | 84 | 68.6 | d |
| `AS-oregon-1` | 11174 | 23409 | 17 | 75 | 42.8 | d |
| `AS-oregon-2` | 11461 | 32730 | 31 | 118 | 54.6 | d |
| `ca-HepPh` | 12006 | 118489 | 238 | 90 | 436.8 | c |
| `ukroad` | 12378 | 15641 | 3 | 5 | 11.2 | d |
| `iscas89-s35932` | 12515 | 15961 | 2 | 2 | 2799.9 | d |
| `foldoc` | 13357 | 91471 | 12 | 63 | 65.1 | d |
| `BioGrid-Affinity-Capture-Rna` | 13765 | 42815 | 54 | 1630 | 230 | d |
| `escorts` | 16730 | 39044 | 11 | 69 | 51.6 | d |
| `marvel` | 19429 | 96662 | 18 | 745 | 204.2 | d |
| `BioGrid-Affinity-Capture-Western` | 21028 | 64046 | 17 | 93 | 80.8 | d |
| `as-22july06` | 22963 | 48436 | 25 | 218 | 85 | d |
| `edinburgh_associative_thesaurus` | 23132 | 297094 | 34 | 205 | 1175.4 | d |
| `ca-CondMat` | 23133 | 93439 | 25 | 27 | 81.6 | d |
| `cora_citation` | 23167 | 89157 | 13 | 70 | 49.8 | d |
| `soc-gplus` | 23628 | 39194 | 12 | 490 | 97.6 | d |
| `google+` | 23629 | 39194 | 12 | 490 | 97 | d |
| `BioGrid-Homo-Sapiens` | 24093 | 369767 | 71 | 802 | 1667.5 | d |
| `cit-HepTh` | 27769 | 352285 | 37 | 1670 | 249.3 | d |
| `digg` | 30399 | 86312 | 10 | 20 | 139.7 | d |
| `linux` | 30834 | 213217 | 23 | 4836 | 3946.5 | d |
| `BioGrid-Chemicals` | 33266 | 28093 | 1 | 2 | 18.7 | d |
| `cit-HepPh` | 34546 | 420877 | 30 | 300 | 198.6 | d |
| `email-Enron` | 36692 | 183831 | 43 | 187 | 752.2 | d |
| `BioGrid-Affinity-Capture-Ms` | 40495 | 321887 | 58 | 843 | 580.6 | d |
| `slashdot_threads` | 51084 | 117378 | 15 | 65 | 212.4 | d |
| `deezer` | 54573 | 498202 | 21 | 83 | 400.2 | d |
| `loc-brightkite_edges` | 58228 | 214078 | 52 | 184 | 213.7 | d |
| `facebook-links` | 63731 | 817090 | 52 | 233 | 1705.6 | d |
| `BioGrid-All` | 75550 | 1316843 | 164 | 1792 | 1738.1 | d |
| `soc-Epinions1` | 75879 | 405740 | 67 | 551 | 1301.5 | d |
| `soc-Slashdot0811` | 77360 | 469180 | 54 | 618 | 832.4 | d |
| `NYClimateMarch2014` | 102378 | 327080 | 34 | 1037 | 2548.4 | d |
| `livemocha` | 104104 | 2193083 | 92 | 1129 | 52801.5 | d |
| `tv_tropes` | 152093 | 3232134 | 115 | 2655 | 25491.8 | d |
| `gowalla` | 196591 | 950327 | 51 | 1132 | 1258.8 | d |
| `bahamas` | 219856 | 246291 | 6 | 58 | 145.9 | d |
| `location` | 225486 | 293697 | 5 | 854 | 486.3 | d |
| `offshore` | 278877 | 505965 | 13 | 5698 | 479.2 | d |
| `dogster_friendships` | 426821 | 8546581 | 249 | 32804 | 48807.8 | d |
| `Cannes2013` | 438089 | 835892 | 27 | 3168 | 2523.8 | d |
| `actor_movies` | 511463 | 1470404 | 14 | 217 | 1036.9 | d |
| `paradise` | 542102 | 794545 | 23 | 4242 | 1317.1 | d |
| `panama` | 556686 | 702437 | 62 | 1948 | 699.5 | d |
| `countries` | 592414 | 624402 | 6 | 11048 | 9415.9 | d |
| `teams` | 935591 | 1366466 | 9 | 351 | 6893.1 | d |
| `mag_geology_coauthor` | 2852295 | 4448428 | 13 | 193 | 3205.6 | d |