Cones of Weighted and Partial Metrics

The format for the .ext and .ine files is described in the cddlib manual.

The format of the .oex and .oin files is based on it, with the difference of the index line, and the last columns representing the incidence number and size of the orbit. Also, the numbers before the index line represent the number of orbits and the number of points, rather than the number of extreme rays/inequalities (which is specified in the total line) and columns. In the case of weighted metrics ((01-)(d)w(wq)met) the i,i columns represent the values wi. When the dimension of the cone is smaller than the dimension of the space it is embedded in, the basis of the orthogonal subspace is given after the nullspace keyword.

The .skg and .rdg files contain two matrices. The first matrix is the list of orbits of edges, represented by a pair of the two extreme rays/inequalities connected by an edge in the orbit. The second matrix is the collapsed adjacency matrix according to the equitable partition of extreme rays or inequalities into orbits, as represented in the .oex and .oin files. There is no particular order in which the orbits appear in the matrix. In some cases they have been manually marked. For some graphs, there are also pictures of the graph or its complement available.

The .ead, .iad, .ecd and .icd files contain a list of extreme rays or inequalities, each followed by a number whose absolute value is the adjacency or incidence number, and a list of adjacent or incident extreme rays/inequalities if the number was positive, or a list of non-adjacent or non-incident extreme rays/inequalities if the number was negative. The extreme rays or inequalities are numbered according to the order in which they appear in the .ext and .ine files.

The data was computed using cddlib by Komei Fukuda, lrslib by David Avis, and the polyhedral package for GAP by Mathieu Dutour Sikirić, with some additional functions for working with cones of metrics.

This is a joint work with Michel Deza and Elena Deza. The article is available at arXiv.

Cone Orbits of extreme rays Orbits of inequalities Skeleton graph Ridge graph Extreme rays Inequalities Adjacence of extreme rays Adjacence of inequalities Incidence of generators and inequalities Incidence of inequalities and generators
01-dwmet2
dim. 3 		oex
4 ext. rays
in 3 orbits 		oin
4 facets
in 2 orbits 		               
01-dwmet3
dim. 6 		oex
10 ext. rays
in 4 orbits 		oin
15 facets
in 4 orbits 		skg
skg.compl.pdf
skg.pdf
diameter 2 		rdg
rdg.pdf
diameter 2 		           
01-dwmet4
dim. 10 		oex
22 ext. rays
in 6 orbits 		oin
62 facets
in 7 orbits 		skg
diameter 2 		rdg
diameter 3 		           
01-dwmet5
dim. 15 		oex
46 ext. rays
in 7 orbits 		oin
1165 facets
in 27 orbits 		skg
diameter 2 		rdg
diameter 3 		           
01-dwmet6
dim. 21 		oex
94 ext. rays
in 9 orbits 		oin
369401 facets
in 806 orbits 		skg
diameter 2 		             
01-pmet3
dim. 6 		oex
13 ext. rays
in 5 orbits 		oin
12 facets
in 3 orbits 		skg
diameter 3 		rdg
diameter 2 		           
01-pmet4
dim. 10 		oex
44 ext. rays
in 9 orbits 		oin
46 facets
in 5 orbits 		skg
diameter 3 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
01-pmet5
dim. 15 		oex
166 ext. rays
in 14 orbits 		oin
585 facets
in 15 orbits 		skg
diameter 3 		rdg
diameter 3 		ext
 ine
 ead
 iad
 ecd
 icd
01-pmet6
dim. 21 		oex
705 ext. rays
in 22 orbits 		                 
01-spmet4
dim. 10 		oex
15 ext. rays
in 4 orbits 		oin
40 facets
in 5 orbits 		               
01-spmet5
dim. 15 		oex
31 ext. rays
in 5 orbits 		oin
210 facets
in 12 orbits 		               
01-spmet6
dim. 21 		oex
63 ext. rays
in 6 orbits 		oin
38780 facets
in 145 orbits 		               
01-wmet2
dim. 3 		oex
3 ext. rays
in 2 orbits 		oin
3 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 1 		           
01-wmet3
dim. 6 		oex
6 ext. rays
in 2 orbits 		oin
6 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 1 		           
01-wmet4
dim. 10 		oex
11 ext. rays
in 3 orbits 		oin
16 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 2 		           
01-wmet5
dim. 15 		oex
20 ext. rays
in 3 orbits 		oin
45 facets
in 3 orbits 		skg
diameter 1 		rdg
diameter 2 		           
01-wmet6
dim. 21 		oex
37 ext. rays
in 4 orbits 		oin
216 facets
in 5 orbits 		skg
diameter 1 		rdg
diameter 2 		           
01-wqmet5
dim. 14 in R20 		oex
110 ext. rays
in 8 orbits 		oin
250 facets
in 5 orbits 		skg
diameter 2 		rdg
diameter 2 		           
01-wqmet6
dim. 30 		oex
802 ext. rays
in 17 orbits 		                 
cut4
dim. 6 		oex
7 ext. rays
in 2 orbits 		oin
12 facets
in 1 orbit 		skg
diameter 1 		rdg
diameter 2 		           
cut5
dim. 10 		oex
15 ext. rays
in 2 orbits 		oin
40 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
cut6
dim. 15 		oex
31 ext. rays
in 3 orbits 		oin
210 facets
in 4 orbits 		skg
diameter 1 		rdg
diameter 3 		ext
 ine
 ead
 iad
 ecd
 icd
dwmet2
dim. 3 		oex
4 ext. rays
in 3 orbits 		oin
4 facets
in 2 orbits 		               
dwmet3
dim. 6 		oex
13 ext. rays
in 5 orbits 		oin
12 facets
in 3 orbits 		               
dwmet4
dim. 10 		oex
62 ext. rays
in 11 orbits 		oin
28 facets
in 3 orbits 		               
dwmet5
dim. 15 		oex
1696 ext. rays
in 44 orbits 		oin
55 facets
in 3 orbits 		               
dwmet6
dim. 21 		oex
337092 ext. rays
in 734 orbits 		oin
96 facets
in 3 orbits 		               
met3
dim. 3 		oex
3 ext. rays
in 1 orbit 		oin
3 facets
in 1 orbit 		skg
diameter 1 		rdg
diameter 1 		ext
 ine
 ead
 iad
 ecd
 icd
met4
dim. 6 		oex
7 ext. rays
in 2 orbits 		oin
12 facets
in 1 orbit 		skg
diameter 1 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
met5
dim. 10 		oex
25 ext. rays
in 3 orbits 		oin
30 facets
in 1 orbit 		skg
diameter 2 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
met6
dim. 15 		oex
296 ext. rays
in 8 orbits 		oin
60 facets
in 1 orbit 		skg
diameter 2 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
nmet2
dim. 1 in R3 		oex
1 ext. ray
in 1 orbit 		oin
1 facet
in 1 orbit 		skg
 rdg
            
nmet3
dim. 3 in R6 		oex
3 ext. rays
in 1 orbit 		oin
3 facets
in 1 orbit 		skg
diameter 1 		rdg
diameter 1 		           
nmet4
dim. 6 in R10 		oex
7 ext. rays
in 2 orbits 		oin
12 facets
in 1 orbit 		skg
diameter 1 		rdg
diameter 2 		           
nmet5
dim. 10 in R15 		oex
25 ext. rays
in 3 orbits 		oin
30 facets
in 1 orbit 		skg
diameter 2 		rdg
diameter 2 		           
nmet6
dim. 15 in R21 		oex
296 ext. rays
in 8 orbits 		oin
60 facets
in 1 orbit 		skg
diameter 2 		rdg
diameter 2 		           
nmet7
dim. 21 in R28 		oex
55226 ext. rays
in 46 orbits 		oin
105 facets
in 1 orbit 		  rdg
diameter 2 		           
nqmet2
dim. 1 in R4 		oex
1 ext. ray
in 1 orbit 		oin
1 facet
in 1 orbit 		skg
 rdg
            
nqmet3
dim. 4 in R9 		oex
5 ext. rays
in 2 orbits 		oin
6 facets
in 1 orbit 		skg
diameter 2 		rdg
diameter 2 		           
nqmet4
dim. 9 in R16 		oex
37 ext. rays
in 5 orbits 		oin
24 facets
in 1 orbit 		skg
diameter 2 		rdg
diameter 2 		           
nqmet5
dim. 16 in R25 		oex
2333 ext. rays
in 33 orbits 		oin
60 facets
in 1 orbit 		skg
diameter 3 		rdg
diameter 2 		           
nqmet6
dim. 25 in R36 		  oin
120 facets
in 1 orbit 		               
nqmet7
dim. 36 in R49 		  oin
210 facets
in 1 orbit 		               
ocut2
dim. 2 		oex
2 ext. rays
in 1 orbit 		oin
2 facets
in 1 orbit 		skg
diameter 1 		rdg
diameter 1 		           
ocut3
dim. 5 in R6 		oex
6 ext. rays
in 2 orbits 		oin
9 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 2 		           
ocut4
dim. 9 in R12 		oex
14 ext. rays
in 3 orbits 		oin
30 facets
in 3 orbits 		skg
diameter 1 		rdg
diameter 2 		           
ocut5
dim. 14 in R20 		oex
30 ext. rays
in 4 orbits 		oin
130 facets
in 6 orbits 		skg
diameter 1 		rdg
diameter 3 		           
ocut6
dim. 20 in R30 		oex
62 ext. rays
in 5 orbits 		oin
16460 facets
in 61 orbits 		skg
diameter 1 		             
owhyp5
dim. 14 in R20 		oex
70 ext. rays
in 6 orbits 		oin
90 facets
in 4 orbits 		skg
diameter 2 		rdg
diameter 2 		           
phyp4
dim. 10 		oex
56 ext. rays
in 10 orbits 		oin
oin.orig
34 facets
in 4 orbits 		skg
diameter 3 		rdg
diameter 2 		           
phyp4a
dim. 10 		oex
50 ext. rays
in 10 orbits 		oin
40 facets
in 4 orbits 		skg
diameter 3 		rdg
diameter 2 		           
pmet2
dim. 3 		oex
4 ext. rays
in 3 orbits 		oin
4 facets
in 2 orbits 		               
pmet3
dim. 6 		oex
13 ext. rays
in 5 orbits 		oin
12 facets
in 3 orbits 		skg
skg.pdf
diameter 3 		rdg
rdg.compl.pdf
rdg.pdf
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
pmet4
dim. 10 		oex
62 ext. rays
in 11 orbits 		oin
28 facets
in 3 orbits 		skg
diameter 3 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
pmet5
dim. 15 		oex
1696 ext. rays
in 44 orbits 		oin
55 facets
in 3 orbits 		skg
diameter 3 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
pmet6
dim. 21 		oex
337092 ext. rays
in 734 orbits 		oin
96 facets
in 3 orbits 		  rdg
diameter 2 		  ine
        
qmet2
dim. 2 		oex
2 ext. rays
in 1 orbit 		oin
2 facets
in 1 orbit 		               
qmet3
dim. 6 		oex
20 ext. rays
in 5 orbits 		oin
12 facets
in 2 orbits 		               
spmet2
dim. 3 		oex
3 ext. rays
in 2 orbits 		oin
3 facets
in 2 orbits 		               
spmet3
dim. 6 		oex
7 ext. rays
in 3 orbits 		oin
12 facets
in 3 orbits 		               
spmet4
dim. 10 		oex
25 ext. rays
in 6 orbits 		oin
30 facets
in 3 orbits 		               
spmet5
dim. 15 		oex
296 ext. rays
in 16 orbits 		oin
60 facets
in 3 orbits 		               
spmet6
dim. 21 		oex
55226 ext. rays
in 178 orbits 		oin
105 facets
in 3 orbits 		               
wpmet2
dim. 3 		oex
3 ext. rays
in 2 orbits 		oin
3 facets
in 2 orbits 		               
wpmet3
dim. 6 		oex
6 ext. rays
in 2 orbits 		oin
6 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 1 		ext
 ine
 ead
 iad
 ecd
 icd
wpmet4
dim. 10 		oex
11 ext. rays
in 3 orbits 		oin
16 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 2 		           
wpmet5
dim. 15 		oex
30 ext. rays
in 4 orbits 		oin
35 facets
in 2 orbits 		skg
diameter 2 		rdg
diameter 2 		           
wpmet6
dim. 21 		oex
302 ext. rays
in 9 orbits 		oin
66 facets
in 2 orbits 		skg
diameter 2 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
wpmet7
dim. 28 		  oin
112 facets
in 2 orbits 		               
wqmet3
dim. 5 in R6 		oex
6 ext. rays
in 2 orbits 		oin
9 facets
in 2 orbits 		skg
diameter 1 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
wqmet4
dim. 9 in R12 		oex
20 ext. rays
in 4 orbits 		oin
24 facets
in 2 orbits 		skg
diameter 2 		rdg
diameter 2 		ext
 ine
 ead
 iad
 ecd
 icd
wqmet5
dim. 14 in R20 		oex
190 ext. rays
in 11 orbits 		oin
50 facets
in 2 orbits 		skg
diameter 2 		rdg
diameter 2 		           
wqmet6
dim. 20 in R30 		oex
18502 ext. rays
in 77 orbits 		oin
90 facets
in 2 orbits 		  rdg
diameter 2 		           
wwqmet3
dim. 6 in R9 		oex
13 ext. rays
in 5 orbits 		oin
12 facets
in 3 orbits 		skg
diameter 3 		rdg
diameter 2 		           
wwqmet4
dim. 10 in R16 		oex
62 ext. rays
in 11 orbits 		oin
28 facets
in 3 orbits 		skg
diameter 3 		rdg
diameter 2 		           
y5
dim. 10 		oex
oex.orig
25 ext. rays
in 3 orbits 		oin
30 facets
in 1 orbit 		skg
diameter 2 		rdg
diameter 2 		           
y6
dim. 15 		oex
oex.orig
206 ext. rays
in 7 orbits 		oin
510 facets
in 3 orbits 		skg
diameter 2 		rdg
diameter 3 		           
Janoš Vidali 2022-12-01