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 w_i. 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	Adjacency of extreme rays	Adjacency 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.pdf skg.compl.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. 20 in R⁶	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. 3 in R²	oex 1 ext. ray in 1 orbit	oin 1 facet in 1 orbit	skg	rdg
nmet3 dim. 6 in R³	oex 3 ext. rays in 1 orbit	oin 3 facets in 1 orbit	skg diameter 1	rdg diameter 1
nmet4 dim. 10 in R⁴	oex 7 ext. rays in 2 orbits	oin 12 facets in 1 orbit	skg diameter 1	rdg diameter 2
nmet5 dim. 15 in R⁵	oex 25 ext. rays in 3 orbits	oin 30 facets in 1 orbit	skg diameter 2	rdg diameter 2
nmet6 dim. 21 in R⁶	oex 296 ext. rays in 8 orbits	oin 60 facets in 1 orbit	skg diameter 2	rdg diameter 2
nmet7 dim. 28 in R⁷	oex 55226 ext. rays in 46 orbits	oin 105 facets in 1 orbit		rdg diameter 2
nqmet2 dim. 4 in R³	oex 1 ext. ray in 1 orbit	oin 1 facet in 1 orbit	skg	rdg
nqmet3 dim. 9 in R⁵	oex 5 ext. rays in 2 orbits	oin 6 facets in 1 orbit	skg diameter 2	rdg diameter 2
nqmet4 dim. 16 in R⁷	oex 37 ext. rays in 5 orbits	oin 24 facets in 1 orbit	skg diameter 2	rdg diameter 2
nqmet5 dim. 25 in R⁹	oex 2333 ext. rays in 33 orbits	oin 60 facets in 1 orbit	skg diameter 3	rdg diameter 2
nqmet6 dim. 36 in R¹¹		oin 120 facets in 1 orbit
nqmet7 dim. 49 in R¹³		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. 6 in R¹	oex 6 ext. rays in 2 orbits	oin 9 facets in 2 orbits	skg diameter 1	rdg diameter 2
ocut4 dim. 12 in R³	oex 14 ext. rays in 3 orbits	oin 30 facets in 3 orbits	skg diameter 1	rdg diameter 2
ocut5 dim. 20 in R⁶	oex 30 ext. rays in 4 orbits	oin 130 facets in 6 orbits	skg diameter 1	rdg diameter 3
ocut6 dim. 30 in R¹⁰	oex 62 ext. rays in 5 orbits	oin 16460 facets in 61 orbits	skg diameter 1
owhyp5 dim. 20 in R⁶	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.pdf rdg.compl.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. 6 in R¹	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. 12 in R³	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. 20 in R⁶	oex 190 ext. rays in 11 orbits	oin 50 facets in 2 orbits	skg diameter 2	rdg diameter 2
wqmet6 dim. 30 in R¹⁰	oex 18502 ext. rays in 77 orbits	oin 90 facets in 2 orbits		rdg diameter 2
wwqmet3 dim. 9 in R³	oex 13 ext. rays in 5 orbits	oin 12 facets in 3 orbits	skg diameter 3	rdg diameter 2
wwqmet4 dim. 16 in R⁶	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