The Combinatorial Structure of Small

Cut and Metric Polytop es

Antoine DEZA

Michel DEZA

Lab oratoire dInformatique URA du CNRS

Departement de Mathematiques et dInformatique

Ecole Normale Superieure

LIENS

January

The Combinatorial Structure of Small Cut and

Metric Polytop es

Antoine DEZA Michel DEZA

January

Abstract We study the combinatorial structure of the cut and metric p olytop es on n

no des for n Those two p olytop es have a complicated geometrical structure but using

their large symmetry group we can completely describ e their face lattices We present

for any n some orbits of faces and give new result on the tightness of the wrapping of

the cut p olytop e by the metric p olytop e disproving a conjecture of on their lattices

Key words metric p olytop e complete bipartite subgraphs p olytop e face lattice

Intro duction

We rst recall the denition of the metric polytope m the cut polytope c and their

n n

relatives the metric cone and the cut cone Then we present some applications to well

known optimization problems and some combinatorial and geometric prop erties of those

p olyhedra The general references are for p olytop es and for graphs For a

complete study of the applications and the combinatorial optimization asp ects of those

p olyhedra we refer resp ectively to the surveys and

For all sets fi j k g f ng we consider the following inequalities

x x x

ij ik j k

x x x

ij ik j k

The inequalities dene the metric cone and the metric p olytop e m is obtained by

n

n

b ounding the latter by the inequalities The facets dened by the inequalities

3

which can b e seen as triangle inequalities for distance x on f ng are called

ij

n

homogeneous triangle facets and are denoted by Tr The facets dened by the

ijk

3

inequalities are called nonhomogeneous triangle facets and are denoted by Tr

ij k

Given a subset S of V f ng the cut determined by S consists of the pairs

n

i j of elements of V such that exactly one of i j is in S S denotes b oth the cut

n

n

2

and its incidence vector in IR that is S if exactly one of i j is in S and

ij

otherwise for i j n By abuse of language we use the term cut for b oth the cut

n

2

itself and its incidence vector so S are considered as co ordinates of a p oint in IR

ij

The cut p olytop e of the c which is also called the complete bipartite

n

n1

subgraphs polytope is the convex hull of all cuts and the cut cone is the conic hull

n1

of all nonzero cuts Those p olyhedra were considered by many authors see for

instance and references therein One of the motivations

for the study of these p olyhedra comes from their applications in combinatorial optimiza

tion the most imp ortant b eing the maxcut and multicommo dity ow problems

Given a graph G V E and nonnegative weights w e E assigned to its edges

n e

P

w is as large the maxcut problem consists in nding a cut S whose weight

e

e (S )

as p ossible It is a wellknown NP complete problem By setting w if e is not an

e

edge of G we can consider without loss of generality the complete graph on V Then the

n

maxcut problem can b e stated as a problem over the cut p olytop e

c as follows

n

T

max w x

x c

n

T

Since the metric p olytop e is a relaxation of the cut p olytop e optimizing w x over c

n

instead of m provides an upp er b ound for the maxcut problem

n

With E the set of edges of the complete graph on V an instance of the multicom

n

modity ow problem is given by two nonnegative vectors indexed by E a capacity ce

and a requirement r e for each e E Let U fe E r e g If T denotes the

subset of V spanned by the edges in U then we say that the graph G T U denotes

n

the suppor t of r For each edge e s t in the supp ort of r we seek a ow of r e units

b etween s and t in the complete graph The sum of all ows along any edge e E must

not exceed ce If such a ow exists we call c r feasible A necessary and sucient

condition for feasibility is given by the Japanese theorem a pair c r is feasible if and

T

only if c r x is valid over the metric cone For example Tr can b e seen as

ijk

an elementary solvable ow problem with cij r ik r j k and ce r e

T

otherwise so the inequalities corresp ond to c r x for x in the metric cone

Therefore the metric cone is the dual cone to the cone of feasible multicommo dity ow

problems

Combinatorial and geometric prop erties of the

cut and metric p olytop es

n

n1

The p olytop e c is a dimensional p olyhedron with vertices and m is

n n

2

n

n

2

a p olytop e of same dimension with facets inscrib ed in the cub e We have

3

1 1 1

c m with equality only for n It is easy to see that the p oint

n n n

2 2 2

is the center of gravity of b oth c and m and is also the center of the sphere of radius

n n

p

1

nn where all the cuts lie Another two geometric characteristics of the r

2

cut p olytop e c are its width and geometric diameter We recall that while the width of

n

a p olytop e P is equal to the minimum distance b etween a pair of parallel hyp erplanes

containing P in the slice b etween them the geometric diameter of P is the maximum

distance b etween a pair of supp orting hyp erplanes The width of c is and

n

p

n 1

2

its geometric diameter is for n even and n for n o dd see Any facet

2 2

resp ectively ridge that is a face of co dimension of the metric p olytop e contains a

facet resp ectively a ridge of the cut p olytop e and the vertices of the cut p olytop e are

vertices of the metric p olytop e in fact the cuts are precisely the integral vertices of the

metric p olytop e Actually the metric p olytop e m wraps the cut p olytop e c very tightly

n n

since in addition to the vertices all edges and faces of c are also faces of m

n n

In other words c is a segment of order of m and its dual m is a segment of order

n n

n

of c in terms of a p olytop e P is a segment of order s of a p olytop e Q if they have

n

the same dimension and if every iface of P is a face of Q for i s The p olytop e

c is neighb ourly Any two cuts are adjacent b oth on c and on m

n n n

in other words m is quasiintegral in terms of that is the skeleton of the convex

n

hull of its integral vertices ie the skeleton of the c is an induced subgraph of the edge

n

is the diameters of graph of the metric p olytop e itself While the diameter of m

n

c and m are resp ectively conjectured to b e and For a detailed study of

n

n

the combinatorial and geometric prop erties of c and m we refer to

n n

The metric p olytop e and the cut p olytop e share the same symmetry group that is the

group of isometries preserving a p olytop e This group is isomorphic to the automorphism

group of the folded ncube that is I sm I sc Aut2 see We recall

p p n

that the folded ncub e is the graph whose vertices are the partitions of V f ng

n

into two subsets two partitions b eing adjacent when their common renement contains

a set of size one see More precisely for n I sm I sc is induced by

n n

p ermutations on V f ng and switching reections by a cut Given a cut S the

n

switching reection r is dened by y r x where y x if i j S and

ij ij

(S ) (S )

y x otherwise These symmetries preserve the adjacency relations and the linear

ij ij

indep endency For the study of their face lattices we frequently use the fact that the

faces of m and c are partitioned into orbits of their symmetry group

n n

We nally mention the following link with metrics there is an evident corre

sp ondence b etween the elements of the metric cone and all the semimetrics on n p oints

and the elements of the cut cone corresp ond precisely to the semimetrics on n p oints

n

m

that are isometrically emb eddable into some l see it is easy to check that m

1

2

Face lattices of small cut and metric p olytop es

Face lattice of the c m for n 

n n

For n we have c m moreover c and c are b oth wellknown p olytop es While

n n 3 4

p

1

c is combinatorially and volume v c is the regular tetrahedron of edge length

4 3 3

3

2

equivalent to the dimensional cyclic p olytop e with vertices and its volume is v

4

45

The f vector of c is obviously f c more precisely all prop er faces

4 4

of c are partitioned into the following orbits of the symmetry group I sc Aut2

4 4 4

1

the vertices of c form the orbit O

4

0

1 2

the orbits O and O of edges f S S g are resp ectively formed by the

1 1

edges with jS S j o dd and the ones with jS S j even that is resp ectively

represented by f g and f g

1

the orbits of faces are O of size which is represented by f g

2

2

and O of size which is represented by f g

2

1

the orbits of faces are O of size which is represented by f

3

2 3

g O of size which is represented by f g and O of

3 3

size and represented by f g

1

the ridges form the orbit O they are the cofaces that is the convex hull of the

4

1

vertices not b elonging to a face corresp onding to the faces from the orbit O

2

1

the facets form the orbit O they are the cofaces corresp onding to the edges

5

1

from the orbit O

1

Remark

i The skeleton of c is the grid which is also the line graph of K 2

44 4

4

ii A set of vertices is not a face of c if and only if it contains one of the fol lowing

4

sets of vertices f g and f g

Face lattices of c and m

5 5

The face lattices of c and m were obtained in the following way We rst got

5 5

all the nonsimplex faces by systematically checking all p ossible pairwise wise etc

intersections of nonsimplex facets Then considering all and sets of vertices and

the remaining p ossible pairwise wise etc intersections of facets we obtained all ifaces

for i Finally noticing that few ifaces contains the complete lower part

of the lattice for example any face of m is a facet of a face b elonging to a single orbit

5

of face we found by a case by case analysis all the remaining simplex and faces

of m and c The dimensions of the faces were computed using the list of all ane

5 5

dep endencies of m and c given b elow

5 5

Using one can that check all ane dep endencies on the vertices of m and c

5 5

P P

that is equations x with are up to p ermutations switchings and the

i i i

bijection S S which clearly preserves ane dep endencie s

X

jS j

S

S f1234g

X

jS j

S

S f12345g jf45gS j=1

5

X

i i

i=2

5

X

i

i=2

The restriction of the face lattices of c and m to their nonsimplex faces are resp ectively

5 5

given in Figures and All simplicial and all maximal under inclusion simplex

faces are given in Prop osition and their complete face lattices are presented in detail

in Section and Section The f vectors of c and m are resp ectively

5 5

f c

5

f m

5

i i th i

By f and g we resp ectively denote a representative j face of the i orbit C resp ectively

j j j

i

M of c and m

5 5

j

Face lattice of c

5

In Figure all the orbits of prop er nonsimplex faces of m are given Each orbit

5

is represented by the set of vertices b elonging to a representative face of the orbit A cut

i resp ectively ij is denoted by a circled p oint i and resp ectively by an edge fi j g

1 1 1 1

5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 1 1 6 2

f 9 f 8 f 7 f 6

Figure Nonsimplices of the face lattice of c

5

The face lattice of c is partitioned into the following orbits of I sc

5 5

1 2

the facets are partitioned into the orbits C and C resp ectively formed by the

9 9

1

triangle facets represented by f Tr and the switchings of the equicut

123

9

X

2

facet f which is dened by the inequality x

ij

9

1ij 5

1 2 3

the ridges are partitioned into the orbits C C and C of size and

8 8 8

1 2

and resp ectively represented by f Tr Tr f Tr Tr and

123 124 123 145

8 8

3 2

f Tr f

123

8 9

1 2 7

the faces are partitioned into the orbits C C C of size

7 7 7

and resp ectively represented by the graphs given in Figure

1 1 2 2 3 3 4 2 2

We have f f Tr Tr f f Tr f f Tr f f f

243 134 234 125

7 8 7 8 7 8 7 8 9

3 5 1 6 1 7 1

f Tr f f Tr f f Tr and f f Tr

145 134 125 135

8 7 8 7 8 7 8

1 1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 4 3 1 2 3 4 5 6 7

f 7 f 7 f 7 f 7 f 7 f 7 f 7

Figure faces of c

5

1 10

the faces are partitioned into the orbits C C resp ectively represented

6 6

by the graphs given in Figure

1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 1 2 3 4 5 6 f 6 f 6 f 6 f 6 f 6 f 6

1 1 1 1

5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 7 8 9 10

f 6 f 6 f 6 f 6

Figure faces of c

5

1 11

the faces are partitioned into the orbits C C resp ectively represented

5 5

by the graphs given in Figure

1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 1 2 3 4 5 6 f 5 f 5 f 5 f 5 f 5 f 5

1 1 1 1 1

5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 7 8 9 10 11

f 5 f 5 f 5 f 5 f 5

Figure faces of c

5

1 8

the faces are partitioned into the orbits C C resp ectively represented by

4 4

the graphs given in Figure

1 1 1 1 1

5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 1 2 3 4 5 f 4 f 4 f 4 f 4 f 4

1 1 1

5 2 5 2 5 2

4 3 4 3 4 3 7 f 6 f f 8

4 4 4

Figure faces of c

5

7 1

resp ectively represented C the faces are partitioned into the orbits C

3 3

by the graphs given in Figure Actually the only sets of cuts which are not

faces of c are the memb ers of the orbit of f g 5

1 1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 4 3 1 2 3 4 5 6 7

f 3 f 3 f 3 f 3f 3 f 3f 3

Figure faces of c

5

1 2 3

the faces are partitioned into the orbits C C and C of size

2 2 2

1 2

and and resp ectively represented by f f g f f

2 2

3

g and f f g

2

1 2

the edges are partitioned into the orbits C and C resp ectively formed by

1 1

the edges f S S g with jS S j or and the ones with jS S j

1 2

or that is resp ectively represented by f f g and f f g

1 1

1

the vertices form the orbit C

0

Face lattice of m

5

In Figure all the orbits of prop er nonsimplex faces of m are given As for c each

5 5

orbit is represented by the set of vertices b elonging to one of its representative face While

a straight line links incident faces a dotted one links faces incident up to a p ermuta

1 2

S tion Besides the cuts the vertices of m are the anticuts S

5

3 3

A anticut i resp ectively ij is denoted by a grey circled p oint i and resp ectively

by a grey edge fi j g The anticut is denoted by a grey Note that a face cannot

contain b oth S and S

The face lattice of m is partitioned into the following orbits of I sm

5 5

1 1

the triangle facets form the orbit M represented by g Tr

123

9 9

1 2

the ridges are partitioned into the orbits M and M b oth of size and

8 8

1 2

resp ectively represented by g Tr Tr and g Tr Tr

123 124 123 145

8 8

1 2 6

the faces are partitioned into the orbits M M M of size

7 7 7

and resp ectively represented by the graphs given in Figure

1 1 2 1 3 1 4 1

We have g g Tr Tr g g Tr g g Tr g g Tr

243 134 354 345 134

7 8 7 8 7 8 7 8

1 6 1 5

Tr g Tr and g g g

135 125

8 7 8 7

1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 g1 g2 g3 g 4 g 5 g6

7 7 7 7 7 7

Figure faces of m

5 1

5 2

4 3 g1 9

1 1

5 2 5 2

4 3 4 3 g1 g2 8 8

1 1 1

5 2 5 2 5 2

4 3 4 3 4 3 g4 g5 g6 7 7 7

1 1 1 1

5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 11 g g9 g2 g10 6 6 6 6

1

5 2

4 3 g13 5

1

5 2

4 3 g12

4

Figure Nonsimplices of the face lattice of m

5

1 11

the faces are partitioned into the orbits M M resp ectively represented

6 6

by the graphs given in Figure

1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 g1 g2 g3 g4 g5 g 6 6 6 6 6 6 6

1 1 1 1 1

5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 g7 g8 g9 g10 g11

6 6 6 6 6

Figure faces of m

5

1 13

the faces are partitioned into the orbits M M resp ectively represented

5 5

by the graphs given in Figure

1 1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 4 3 g1 g2 g 3 4 5 6 g7 5 5 5 g 5 g5 g 5 5

1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 g8 g9 g10 11 12 13

5 5 5 g5 g5 g 5

Figure faces of m

5

1 12

the faces are partitioned into the orbits M M resp ectively represented

4 4

by the graphs given in Figure

1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 g1 2 g3 4 5 g 6 4 g 4 4 g 4 g 4 4

1 1 1 1 1 1

5 2 5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 4 3 8 g 7 g g 9 g10 g11 g12

4 4 4 4 4 4

Figure faces of m

5

1 10

the faces are partitioned into the orbits M M resp ectively repre

3 3

sented by the graphs given in Figure

1 1 1 1 1

5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 g1 2 3 4 5 3 g3 g3 g 3g 3

1 1 1 1 1

5 2 5 2 5 2 5 2 5 2

4 3 4 3 4 3 4 3 4 3 6 7 8 9 g10

g 3 g3 g3 g3 3

Figure faces of m

5

1 2 5

the faces are partitioned into the orbits M M M of size

2 2 2

1

and and resp ectively represented by g f g

2

2 3 4

g f g g f g g f g

2 2 2

5

and g f g

2

1 2

the edges are partitioned into the orbits M M resp ectively formed by the

1 1

edges f S S g with jS S j or and the ones with jS S j or

3

and the orbit M formed by the edges f S S g with jS S j or

1

1 2

that is resp ectively represented by g f g g f g and

1 1

3

g f g

1

1 2

the vertices are partitioned into the orbits M and M resp ectively formed by

0 0

the cuts and the anticuts

Prop osition

i The maximal under inclusion simplex faces of c are the faces belonging to the orbits

5

2 2 1 5

C C C and C

9 8 7 7

ii The maximal under inclusion simplex faces of m are the faces belonging to the

5

1 2 3

orbits M M and M

7 7 7

iii The simplicial faces of c and m are respectively the faces the belonging to the orbit

5 5

2 11 2 12

C and M M and M

6 6 6 4

2 2

iv The faces belonging to C and M are combinatorial ly equivalent to m c and so

4 4

6 6

to the dimensional cyclic polytope with vertices

Prop osition

i The maximal under inclusion simplex faces of m containing an anticut are the faces

5

3

belonging to the orbit M

7

ii The number of simplex ifaces of m containing an anticut is for each anticut

5

f for i

i

Proof Let g b e a iface of m with i containing an anticut for example

5

and denote g Tr Clearly we have g g with equality if and only if g is a

ij k

(ij k ) g

simplex Supp ose that i g g it will mean that i is a universal vertex that is the

graph Gg representing g is one of the following graphs

It turns out that those graphs are resp ectively subgraphs we require only inclusion

9 10 13 12

of the edgeset of the graphs representing the nonsimplex faces g g g g see

6 6 5 4

Figure Now since g is the intersection of homogeneous triangle facets ij g

implies that i j g Then i g g if and only if Gg contains the clique K

4

11

which turns out to b e a subgraph of the nonsimplex face g 2

6

Wrapping of c by m

n n

Let call extra iface of c resp ectively m a iface of c resp m which is not a

n n n n

iface of m resp c We recall that all ifaces of c are also ifaces of m for i

n n n n

i

and moreover it was conjectured in that for n large enough n all ifaces of

c are also ifaces of m We disprove this conjecture by exhibiting an extra face of c

n n n

For n let consider the following face of c

n

f f g

3

Prop osition For n the face f is an extra face of c

3 n

7

Proof For n the face f is the face f of c which is itself a face of c

3 5 n

3

T

n

c Tr Tr that is f is a face of c Now let supp ose that f is also a

5 12i 2i1 3 n 3

i=6

face of m it would implies that f is a face of the following face of m

n 3 n

n

g Tr Tr Tr

12i 2i1 1ij

i=6

2ij 5

12

where the triangle facets are seen as facets of m For n we have g g One

n

4

can easily check that g contains b esides the cuts and the

2

vertex x which co ordinates x i j n are x for i j and

ij ij

3

otherwise Then to remove x we should intersect g with some Tr with i j k

ijk

but doing so will also eliminate one of the cuts or of f as well

5

which implies that f cannot b e a face of m 2

3 n

Corollary For n al l ifaces of c are also ifaces of m for exactly i and

n n

Proof Let supp ose that all i faces of c are also i faces of m with i Then

0 n 0 n 0

the face g of m equal to a face f of c containing f will contain a face g equal to

i n i n 3 3

0 0

f which contradicts Prop osition 2

3

Prop osition

2 3 3 4 6 7 8 8 9

i The representative extra ifaces of c belong to F ff f f f f f f f f

5

9 8 7 7 6 6 6 5 5

10 7 7 2 1 1 1 2 5 6

f f f g that is al l extra faces belonging to f and to F fc f f f f f f

5

5 4 3 9 10 9 8 8 7 7

7 9 10 11 8

f f f f f g

7 6 6 5 4

3 6 7 8 8 9 10 11

ii The representative extra ifaces of m belong to G fg g g g g g g g

5

7 6 6 6 5 5 5 5

12 7 8 9 10 11 7 8 9 10 4 5 3 2

g g g g g g g g g g g g g g g that is al l simplex extra faces ie al l

5 4 4 4 4 4 3 3 3 3 2 2 1 0

3 1 1 1 2 4 5 6 9 10 11 13 12

extra faces belonging to g and G fm g g g g g g g g g g g g g

5

7 10 9 8 8 7 7 7 6 6 6 5 4

2 2

that is al l faces given in Figure except g f which is the unique nonsimplex com

6 6

mon face of c and m

5 5

iii Al l remaining faces besides the ones given in i and ii are common faces of c and

5

i i 1 1

m we ordered them so that f g Their representative ifaces belong to H ff f

5

j j 6 5

2 3 1 2 3 4 5 1 2 3 4 5 1 2 3 1 2 1 1

f f f f f f f f f f f f f f f f f f f g that is al l

5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 1 1 0 1

1 1 2 2 3 4 5 4 5 6 7 6 6

common faces belonging to f and to H ff f f f f f f f f f f f g

6 7 7 6 6 6 6 5 5 5 5 4 3

2 11

Remark We observed that F fg f g G g F fg c g G fg gg

5

9 6

10 2

H fg c g Gg and F ff g ff f f F g and that the dimension of

5

5 9

those intersections is one less than the one of the corresponding face The four above

7 10 11 10

equalities g c g c and g c f give the fol lowing four bijections F G

5 5 5

3 3 6 5

10 7 11

g g and F F ff g H G fg F G fg

5 3 6

dimension

total of orbits in c

5

of orbits in F

of orbits in F

of orbits in H

of orbits in H

of orbits in G

of orbits in G

total of orbits in m

5

Figure Numb er of orbits of ifaces

Remark

7 10 2 7

i Al l minimal by inclusion faces from F belong to the orbits of f or f f f

3 5 9 3

6 7

ii Al l minimal by inclusion faces from H belong to the orbits of f or f

3 5

7 6

iii Besides f and the cofacet f Tr Tr Tr Tr any of the remaining

134 134 143 341

3 3

representative faces of c belongs to exactly either or triangle facets

5

Some orbits of faces and nofaces of c and m

n n

In Figure we present orbits of c and m For each orbit a representative

n n

facet the co dimension the size and the numb er of cuts and anticuts for orbits of m

n

5 5 5 5

b elonging to a face of the orbit are given The orbit O resp ectively O O and O

2 3 4 5

1 1 2 2 1 1 2 2

corresp onds to C and M resp ectively C and M C and M and C and M In

8 8 8 8 7 7 6 6

n

m the faces b elonging to O are combinatorially equivalent to m

n n1

5

orbit representative co dimension size cuts anticuts

n

n n5 n5

O Tr Tr

135 246

1

6

n

n n4 n4

O Tr Tr

123 124

2

4

n

n n5 n5

O Tr Tr

123 145

3

5

n

n n2

O Tr Tr

135 453

4

4

n

n n2

O Tr Tr n

135 153

5

2

Figure All pairwise intersections of facets of m

n

We then consider the following set of cuts A f i i ng It was remarked

n

in that no triangle facet contains A Moreover we have

n

Prop osition

i No proper face of c contains A

n n

ii For n any n subset of A forms an extra n face of c

n n

n1

iii The size of the orbit represented by A is jO A j for n

n n

X

Proof Let F b e a facet induced by the inequality f x a and F S the

ij ij

1ij n

value of the left hand side of the inequality on the cut S Since we have

a

F ij F i F j f a A b elongs to F will implies that f

ij n ij

2

ajS j(njS j)

So we have a F S this implies a and therefore for n all

2

a

which contradicts F a To prove ii remark that A f g is f

n ij

2

the following face of c A f g E Tr where E is the switch

n n ij1

(1) (1)

2ij n

X

n n

ing by the cut of the face dened by the inequality x d e b c On

ij

1ij n

the other hand Tr contains which can b e removed only by intersecting

ij1

2ij n

A f g with a nonhomogeneous triangle facet but this will also eliminate some cuts

n

i that is A f g cannot b e a face of m Since for n all switchings of A are

n n n

n1

dierent and any p ermutation amounts to a switching jO A j cases n and

n

are clear 2

Remark Proposition means that A is a minimal by inclusion blocker that

n

is A f for the clutter T ff f is a facet of c g of complements of facets of

n c n

c and a fortiori for the clutter T fg g is a facet of m g Perhaps A is also a

n m n n

minimum that is of minimal cardinality blocker for T Remark that minimum blockers

c

for T are exactly the pairs f S S g

m

We call noface of c a set of cuts which do es not form a face of c

n n

Prop osition

i For n any set containing member of O A is a noface see Proposition

n

ii For n any noface contains a member of O A the only nofaces which are cofaces

4

2 2

belong to the orbits represented by f and f

1 2

iii For n there are exactly orbits of isets of cuts in c for i

5

and among there is exactly orbits of nofaces represented by the fol lowing graphs

One can check that the following holds for any n

n1

the vertices of c form the orbit represented by f g

n

n1

2

n

edges of c are partitioned into the b the c orbits resp ectively represented by

n

2 2

n

n n

i i n2

f f ig for i b c with jO j c and for i b

1 1

2 2 i

n

i n3

for n even jO j

n

1

2

n1

2

the faces of c are partitioned into the orbits resp ectively represented by

n

3

rst

f r s r r s tg for all triplets of integers fr s tg f

2

nr n n

c s r r t minb c b c s n r s such that r b

3 2 2

n

2

16(n 7)

n

4

ridges that is face of m are partitioned into for n the

n

3 2

n n n

the orbits O O and O given in Figure see

1 2 3

n

the facets of m form the orbit represented by Tr

n 123

3

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ANTOINE DEZA

Tokyo Institute of Technology department of mathematical and computing sciences Meguro

ku Ookayama Tokyo Japan Email dezaistitechacjp

MICHEL DEZA

CNRS Ecole Normale Superieure departement mathematiques et informatique rue

dUlm Paris Email dezadmiensfr