Generalized cut and metric polytopes of graphs and simplicial complexes
aa r X i v : . [ m a t h . M G ] J un GENERALIZED CUT AND METRIC POLYTOPES OF GRAPHSAND SIMPLICIAL COMPLEXES
MICHEL DEZA AND MATHIEU DUTOUR SIKIRI´C
Abstract.
Given a graph G one can define the cut polytope CUTP( G ) andthe metric polytope METP( G ) of this graph and those polytopes encodein a nice way the metric on the graph. According to Seymour’s theorem,CUTP( G ) = METP( G ) if and only if K is not a minor of G .We consider possibly extensions of this framework:(1) We compute the CUTP( G ) and METP( G ) for many graphs.(2) We define the oriented cut polytope WOMCUTP( G ) and oriented mul-ticut polytope OMCUTP( G ) as well as their oriented metric versionQMETP( G ) and WQMETP( G ).(3) We define an n -dimensional generalization of metric on simplicial com-plexes. Introduction
The cut polytope [23] is a natural polytope arising in the study of the maximumcut problem [10]. The cut polytope on the complete graph K n has seen muchstudy (see [23]) but the cut polytope on a graph was much less studied [20, 4, 2].Moreover, generalizations of the cut polytope on graphs seems not to have beenconsidered.Given a graph G = ( V, E ), for a vertex subset S ⊆ V = { , . . . , n } , the cutsemimetric δ S ( G ) is a vector (actually, a symmetric { , } -matrix) defined as(1) δ S ( x, y ) = (cid:26) xy ) ∈ E and | S ∩ { x, y }| = 1 , cut polytope CUTP( G ), respectively cut cone CUT( G ), are defined as the convexhull of all such semimetrics, respectively positive span of all non-zero ones amongthem. The dimension of CUTP( G ) and CUT( G ) is equal to the number of edgesof G .The metric cone MET( K n ) is the set of all semimetrics on n points, i.e., thefunctions d : { , . . . , n } → R ≥ (actually, symmetric matrices over R ≥ havingonly zeroes on the diagonal), which satisfy all 3 (cid:0) n (cid:1) triangle inequalities d ( i, j ) + d ( i, k ) − d ( j, k ) ≥
0. The bounding of MET( K n ) by (cid:0) n (cid:1) perimeter inequalities d ( i, j ) + d ( i, k ) + d ( j, k ) ≤ metric polytope METP( K n ).For a graph G = ( V, E ) of the order | V | = n , let MET( G ) and METP( G ) denotethe projections of MET( K n ) and METP( K n ), respectively, on the subspace R | E | indexed by the edge set of G . Clearly, CUT( G ) and CUTP( G ) are projections of,respectively, CUT( K n ) and CUTP( K n ) on R E . It holdsCUT( G ) ⊆ MET( G ) and CUTP( G ) ⊆ METP( G ) . Key words and phrases. max-cut problem, cut polytope, metrics, graphs, cycles, quasi-metrics,hemimetrics.
In Section 2 we consider the structure of those polytopes and give the descriptionof the facets for many graphs (see Tables 1 and 2). The data file of the groups andorbits of facets of considered polytopes is available from [24].The construction of cuts and metrics can be generalized to metrics which are notnecessarily symmetric are considered in Section 3 (see also [19, 16]). The triangleinequality becomes d ( i, j ) ≤ d ( i, k ) + d ( k, j ) and the perimeter inequality becomes d ( i, j ) + d ( j, k ) + d ( k, i ) ≤ ≤ i, j, k ≤ n . We also need the inequalities0 ≤ d ( i, j ) ≤
1. The quasi metric polytope QMETP( K n ) is defined by the aboveinequalities and the quasi metric cone QMET( K n ) is defined by the inequalitiespassing by zero. The quasi metric cone QMET( G ) and polytope QMETP( G ) aredefined as projection of above two cone and polytopes. In Theorem 3 we give aninequality description of those projections.Given an ordered partition ( S , . . . , S r ) of { , . . . , n } we defined an orientedmulticut as: δ ′ ( S , . . . , S r ) x,y = (cid:26) x ∈ S i , y ∈ S j and i < j, . The convex cone of the oriented multicut is the oriented multicut cone OMCUT( K n ).The convex polytope can also be defined but there are vertices besides the orientedmulticuts. A smaller dimensional cone WQMET( G ) and polytope WQMETP( G )can be defined by adding the cycle equality d ( i, j ) + d ( j, k ) + d ( k, i ) = d ( j, i ) + d ( k, j ) + d ( i, k )to the cone QMET( G ) and polytope QMETP( G ). A multicut satisfies the cycleequality if and only if r = 2. We note the corresponding cone WOMCUT( G ) andWOMCUTP( G ). In Section 3 we consider those cones and polytopes and theirfacet description.The notion of metrics can be generalized to more than 2 points and we obtainthe hemimetrics . Those were considered in [15, 14, 17, 21]. Only the notion ofcones makes sense in that context. The definition of the above papers extends thetriangle inequality in a direct way: It becomes a simplex inequality with the areaof one side being bounded by the sums of area of the other sides. In [12] we arguedthat this definition was actually inadequate since it prevented right definition ofhemimetric for simplicial complex. In Section 4 we give full details on what weargue is the right definition of hemimetric cone.There is much more to be done in the fields of metric cones on graphs and sim-plicial complexes. Besides further studies of the existing cones and the ones definedin this paper, two other cases could be interesting. One is to extend the notion ofhypermetrics cone HYP( K n ) to graphs; several approaches were considered in [18],for example projecting only on the relevant coordinates, but no general results wereproved.Another generalization that could be considered is the diversities considered in[7, 8]. Diversity cone
DIV n is the set of all diversities on n points , i.e., the functions f : { A : A ⊆ { , . . . , n }} → R ≥ satisfying f ( A ) = 0 if | A | ≤ f ( A ∪ B ) + f ( B ∪ C ) ≥ f ( A ∪ C ) if B = ∅ . The induced diversity metric d ( i, j ) is f ( { i, j } ). ENERALIZED CUT AND METRIC POLYTOPES OF GRAPHS AND SIMPLICIAL COMPLEXES3
Cut diversity cone
CDIV n is the positive span of all cut diversities δ ( A ), where A ⊆ { , . . . , n } , which are defined, for any S ⊆ { , . . . , n } , by δ S ( A ) = (cid:26) A ∩ S = ∅ and A \ S = ∅ , CDIV n is the set of all diversities from DIV n , which are isometrically embed-dable into an l -diversity , i.e., one, defined on R m with m ≤ (cid:0) n ⌊ n ⌋ (cid:1) by f m ( A ) = m X i =1 max a,b ∈ A {| a i − b i |} . These two cones are extensions of the MET( K n ) and CUT( K n ) on a completehypergraphs and it would be nice to have a nice definition on any hypergraph.2. Structure of cut polytopes of graphs
The cut metric δ S defined at Equation (1) satisfies the relation δ { ,...,n }− S = δ S .The cut polytope CUTP( K n ) is defined as the convex hull of the metrics δ S andthus has 2 n − vertices.For a given subset S of { , . . . , n } we can define the switching operation F S by F S ( d )( i, j ) = (cid:26) − d ( i, j ) if | S ∩ { i, j }| = 1 ,d ( i, j ) otherwise . The operation on cuts is F S ( δ T ) = δ S ∆ T with ∆ denoting the symmetric difference(see [23] for more details). For a graph G we define CUTP( G ) to be the projectionof CUTP( K n ) on the coordinates corresponding to the edges of the graph G . If G is connected then CUTP( G ) has exactly 2 n − vertices. Then δ S can be seen also asthe adjacency matrix of a cut (into S and S ) subgraph of G . The cut cone CUT( G )is defined by taking the convex cone generated by the metrics δ S but it is generallynot used in that section.In fact, CUT( K n ) is the set of all n -vertex semimetrics, which embed isometri-cally into some metric space l , and rational-valued elements of CUT( K n ) corre-spond exactly to the n -vertex semimetrics, which embed isometrically, up to a scale λ ∈ N , into the path metric of some m -cube K m . It shows importance of this conein Analysis and Combinatorics. The enumeration of orbits of facets of CUT( K n )and CUTP( K n ) for n ≤ n = 5, 6, 7 respectively, andin [9], completed by [20], for n = 8.2.1. Automorphism group of cut polytopes.
The symmetry group Aut( G )of a graph G = ( V, E ) induces symmetry of CUTP( G ). For any U ⊂ { , . . . , n } ,the map δ S δ U ∆ S also defines a symmetry of CUTP( G ). Together, those formthe restricted symmetry group ARes (CUTP( G )) of order 2 | V |− | Aut( G ) | . The fullsymmetry group Aut(CUTP( G )) may be larger. In Tables 1, 2, such cases aremarked by ∗ . Denote 2 −| V | | Aut(CUTP( G )) | by A ( G ).For example, | Aut(CUTP( K n )) | is 2 n − n ! if n = 4 and 6 ×
4! if n = 4 ([13]). Remark 1. (i) If G = ( V, E ) is P rism m ( m = 4 ), AP rism m ( m > ), M¨obiusladder M m and P yr ( C m ) ( m > ), then Aut ( G ) = 4 m .(ii) If G is a complete multipartite graph with t parts of size a , . . . , t r parts ofsize a r , with a < a < · · · < a r and all t i ≥ , then | Aut ( G ) | = Q ri =1 t i !( a i !) t i . MICHEL DEZA AND MATHIEU DUTOUR SIKIRI´C (iii) Among the cases considered here, all occurrences of A ( G ) > | Aut ( G ) | are: A ( G ) = m !2 m − | Aut ( G ) | for G = K ,m> , K , ,m> and A ( G ) = 6 | Aut ( G ) | , i.e., m ! = 48 , m ! for G = K , and K , , , , respectively.(iv) If G = P m ( m ≥ edges), then | Aut ( G ) = 2 , while A ( G ) = m ! = ( | V | − .If G = C m ( m > ), then | Aut ( G ) | = 2 m , while A ( G ) = 2 m ! for m = 4 and A ( G ) = m ! = | V | ! for m ≥ . Edge faces, s -cycle faces and metric polytope.Definition 1. Let G = ( V, E ) be a graph.(i) Given an edge e ∈ E , the edge inequality (or ) is x ( e ) ≥ . (ii) Given a s -cycle c = ( v , . . . , v s ) , s ≥ , of G , the s -cycle inequality is: x ( c, ( v , v s )) = s − X i =1 x ( v i , v i +1 ) − x ( v , v s ) ≥ . The edge inequalities and s -cycle inequalities are valid on CUTP( G ), since theyare, clearly, valid on each cut: a cut intersects a cycle in the set of even cardinality.So, they define faces, but not necessarily facets. In fact, it holds Theorem 1. (i) The inequality x ( e ) is facet defining in CUTP( G ) (also, in CUT( G ) )if and only if e is not contained into a -cycle of G .(ii) An s -cycle inequality is facet defining in CUTP( G ) (also, in CUT( G ) ) if andonly corresponding s -cycle is chordless.(iii) METP( G ) is defined by all edge and s -cycle inequalities, while MET( G ) isdefined by all s -cycle inequalities. In fact, (i) and (ii) above were proved in [6], (iii) was proved in [5]; see alsoSection 27.3 in [23].The following Theorem, proved in [30] for cones and in [4] for polytopes, clarifieswhen the metric and cut polytope coincides:
Theorem 2.
CUT( G ) = MET( G ) or, equivalently, CUTP( G ) = METP( G ) if andonly if G does not have any K -minor. As a corollary of Theorem 2, we have that the facets of CUTP( G ) (also, inCUT( G )) are determined by edge inequalities and s -cycle inequalities if and onlyif G does not have any K -minor.3-cycle inequality is usual triangle inequality; in fact, it is unique, among edgeand all s -cycle inequalities to define a facet in a CUTP( K n ).The girth and circumference of a graph, having cycles, are the length of theshortest and longest cycle, respectively. In a graph G , a chordless cycle is anycycle, which is induced subgraph; so, any triangle, any shortest cycle and anycycle, bounding a face in some embedding of G , are chordless. Let c ′ s and c s denotethe number of all and of all chordless s -cycles in G , respectively.There are 2 | E | edge faces, which decompose into orbits, one for each orbit ofedges of G under Aut( G ). There are 2 s − c ′ s s -cycle faces, which decompose intoorbits, one for each orbit of s -cycles of G under Aut( G ).The incidence of edge faces is 2 | V |− and the size of each orbit is twice the sizeof corresponding orbit of edges. The incidence of s -cycle faces is 2 | V |− s s and thesize of each orbit is 2 s − times the size of corresponding orbit of s -cycles in G . ENERALIZED CUT AND METRIC POLYTOPES OF GRAPHS AND SIMPLICIAL COMPLEXES5
Table 1.
The number of facets of CUTP( G ) of some K -minor-free graphs G ; ∗ shows A ( G )=2 −| V | | Aut(CUTP( G )) | > | Aut ( G ) | G = ( V, E ) | V | , | E | A ( G ) Number of facets Orbit’s s M¨obius ladder M , , , , M = K , , , K , , ,m , m > m +3 , m +3 3! m ! 4 + 12 m (2) 3 , K , ,m , m > m +3 , m +2 | Aut ( K , ,m ) | m + 8 (cid:0) m (cid:1) (2) 3 , K ,m , m ≥ m +3 , m | Aut ( K ,m ) | m + 24 (cid:0) m (cid:1) (2) 2 , K ,m , m > m + 2 , m m − m ! | Aut ( K ,m ) | ∗ m (1) 2 with 4 K , , | Aut( K , ) | ∗ K , ,m , m > m +2 , m +1 2 m − m ! | Aut ( K , ,m ) | ∗ m (1) 3 K m +1 - K m = K ,m , m> m +1 , m m ! 2 m (1) 2 AP rism ,
24 24 2 , , , , , AP rism ,
20 20 552(4) 3 , , , AP rism ,
16 16 176(3) 3 , , P rism ,
21 28 7 , , , , , , P rism ,
18 24 2 , , , , , , P rism ,
15 20 742(5) 2 , , , , P rism , , , ,
18 24 540(4) 2 , , , ,
24 48 1 , , , , , ,
30 120 23 , , , , , ,
30 120 1 , , , , K ,
12 48 200(3) 2 , , K , , ,
12 48 56(2) 3 , K , | Aut ( K ) | ∗ Wagner’s theorem [32], a finite graph is planar if and only if it has no minors K and K , . For embeddability on the projective plane P , there are exactly 103forbidden topological minors and exactly 35 forbidden minors (see [1, 27]). Forembeddability on the torus T , 16629 forbidden minors are known (see [26]) butthe list is not necessarily complete. Closely related Kuratowski’s theorem [29] statesthat a finite graph is planar if and only if it does not contain a subgraph that is asubdivision of K or of K , .2.3. Skeletons of Platonic and semiregular polyhedra.
Let G be embeddedin some oriented surface; so, it is a map ( V, E, F ), where F is the set of faces of G . Let ~p = ( . . . , p i , . . . ) denote the p -vector of the map, enumerating the number p i > i , existing in G .Call face-bounding any s -cycle of G , bounding a face in map G . Call an s -cycleof G i -face-containing , edge-containing or point-containing , if all its interior pointsform just i -gonal face, edge or point, respectively. Call equator any cycle C , theinterior of which (plus C ) is isomorphic to the exterior (plus C ).The chordless 4 , , , , , , MICHEL DEZA AND MATHIEU DUTOUR SIKIRI´C
Table 2.
The number of facets of CUTP( G ) for some graphs G with K -minor G = ( V, E ) | V | , | E | A ( G ) Number of facets (orbits) Orbit’s s Heawood graph 14 ,
21 336 5 , , , , ,
15 120 3 , , , M ,
15 20 1 , , , , M ,
18 24 26 , , , , , M ,
21 28 369 , , , , , K , ,
25 2(5!) , , , , , K , ,
28 4!7! 271 , , , K , ,
24 4!6! 23 , , , K , ,
20 4!5! 983 , , K , ,
16 2(4!) , , K , , ,
27 (3!) , , , , K , , ,
24 2(4!) , , , K , , ,
23 3!5! 71 , , K , , ,
19 3!4! 12 , , K , , ,
15 2(3!) , K , , , ,
21 4(3!) , , , K , , , ,
14 3!(2!) , , , K , , , ,
13 4(2!) , , K , , ,m , m > m +4 , m +5 4 m ! 8+20 m +8 (cid:0) m (cid:1) (16 m − , , , K , , , ,m , m > m +4 , m +6 4! m ! 8(8 m − m + 2)(4) 3 , K , , , , , = K − K .
25 360 2 , , , K , , , , , = K − K ,
20 240 31 , , K − C ,
18 144 520(4) 3 , K − C ,
17 48 108(4) 3 , , K − C = P yr ( C ) 7 ,
16 20 780(6) 3 , , K − C = P yr ( P rism ) 7 ,
15 12 452(5) 3 , , , K − C ,
14 14 148(3) 3 , P yr ( P rism ) 9 ,
20 48 10 , , , P yr ( P rism ) 11 ,
25 20 208 , , , , , P yr ( AP rism ) 9 ,
24 16 389 , , , , , P yr ( C ) 8 ,
19 24 3 , , , P yr ( C ) 9 ,
22 28 14 , , , P ,
18 48 62 , , , , , central circuits and zigzags (see [22]), respectively.All c chordless 6-cycles of Icosahedron are exactly their 30 edge-containing onesand 10 face-containing ones, which are exactly the 10 equators and the weak zigzags ([22]). All c chordless 10-cycles of Dodecahedron are 30 edge-containing ones and6 face-containing ones, which are exactly all 6 equators and the zigzags. Proposition 1. If G is the skeleton of a Platonic solid, then all possible facets of CUTP( G ) are: edge facets and s -cycle facets, coming from all face-bounding cycles ENERALIZED CUT AND METRIC POLYTOPES OF GRAPHS AND SIMPLICIAL COMPLEXES7 and from all (if they exist and not listed before) vertex-, edge-, face-containingcycles.For instance:(i) If G = K (Tetrahedron), then CUTP( G ) has unique orbit of p = 16 (simplicial) -cycle facets (from all | F | = p = 4 face-bounding cycles of G ).(ii) If G = K , , (Octahedron), then CUTP( G ) has facets in orbits, namely:orbit of p -cycle facets (from all | F | = p = 8 face-bounding cycles, orbit of c -cycle facets (from all c = | V | = 3 vertex-containing -cycles).(iii) If G = K (Cube), then CUTP( G ) has facets in orbits, namely:orbit of | E | = 24 edge facets, orbit of p -cycle facets (from all | F | = p = 6 face-bounding cycles),orbit of c = 128 6 -cycle facets (from all c = 4 vertex-containing -cycles).(iv) If G is Icosahedron, then CUTP( G ) has , facets in orbits, namely:orbit of p = 80 3 -cycle facets (from all | F | = p = 20 face-bounding cycles),orbit of c = 192 5 -cycle facets (from all c = 12 vertex-containing -cycles),orbit of | E | = 960 6 -cycle facets (from | E | = 30 edge-containing -cycles),orbit of
320 6 -cycle facets (from | F | = 10 face-containing -cycles).(v) If G is Dodecahedron, then CUTP( G ) has , facets in orbits, namely:orbit of | E | = 60 edge facets, orbit of p = 192 5 -cycle facets (from all | F | = p = 12 face-bounding cycles), orbit of c = 5 ,
120 9 -cycle facets (from all c = 20 vertex-containing -cycles), orbit of | E | = 15 ,
360 10 -cycle facets (from edge-containing -cycles), orbit of × ,
072 10 -cycle facets (from | F | = 6 face-containing -cycles). In a Truncated Tetrahedron, call ring-edges those bounding a triangle, and rung-edges all 6 other ones.
Proposition 2. (i) If G is Truncated Tetrahedron, then CUTP( G ) has facets: (1) orbit of × edge facets (from all rung-edges), (2) orbit of p -cycle facets (from all p = 4 3 -face-bounding cycles), (3) orbit of p -cycle facets (from all p = 4 6 -face-bounding cycles), (4) orbit of × -cycle facets (from (cid:0) (cid:1) rung-edge-containing -cycles, whichare also the equators).(ii) If G is Cuboctahedron, then CUTP( G ) has , facets, namely: (1) orbit of p -cycle facets (from all p = 8 3 -face-bounding cycles), (2) orbit of p -cycle facets (from all p = 6 4 -face-bounding cycles), (3) orbit of | V | -cycle facets (from all vertex-containing -cycles), (4) orbit of × -cycle facets (from all p -face-containing -cycles,which are also equators and the central circuits), (5) orbit of p = 768 8 -cycle facets (from all -face-containing -cycles,which are also zigzags). Given a
P rism m ( m = 4) or an AP rism m ( m = 3), we call rung-edges the edgesconnecting two m -gons, and ring-edges other 2 m edges.Let P be an ordered partition X ∪ · · · ∪ X t = { , . . . , m } into ordered sets X i of | X i | ≥ P -cycle of P rism m the chordless ( m + 2 t )-cycleobtained by taking the path X on the, say, 1-st m -gon, then rung edge (in thesame direction, then path X on the 2-nd m -gon, etc. till returning to the path MICHEL DEZA AND MATHIEU DUTOUR SIKIRI´C X . Any vertex of P rism m can be taken as the 1-st element of X , in order tofix a P -cycle. So, a P -cycle defines an orbit of 2 m +2 t − m ( m + 2 t )-cycle facetsof CUTP( P rism m ), except the case ( | X | , . . . , | X m | ) = ( | X | , . . . , | X t | , | X | ) whenthe orbit is twice smaller.A P -cycle of AP rism m is defined similarly, but we ask only | X i | ≥ m -gon, should be selected, in the cases | X i | = 2 , P . Clearly, P -cycles are are all possiblechordless t -cycles with t = 4 , m for P rism m and with t = 2 , m for AP rism m . Proposition 3. (i) If G is P rism m ( m ≥ ), then all facets of CUTP( G )) are: (1) orbit of m edge facets (from all m rung-edges) (2) orbit of m edge facets (from all m ring-edges); (3) orbit of p = 8 m -cycle facets (from all m -face-bounding -cycles); (4) orbit of m − p m of m -cycle facets (from both m -face-bounding m -cycles); (5) orbits of cycle facets for all possible P -cycles.(ii) If G is AP rism m ( m ≥ ), then all facets of CUTP( G )) are: (1) orbit of p = 8 m -cycle facets (from all m -face-bounding -cycles); (2) orbit of m − p m of m -cycle facets (from both m -face-bounding m -cycles); (3) orbits of cycle facets for all possible P -cycles. M¨obius ladders and Petersen graph.
All M¨obius ladders M m are toroidal.M¨obius ladder M = K , , Petersen graph and Heawood graph are both, toroidaland 1-planar.Given the M¨obius ladder M m , call ring-edges m those belonging to the 2 m -cycle C ,..., m , and rung-edges all other ones, i.e., ( i, i + m ) for i = 1 , . . . , m .For any odd t dividing m , denote by C ( m, t ) the ( m + t )-cycle of M m , having,up to a cyclic shift, the form1 , . . . , mt , mt + m, . . . , mt + m, mt + 2 m, . . . , mt + 2 m, . . . , i.e., t consecutive sequences of mt − C ( m,
1) exists for any m ≥
3; for t >
1, their existence requires divisibility of m by t . Clearly, the number of ( m + t )-cycles C ( m, t ) is mt . Conjecture 1. If G = M m ( m ≥ ), then among facets of CUTP( G ) there are:two orbits of m and m edge facets (from all m ring- and m rung-edges),orbit of c = 8 m -cycles facets (from all m -cycles),orbit of m m ( m + 1) -cycle facets (from all m ( m + 1) -cycles C ( m, ),for any odd divisor t > of m , orbit of m + t mt ( m + t ) -cycle facets (from all ( m + t ) -cycles C ( m, t ) ). There are no other orbits for m = 3 , m = 3 first two orbits unite intoone of 18 edge facets, while all other orbits unite into one of 2 c = 72 4-cyclefacets. CUTP( M ) has only one more orbit: the orbit of 2 facets of incidence 15(i.e., simplicial facets), defined by a cyclic shift of X i =1
12 (3 − ( − i ) x i,i +1 + m X i =0 x i,i + m − x , + 2 x , + x , ) . CUTP( M ) also has only one more orbit: 2 ENERALIZED CUT AND METRIC POLYTOPES OF GRAPHS AND SIMPLICIAL COMPLEXES9
Petersen graph has three circuit double covers: by six 5-gons (actually, zigzags),by five cycles of lengths 9 , , , , , , , ,
5. It can beembedded in projective plane, in torus and in Klein bottle with corresponding setsof six, five and five faces.Petersen graph have only 5 − , − , − and 9-cycles; it has c = 12 and c = 10.Heawood graph, i.e., (3 , -cage , have the girth 6 and c = 28, c = | E | = 21. Proposition 4.
CUTP(
P etersen graph ) has , facets in orbits: (1) orbit of | E | = 30 edge facets, (2) orbit of c = 192 5 -cycle facets, (3) orbit of c = 320 5 -cycle facets, (4) orbit of simplexes, represented by ( C − x ) − ( C ′ ′ ′ ′ ′ − x ′ ′ − x ′ ′ ) + 2 X ≤ i ≤ x ii ′ , where Petersen graph is seen as C + C ′ ′ ′ ′ ′ + P ≤ i ≤ x ii ′ . Remark 2.
Three of all orbits of facets of CUTP(
Heawood graph ) , are: (1) 2 | E | = 42 edge facets, (2) 2 c = 896 6 -cycle facets and (3) 2 c = 2 ,
688 8 -cycle facets.
Complete-like graphs. K n is toroidal only for n = 5 , ,
7, while it is 1-planar only for n = 5 ,
6. Among complete multipartite graphs G , the planar onesare: K ,m ; K , ,m ; K , , ; K , , , = K and their subgraphs. The 1-planar G are,besides above: K ; K , , , ; K , , , ; K , , , ; K , , , , and their subgraphs ([11])Given sets A , . . . , A t with t ≥ ≤ | A | ≤ · · · ≤ | A t | , let G be completemultipartite graph K a ,...,a t with a i = | A i | for 1 ≤ i ≤ t .All possible chordless cycles in G are c = P ≤ i
3. Thefacets of CUTP( G ) are the orbit of 2 m edge facets for i = 1, the orbit of 2 m i = 2 and two orbits (of sizes 12 m and 4) of 3-cycle facets for i = 3.Some of remaining cases presented in Table 2. For G = K m +4 − K m = K , , , ,m> and K , , ,m> , the number of orbits stays constant for any m : 4 and 7, respectively.Given sequence b , . . . , b n of integers, which sum to 1, let us call hyp ( b ) = X ≤ i,j ≤ n x ij b i b j ≤ hypermetric inequality . Note that hyp (1 , , − , , . . . ,
0) isusual triangle inequality. Denote hyp ( b ) with all non-zero b i being b x = b y = 1 = − b z by T r ( x, y ; z ) and hyp ( b ) with all non-zero b i being b x = b y = b z = 1 = − b u = − bv by P ent ( x, y, z ; u, v ).If G = K , , ,m with m ≥
3, then CUTP( G ) has 8 + 20 m + 8 (cid:0) m (cid:1) (16 m −
15) facetsin 7 orbits: 3 orbits of 8 , m, m (cid:0) m (cid:1) (cid:0) m (cid:1) , (cid:0) m (cid:1) , (cid:0) m (cid:1) { , ± } -valued non-s-cycle facets, having 4 values − , ,
12 values of 1. The partition is { } , { } , { , } , { , . . ., m + 4 } .CUTP( K , , , ) has 184 facets in 4 orbits: 2 orbits of 8 + 8 ,
32 3-cycle facets, oneorbit of 8 4-cycle facets and one orbits of 2 facets, represented by hyp (1 , , , − , − ,
0) + hyp (0 , , , , , − , − ≤ . The graph G = K m + t − K m = K ,..., ,m has a K -minor only if t ≥
4. If m ≥ G ) has 2 orbits of 4 m (cid:0) t (cid:1) and 4 (cid:0) t (cid:1) t < { } , . . . , { t } , { t + 1 , . . . , t + m } .If G = K m +4 − K m , then CUTP( G ) has 8(8 m − m + 2) facets in 4 orbits: 2orbits of 24 m , 16 3-cycle facets and 2 orbits of sizes 16 m, (cid:0) m (cid:1) , represented by hyp (1 , , − , − , , , . . . , ≤ , i.e., P ent (1 , ,
5; 3 ,
4) and hyp (1 , , − , , , − , , . . . ,
0) + hyp (0 , , , − , , , , . . . , ≤ . If G = K m +5 − K m , then among many orbits of facets of CUTP( G ), thereare 2 orbits of 40 , m , m, m ( m −
1) facets,represented, respectively, by(1) hyp (1 , , , − , − , , . . . , ≤ hyp (1 , , − , − , , , , . . . , ≤ hyp (1 , − , − , , , , , , . . . ,
0) + hyp (0 , , , , , − , , . . . , ≤ K − K , two (of sizes 2 ,
60) are { , ± } -valued; they are represented, respectively, by(1) hyp (1 , , − , − , , , −
1) + ( x + x − x , + x − x ) ≤ x + x + x + x ) + ( x + x + x + x ) − ( x + x − x + x + x + x ).Let G = P yr ( C m ). Clearly, it is K , K if m = 2 ,
3, respectively. For m ≥ A ( G ) = 4 m and all chordless cycles 3 m triangles and unique m -cycle. Anyof 3 m + 1 edges belongs to a triangle. So, among orbits of facets of CUTP( G ),there are two (of size 8 m and 4 m ) orbits of 3-cycle facets and orbit of 2 m − m -cyclefacets. All other facets for m ≤ { , ± } -valued.For P yr ( C ...m ) with m = 4, unique remaining orbit consists of 2 facets,represented by P ent (3 , ,
5; 1 ,
2) +
T r (1 ,
2; 4). Among remaining orbits for m = 5and 7, there is an orbit of 2 m +1 facets represented by(1) P yr ( C ) − x + x )+ ( x + x + x + x )) ≤ P yr ( C ) − x + x ) + ( x + x + x + x x + x )) ≤ m = 5, two remaining orbits (each of size 2
5) are represented by(1) C − x + x + (( x − x ) − ( x − x ) + ( x − x )) ≤ C − x + x +(( x − x ) − ( x − x )+( x − x )) ≤
0, respectively.For m = 6, one of 4 remaining orbits (of size 2
6) is represented by C − x + x + (( x − x ) + ( x − x ) − ( x − x )) ≤ K − C = P yr ( C ). Now, G = K − C = K { } , { } , { } , { , } , { , } has c = 19; CUTP( G ) has four orbits of facets: three (of sizes 48 , ,
4) of 3-cyclefacets and one orbit of size 32, represented by
P ent (4 , ,
6; 2 , K -minors, K { , , , , } and K { , , , , } provides 16 of above 32 facets. ENERALIZED CUT AND METRIC POLYTOPES OF GRAPHS AND SIMPLICIAL COMPLEXES11 G = K − C has c = c = 7; CUTP( G ) has three orbits of facets: one (of size28) of 3-cycle facets, one (of size 56) of 4-cycle facets and one of size 64, representedby ( K − C ) − x + P ath ).3.
Quasi-metric polytopes over graphs
We first define the inequalities satisfied by quasi-metrics on n -points. Definition 2.
Given a fixed n ≥ we define:(i) The oriented triangle inequality for all ≤ i, j, k ≤ nd ( i, j ) ≤ d ( i, k ) + d ( k, j ) (ii) The non-negativity inequality for all ≤ i, j ≤ n is d ( i, j ) ≥ (iii) A bounded oriented metric is a metric satisfying for all ≤ i, j, k ≤ n theinequalities d ( j, i ) + d ( i, k ) + d ( k, j ) ≤ and d ( i, j ) ≤ . Using this we can define the cone of quasimetrics QMET( K n ) (see [19, 16] formore details) to be the cone of oriented metrics satisfying the inequalities (i), (ii)of 2. We define the polytope QMETP( K n ) to be the set of metrics satisfying theinequalities of 2.Given a subset S ⊂ { , . . . , n } we define the oriented switching : F S ( d )( i, j ) = (cid:26) − d ( j, i ) if | S ∩ { i, j }| = 1 ,d ( i, j ) otherwise . The symmetric group
Sym ( n ) acts on QMET( K n ) and define a group of size n !.The oriented switchings determine and Sym ( n ) act on QMETP( K n ) and determinea group of size 2 n − n !.The cone MET( K n ) and polytope METP( K n ) are embedded into QMET( K n )and QMETP( K n ) but we have another interesting subset: Definition 3.
Given n ≥ and an oriented metric d ∈ QMET( K n ) , d is called weightable if it satisfies the following equivalent definitions:(i) An oriented metric is called weightable if there exist a function w i such thatfor all ≤ i, j ≤ n d ( i, j ) + w i = d ( j, i ) + w j (ii) For all ≤ i, j, k ≤ n we have d ( i, j ) + d ( j, k ) + d ( k, i ) = d ( j, i ) + d ( k, j ) + d ( i, k )We thus define the cone WQMET( K n ) and polytope WQMETP( K n ) to be theset of weightable quasimetrics of the cone QMET( K n ) and polytope QMETP( K n ).Clearly, the oriented switching preserves WQMETP( K n ).With all those definitions we can now define the corresponding objects on graphs: Definition 4.
Let G be an undirected graph; we define E ( G ) the set of edges and Dir ( E ( G )) to be the set of directed edges of G :(i) We define the cones QMET( G ) and WQMET( G ) to be the projections of thecones QMET( K n ) and WQMET( K n ) on R Dir ( E ( G )) .(ii) We define the polytopes QMETP( G ) and WQMETP( G ) to be the projectionsof the polytopes QMETP( K n ) and WQMETP( K n ) on R Dir ( E ( G )) . We can now give a description by inequalities of QMET( G ): Theorem 3.
For a given graph G the polyhedral cone QMET( G ) is defined as theset of functions R Dir ( E ) such that(i) For any directed edge e = ( i, j ) of G the inequality ≤ d ( i, j ) .(ii) For any oriented cycle e = ( v , v , . . . , v m ) of G (2) d ( v , v m ) ≤ d ( v , v ) + d ( v , v ) + · · · + d ( v m − , v m ) The same results holds for
WQMET( G ) by adding the extra condition that thereexist a function w such that d ( i, j ) − d ( j, i ) = w i − w j .Proof. Our proof is adapted from the proof of [23, Theorem 27.3.3]. It is clear thatthe cycle inequalities (i) and (ii) are valid for d ∈ QMET( K n ) and that edges of G do not occur in their expression. Therefore, the inequalities are also valid for theprojection.The proof of sufficiency is done by induction and is more complicated. Supposethat the result is proved for G + e , i.e. G to which an edge e = ( i, j ) has beenadded. Suppose we have an element x of R Dir ( E ( G )) satisfying all oriented cycleinequalities.We need to find an antecedent of x , i.e. a function y ∈ R Dir ( E ( G )+ e ) . That is weneed to find y ( i, j ) and y ( j, i ).We write P i,j to be the set of directed paths from i to j in G . Assume first that P i,j = ∅ . We write u i,j = min u ∈ P i,j x ( u )since x is non-negative, we have u i,j ≥
0. We then write l i,j = max v ∈ P i,j ,f ∈ v x ( r ( f )) − x ( v − f )with r ( f ) the reversal of the directed edge f . If P i,j = ∅ , i.e. if the edge e isconnecting two connected components of G then we set l i,j = u i,j = 0.We have l i,j ≤ u i,j since otherwise we could take a path u realizing the minimum u i,j , a path v and directed edge f realizing the maximum l i,j put it together andget a counterexample to the oriented cycle inequality (ii).So, we can find a value y i,j such that l i,j ≤ y i,j ≤ u i,j and since u i,j ≥ y i,j ≥
0. The same holds for y j,i . Therefore wefound an antecedent of x in R Dir ( E ( G )+ e ) and this proves the result for QMET( G )and so the stated theorem.For WQMET( G ) we have to adjust the induction construction. If P i,j = ∅ thenwe can adjust the values of the weights w such that w i = w j . This is possible sincethe weights are determined up to a constant term.On the other hand if P i,j is not empty then the weight is already given and weshould get in the end y i,j − y j,i = w i − w j . Actually this is not a problem since itcan be easily be shown that u i,j − u j,i = w i − w j and l i,j − l j,i = w i − w j and sothe inductive construction works. (cid:3) Now we turn to the construction for the polytope case.
Theorem 4.
For a given graph G the polytope QMETP( G ) is defined as the set offunctions R Dir ( E ) such that(i) For any directed edge e = ( i, j ) of G the inequality ≤ d ( i, j ) ≤ holds ENERALIZED CUT AND METRIC POLYTOPES OF GRAPHS AND SIMPLICIAL COMPLEXES13 (ii) For any oriented cycle C = ( v , v , . . . , v m ) of G and subset F of odd size (3) X f =( v,v ′ ) ∈ F d ( v ′ , v ) − X f =( v,v ′ ) ∈ C − F d ( v, v ′ ) ≤ | F | − The same results holds for
WQMETP( G ) with the extra condition that thereexist a function w such that d ( i, j ) − d ( j, i ) = w i − w j .Proof. The proof follows by remarking that the inequalities (i) and (ii) are theoriented switchings of the non-negative inequality and oriented cycle inequality 2.Thus the proof follow from Theorem 3 and the same proof strategy as [23, Theorem27.3.3]. (cid:3)
The oriented multicut cones defined in the introduction are very complicated.In particular the oriented multicuts are not stable under oriented switchings. How-ever, we have WOMCUTP( K n ) = WQMETP( K n ) for n ≤
4. Based on that andanalogy with Theorem 2 a natural conjecture would be that WOMCUTP( G ) =WQMETP( G ) if G has no K minor. But it seems that for some other graphs withno K minor we have WOMCUTP( G ) = WQMETP( G ).4. hemi-metric polytopes over simplicial complexes We can also generate metrics to a measure of distance of more than 2 objects.Our approach differs from [15, 14, 17, 21] and has the advantage of allowing todefine it on complexes.We consider by
Set n,m the set of subsets of m + 1 points of { , . . . , n } . Definition 5.
Let us fix m ≥ and n :(i) A m -dimensional complex is formed by a subset of Set n,m .(ii) A closed manifold of dimension m is formed by a subset S of Set n,m such thatfor each subset S of m points of { , . . . , n } the number of simplices of S containing S is even. For the case m = 1 the closed manifold of above definition corresponds to theclosed cycles. We now proceed to defining the corresponding cycle inequalities: Definition 6.
Let us fix m ≥ and n . Given a m -dimensional complex K on { , . . . , n } , the hemimetric cone HMET( K ) is formed by the functions d on K satisfying(i) the non-negative inequalities d (∆) ≥ for all ∆ ∈ K .(ii) For all closed manifolds (∆ , . . . , ∆ r ) formed by simplices ∆ i ∈ K the in-equalities d (∆ i ) ≤ X ≤ j ≤ r,i = j d (∆ j ) for all ≤ i ≤ r . For m = 1 the definition corresponds to the one of MET( G ). Theorem 5.
Let us fix m ≥ and n . Let us take K a m -dimensional complex on n points. The cone HMET( K ) is the projection of HMET(
Set n,m ) on the simplicesincluded in K . Proof.
Our proof is adapted from the proof for metric of [23, Theorem 27.3.3]. Theinequalities for HMET( K ) are clearly valid on HM ET ( Set n,m ) which proves oneinclusion.We want to prove it by induction the other inclusion. Suppose that we havea metric d ∈ HMET( K ) and a simplex ∆ / ∈ K . We want to find a metric d ′ onHMET( K + ∆). That is we need to find a value of d (∆) that extends the inequality.For a subset S ⊂ Set n,m we define d ( S ) = X ∆ ′ ∈ S d (∆ ′ ) . Let us consider the W K, ∆ = { U ⊂ K : U ∪ { ∆ } is a closed manifold } . We now define the upper bound u K, ∆ = min U ∈ W K, ∆ d ( U ) . We have u K, ∆ ≥ d ∈ HMET( K ) implies d (∆ ′ ) ≥ l K, ∆ = max P ∈ W K, ∆ ,F ∈ P d ( F ) − d ( P − F ) . Suppose that l K, ∆ > u K, ∆ . We have u K, ∆ realized by U and l K, ∆ is realized by L and a face F ∈ L . The union L ∪ U is not necessarily a closed manifold since L ∪ U may share simplices. If that is so we remove them and consider instead W = L ∪ U − L ∩ U .The inequality l K, ∆ > u K, ∆ implies then d ( F ) > d ( L − F ) + d ( U ) = d ( W − F ) + 2 d ( L ∩ U ) ≥ d ( W − F )which violates the fact that d ∈ HMET( K ). Thus we can find a value α with l K, ∆ ≤ α ≤ u K, ∆ and α ≥ . Thus we can find a value for d (∆) that is compatible with an extension. (cid:3) The inequality set defining HMET( K ) is highly redundant but is still finite so,the cone HMET( K ) is actually polyhedral.On the other hand, using the inequalities obtained from the simplex does notwork. Consider for example the complex Set , . The Octahedron has 6 verticesand 8 faces and is a closed manifold. Thus it determines an inequality of the form x ≤ x + x + x + x + x + x + x which is not implied by the inequality on the simplices. The proof can be doneby linear programming using our software polyhedral ([25]). This proves that ourconstruction is different from the one of [15, 14, 21] and it would be interesting toredo the computations of those works.5. Acknowledgments
Second author gratefully acknowledges support from the Alexander von Hum-boldt foundation.
ENERALIZED CUT AND METRIC POLYTOPES OF GRAPHS AND SIMPLICIAL COMPLEXES15
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