Facets for the cut cone II

We study new classes of facets for the cut coneCn generated by the cuts of the complete graph onn vertices. This cone can also be interpreted as the cone of all semi-metrics onn points that are isometricallyl1-embeddable and, in fact, the study of the facets of the cut polytope is in some sense equivalent to the study of the facets ofCn. These new facets belong to the class of clique-web inequalities which generalize the hypermetric and cycle inequalities as well as the bicycle odd wheel inequalities.


Introduction
This paper is a follow-up to [16] dealing with valid inequalities and facets of the cut cone C,.The cut cone (7, is the cone generated by the cuts of the complete graph K,, on n vertices.The cut polytope Pn is the polytope whose vertices are the cuts of K,.There are several motivations for the study of the cut cone.First, the study of the facial structure of the cut cone is relevant to the polyhedral approach to the max-cut problem which is a notorious NP-hard problem.The max-cut problem can be formulated as an optimization problem over the cut polytope, but a remarkable property of the cut polytope ( [7]) is that all its facets can be deduced from the facets of the cut cone via a "switching" property, that we shall recall below.On the other hand, the cut cone Cn is also relevant to the theory of finite metric spaces; namely, the elements of C, can be interpreted as the semi-metrics on n points which are isometrically/1-embeddable or, in other words, a vector d of ~n(,,-~)/2 belongs to C~ if and only if there exist some vectors x~ .... , x,, in Nm, for some m ~> 1, such that d~ = I]xi -xj I]~ for 1 <~ i <j <~ n (recall that IIx [I, = 21 .... [xu] for x c ~m).We refer e.g., to [2,3] for more information on this connection.
In [16], we introduced, in particular, cycle inequalities and briefly announced their generalization to clique-web inequalities (CW inequalities, for short), This paper is devoted to the study of clique-web inequalities and is organized as follows.
In the first section, we introduce CW inequalities (in their pure and collapsed form) and we give a description of their roots, i.e., of the cuts realizing equality.In the second section, we give several new classes of facets of Cn arising from CW inequalities.We also show that the pure CW inequalities yield facets of the equicut polytope.We group in Section 3 the proofs for the results on CW facets stated in Section 2.
We now give all notation and preliminaries needed for the paper.Our graph notation is classical, as well as the notions of polyhedral combinatorics that we will use.All graphs are simple and undirected.We denote by [1, n] the set of the n integers 1, 2 . . . ., n. K~ denotes the complete graph on the n nodes 1, 2 , . . ., n.Given a subset S of [1, n], the set ~(S) of all the edges of K~, having exactly one endnode in S is called the cut determined by S.Then, the incidence vector of the cut ~(S) is the vector X ~(s), called the cut vector determined by S, and defined by Xa(s) =1 if /j is an edge of 3(S), i.e., [ S ~{ i , j } ] = l , and x~(s)---O otherwise, for 1 <~ i <j<~ n.Since 6(S) = 6([1, n] -S), there are 2 "-1 -1 nonzero cut vectors.The cut cone C~ is the cone generated by all cut vectors of Kn.The cone Cn is a full dimensional polyhedral cone in ~n(~-~)/2 containing the origin.
Given a vector v of R "(n ~/2, the inequality v. x~<0 is called valid for C, if it is satisfied by all vectors of C,, or, equivalently, by all cut vectors.Then, the set F~ = {x ~ C, : v. x = 0} is the face generated by the valid inequality v. x ~< 0, or simply by v.For a cut 6(S), we set v ( 6 ( S ) ) = ~j ~a ( s ) v ~.The cuts ~(S) whose incidence vectors belong to F~, i.e., the cuts 6(S) satisfying v(6(S)) = 0, are called the roots of v; we also say, for short, that the set S itself, defines a root of v. The" set of roots of v is denoted by R ( v ) .The dimension of the face Fo is the maximum number of affinely independent points in F~ minus one, or, equivalently, since F~ contains the origin, it is the maximum number of roots of v whose incidence vectors are linearly independent.A facet is a maximal face of C,, i.e., a face of dimension ½ n ( n -1 ) -1; if F~ is a facet, one also says that v is facet inducing.
Given a vector v of ~n(n--l)/2, its supporting graph G ( v ) is the weighted graph with nodeset [1, n] and whose edges are the pairs (i,j) for which v~j # 0, the edge ij being assigned weight vii.Conversely, if G is a (edge) weighted graph on n nodes, its edgeweight vector is the vector v of length ½ n ( n -1 ) where, for the edges /j of G, v~j is the weight of edge /j and vo = 0 i f / j is not an edge of G.If G is a graph, then, when the weights are not specified, they are assumed to be the unit weights, i.e, we have v;/= 1 if/j is an edge of G and v o = 0 otherwise.We call an inequality v. x<~O pure if the components v~i of v take only values +1, -1 , 0.
Known classes of valid inequalities for the cone C~ include hypermetric inequalities introduced in [ 10] and later independently in [ 18 ] and cycle inequalities introduced in [ quasilinear); we correspondingly define linear and quasilinear hypermetric and cycle inequalities.
Given a vector v of ~(~-~/2 and a cut 6(S) of K,, consider the vector vS of N,(,-~I/2 defined by v~ = -v~j if 0" is an edge of the cut 6(S) and v~ = vii otherwise; one says that v ~ is obtained by switching ofv by the cut 6(S).Let P,~ denote the cut polytope of the complete graph K,,, i.e., P~ is the convex hull of all cut vectors.If v.x<~vo is a valid inequality for P,,, then v~.x<~vo-v(6(S)) is also a valid inequality for P, and v.x<~ vo is facet inducing for P, if and only if v ~. x~< Vo-v(6(S)) is facet inducing for P,, [7]; one then says that both facets are switching equivalent.Therefore, all facets of P,, can be obtained by switching by cuts of the facets of C,.This connection between the cut cone and the cut polytope remains valid for the general case of non complete graphs [7]; the switching operation on facets of C. was introduced in [11].See [13] for a more detailed information on the symmetries of the cut polytope pn.
Finally, recall that zero-lifting preserves facets of the cut cone.Namely, for v ~ ~-1~/2, define v'~ ~,,(~+/~/2 by v,~ = v U if 1 ~ i <j ~ n and v~,,+l = 0 if 1 ~< i ~< n; then, the inequality v-x ~< 0 defines a facet of C, if and only if the inequality v'.x<~0 defines a facet of C,,+~ [11,16].Therefore, when we have an inequality whose supporting graph spans n nodes, if we can show that it defines a facet of the cone C~, then it also defines a facet of the cone Cm for any m ~> n.
Remark 1.2.We have chosen the terminology "clique-web" inequality, since (1.3) can also be written as Hence, there is a web on the first p nodes (the nodes for which b, = +I) and a clique on the remaining q = n-p nodes (the nodes for which b~ =-1).We shall denote in the remainder of the paper the q "negative" nodes p + 1,..., n (i.e., those for which bi=-l) by 1',..., q'.
Note that, if we relax the equality condition p-q=2k+l from (1.1) to the condition p-q>~2k+l, then (1.3) is no more valid for C,; however, a suitable positive value for the right hand side of (1.3), namely, the value ½(p-q)x (p-q-2k-1), restores validity and actually gives validity for general multicut polytopes (see [15]).
As an example, Figure 1 shows the supporting graph of the CW~I inequality (edges with weight +1 are indicated by a plain line while edges with weight -1 are indicated by a dotted line and every node of the triangle is joined to every node of the web).and, therefore, is the sum of k + l triangle inequalities, i.e., except for k =-O, it is not facet inducing.
If G is a (edge) weighted graph on nodeset [1, h] and if v denotes its edgeweight vector, then the 7r-collapse of G is the weighted graph G~; on nodeset [1, p] whose edgeweight vector is v~.In other words one obtains G,~ from G by contracting all nodes from a common partition class into a single node and correspondingly adding the edge weights.
Given a subset S of [1, p], set S ~ = U~s/~ ; so S ~ is a subset of [1, h] and one checks easily that

v~(3(S))=v(fi(S~))
for all S_~ [1, p].(1.4)In consequence, if v. x ~< 0 defines a valid inequality of the cone Cr~ (defined on the h nodes 1, 2,..., h), then its ~--collapse v~.x~<0 defines a valid inequality of the cone C, (defined on the p nodes 1, 2,..., p).Furthermore, the set of roots of v~.x <~ 0 is given by Hence, collapsing is an operation that preserves validity and so it is a useful tool for producing large new classes of valid inequalities (see [9] for a general study of the collapsing operation for the cut cone).Actually, we checked that most of the known facets of C~ are pure or collapses of some pure facets.We conjecture that any non pure facet of C, is the collapse of some pure facet.Note that collapsing does not always preserve facethood, also it may be that the collapse of a non facet inducing inequality be facet inducing.For example, the inequality: (x23 -x12 -x~3) + (x45-x~4-x~5)<~O is not facet inducing for C5 (it is the sum of two triangle inequalities), while the inequality obtained by collapsing both nodes 1, 5 into the node 1: x23-x12-x13 <~ 0 is indeed facet inducing.

Validity and roots 1.3.1. Validity
For k = 0, 1, CW inequalities are the known valid hypermetric and cycle inequalities.We established validity of pure CW inequalities CW~ for the cases k = 2 and p > (k -1)(k2+ k -2) and conjectured the general result in [16].We do not give the proof here, since Alon [ 1 ] proved validity of all pure CW inequalities.As mentioned in Section 1.2, the collapsing operation preserves validity; therefore, validity of general CW inequalities follows immediately from the pure case.The general CW inequality CWnk(b) can also be interpreted as arising from the corresponding pure inequality in which nodes are allowed to be repeated (the b~'s being the repetition numbers); in this form, validity of general CW inequalities was also noticed in [1].But our way of defining general CW inequalities from pure ones via collapsing is necessary for obtaining explicit expression of collapsed antiweb, roots, etc. and, therefore, for studying CW facets.
Remark 1.9.When we say that a subset S of [1, p] is an interval, we mean, more precisely, that S is a circular interval of the circularly ordered set [1, p].There are some redundancies in the presentation of the roots given in Theorem 1.8; it is easy to see that a description of the roots in which each root occurs exactly once can be obtained by replacing the family of sets of Type 2 by the family of sets of Type 2' where: Type 2': S+ is an interval of [1,p] of size s+, k+l<~s+<~p-k-2, and s_ is a subset of [l', q'] of size s , where s_ = s+ -k.Remark 1.10.It is easy to see that, if Soil,p] induces a clique of the antiweb AWe k, then IS[ <~ k+ 1 holds.For example, the interval [1, k+ 1] = {1, 2,..., k, k+ 1} induces a clique of AWpk; therefore, any subset S of the following Type 1" induces a clique of AWp k.
Type 1": S is contained in an interval of size k+ 1 of [1,p].There may exist other subsets of [1,p] inducing cliques of AW~, for example, for k = 3, p = 9, the set {1, 4, 7} induces a clique of AW 3. In fact, there exist additional subsets of [1, p] inducing cliques of AWp k besides those of Type 1" only for p ~< 3k.Actually, all our proofs for CW facets use only the roots 6(S) for which S is of Type 1" or 2.
In the remaining of this section, we give the proof of Theorem 1.8 through Claims 1.11, 1.13, 1.14, 1.15 and 1.16.Claim 1.11.If S is of Type 1 or 2, then 6(S) is indeed a root of CW~.
Proof.Since every node in the antiweb AWp k has degree 2k, then the number of edges in 6(S)c~ AW~ that are incident to each node x c S is greater or equal to 2k -(s -1) = 2k + 1 -s.By assumption, 16(S) c~ Awkl = s(2k + 1 -s), implying that there are exactly 2k+ 1-s edges in /~(S)c~ AWp k incident with each node x c S. Therefore, each node x ~ S is adjacent in AWp k to all other nodes of S, i.e., S induces a clique of AWp k. [] So, in Claim 1.14, we have identified the roots of Type 1; we now turn to the case of the roots of Type 2. We prove by induction on p >~2k+3 that, for any subset S of size s contained in [1, p], the following assertion hold: (H)p Ifk+l<-s<~½p and 16(S)¢~AW~l=k(k+l), then S is an interval.Theorem 1.8 will then follow easily using Claims 1.13, 1.14.Note that, if (H)p holds for k+l<~s<~½p as stated, then it also holds for k+l<.s~p-k-1(simply by considering the complement of S).Observe that the base of induction is p = 2k + 3, i.e., q=2.and ]3(S)c~AW~k+31=k(k+a), then we deduce that 16(S)r~CI = 2(k+ l)= 2s.This implies easily that no two nodes of S are adjacent on the cycle C.If S is not an interval, then S contains two nodes, say 1 and x, which are not contained in any interval of size k + 1, i.e., x = k + 2 or k + 3. Since (l, k + 2), (1, k + 3) are both edges of C, we obtain a contradiction.[~ Take p~>2k+3 and assume that (H)p holds.We want to prove that (H)p+~ still holds; for this, we first establish some connections between the graphs AW~ and AWpk+~.Considering that p+l is a new node inserted between nodes 1 and p on the cycle (1, 2,..., p), one observes easily that the edges belonging to AWp k but not to AWpk+~ are the pairs (p -k+ i, i) with i ~ [1, k], and the edges belonging to AWpk+~ but not to AW~ are the pairs (p + 1, i), (p + 1, p -k + i) with i c [1, k] We can assume w.l.o.g, that p+l~S.From (1.8), we have that: [6(S)&AW~I= k(k+l)+~l.<_i<kai(6(S))<~k(k+l),which, together with Proposition 1.12(ii), implies that ]6(S) c~ AWpkl = k(k+ 1) and, therefore, di(6(S)) = 0 for all i ~ [1, k].From (H)p, we deduce that S is an interval of [1, p].If the pair {1, p} is not contained in S, then S is still an interval of [ (1.9) We derive the following property: Proposition 1.17.For illustration, we describe below the roots of the CW inequality CW~(2, 1,..., 1, -1,..., -1) (cf.Example 1.6); the proof is a direct application of (1.9).

, defines a facet of the cut cone Cn. []
Recall that, since zero-lifting preserves facethood, then any pure CW inequality CW~ defines a facet of the cut cone Cm for any m ~> n.
In fact, as we explain in Remark 2.6, Theorem 2.1 will follow from Theorem 2.5 which states facethood of pure CW inequalities for the equicut polytope; so we do not need to prove Theorem 2.1.

CW facets for the equicut polytope
In this section, we show that a suitable switching of the pure CW inequalities also yields facets of the equicut polytope.Equicuts correspond to partitions of the nodes into (almost) equal parts and they have applications in various domains, in particular, in statistical physics (for the determination of ground states of spin glasses with zero magnetization ) (see [5]).
Our study of CW facets for the equicut polytope is motivated, first, by the intrinsic interest of the equicut polytope in view of its applications, and second, by the fact that, if we can prove that some switching of the CW inequalities defines a facet of the equicut polytope, then, necessarily, it also defines a facet of the cut polytope.
A cut 3(S) of Kn is called an equicut if IS] = [½hi or [½n]; a cut which is not an equicut is called an inequicut.We denote by EP,, the equicutpolytope, i.e., the convex hull of all equicuts of Cn and we denote by IC, the inequieut cone, i.e., the cone generated by all inequicuts.The inequicut cone and its relatives are considered in [12]; in particular, IC, is full dimensional for n >/5 and it is shown in [12] that any facet of C, (hence any CW facet) defines, in fact, a facet of ICm for all m >~ 2n.The equicut polytope was studied in [8]; for n odd, EP~ has dimension ½n(n -1) -1 and, in fact, is the facet of the cut polytope P, induced by the valid inequality ~2~<j~n x0 ~< ¼( n2-1) [8], the inequality CW~ itself is valid for EPn, but is not facet defining.For this, observe that, with the notation of Theorem 1.8 and Remark 1.9, no root of Type 1 of CW~ is an equicut and the only roots of Type 2' which are equicuts are the cuts 6(S) with S=S+wS and s+=s_+k, s_=½q or ½(q+l).
Therefore, the face of EPn induced by the inequality CW~ is contained in the face of EPn induced by the valid inequality Y.p+l<~i<j~n x~ <~ [½qJ [½q] and, hence, is not a facet of C,.However, the inequality CW~ defines a facet of EPm for all m odd, m I> 2n + 1 ( [12]).On the other hand, some suitable switching of CW~ will give a facet of EP,, for all m odd, m ~> n + 2.
We consider now the following inequality (2.1), also denoted by (CW~) ~v'o'], obtained by switching the inequality (1.3) by the cut 6([1', q']): (2.1) As a corollary of Theorem 1.8, we can describe the roots of (CW~)[V'q']; they are exactly the cuts 6(SAIl', q']) for which 6(S) is root of CW~, hence they are the cuts fi(S) for which S is of one of the following two types: Type I: S = S+ u [1', q'] where S+ ~ [1, p] induces a clique of AWp k.Type 2: S : S+ u S_ where S+ is an interval of size s+ of [1, p], k+ 1 ~< s+ ~<p -k, S is a subset of [1', q'] of size s_ and s_=q+k-s+ or q+k+l-s+.Since n =2(q+k)+l, we observe that, for all S of Type 2, 8(S) is, in fact, an equicut, while the sets S of type 1 for which 6(S) is an equicut are those having s+=k or k+l.This result extends Theorems 6.2, 6.3 and 6.4 from [8] which correspond to the extremal cases k= 1 (i.e., up to switching, pure cycle inequality) and k=~(n-5) or, equivalently, p=2k+3 (i.e., the bicycle odd wheel inequality).The bound m ~> n + 2 in Theorem 2.5 can probably be improved to m ~> n for some parameters n, k.The proof of Theorem 2.5 is given in Section 3.1.
Remark 2.6.Theorem 2.1 follows from Theorem 2.5, as we now indicate (see [12] for more details on the connections between facets of the cut and equicut polytopes).
(i) Since the inequality (2.1) is a switching of the inequality (1.3), if we can prove that (2.1) defines a facet of the cut polytope Pn, then this implies that (1.3) defines a facet of the cut cone Cn.
(ii) If we can prove that (2.1) defines a facet of the cut polytope Pn+2, then this implies that (2.1) defines a facet of P,, (recall the property of zero-lifting mentioned in the Introduction).

A necessary condition for a CW inequality to be facet inducing
Facets are, by definition, maximal faces of the cut cone which means that they are not contained in any valid inequality.The CW inequality CW~ (b) is called quasilinear if all negative coefficients b~ of b except at most one are equal to -1.We derive below a necessary condition for quasilinear CW inequalities to be facet inducing; it is based on the trivial fact that a CW facet cannot be contained in any triangle facet.
Consider a quasilinear CW inequality CW~(b), i.e., b=(b~,..., bp,-1,..., -1, b~) with b~,..., bp>0, b,<0, ~ b~ =2k+l and consisting of q-1 coefficients -1.We denote by b~, bJ the two largest coefficients among bl,..., bp.Observe that, what we proved is, that if (2.2) does not hold, then the face defined by CW~(b) is contained in some triangle facet.Remark 2.8.For k = 0, condition (2.2) is bi + bJ ~< q = n -p.Theorem 3.12 [16] gives a complete characterization of quasilinear hypermetric facets and can be rephrased as follows.In the uniform case, i.e., b~ ..... bp, the necessary and sufficient condition for facethood is bi + bj <~ q -1 and, otherwise (in non-uniform case), the necessary and sufficient condition is exactly (2.2) b~ + bj ~< q.In fact, as a biproduct of the proof of Proposition 2.7, one obtains that any non-uniform quasilinear hypermetric face is, either a facet, or is contained in a triangle facet.We will see in the next section other classes of hypermetric inequalities for which the same phenomenon occurs: they are either facets, or contained in some pure hypermetric facet.
[~ The bound from Proposition 2.9 is exponential in n; it would be interesting to find a polynomial upper bound.The problem of finding good estimates of gk(n) is also related to the problem of determining whether the cone CW(n, k) which is defined as the solution set of the CW inequalities CW~(b) for all possible b whose sum is 2k+ 1 is polyhedral.Polyhedrality of the cone CW(n, k) means that IIbll is bounded for all b such that CW~(b) is facet inducing for the cone CW(n, k).For k = 0, the cone CW(n, 0) is indeed polyhedral [14].
(iii) If la -cd I is an odd number, then v is facet defining if and only if [~ -cd I >~ 3 or (]a -~'[ = 1 and min(fl, fi') i> w).
Proof.As for the proof of Theorem 2.10, we use the lifting technique, i.e., if 1, n +2 are the positions corresponding respectively to the new coefficients -w -1 and w + 1,

ij ~ K m
We use the following characterization for proving that v. x <~pq defines a facet of EPm.Let a.x <<-ao be a valid inequality for EPm such that {x ~ EPm : v" x =pq} {x c EPm : a. x = a0}.We show that there exist some scalars A > 0, B such that: The roots in EP,~ of the inequality v.x<~pq are the cuts 8(Su T), where S is a subset of [1, p] u [l', q'] such that 8(S) is a root in P, of the inequality v. x <~ pq, and T is an arbitrary subset of [1", r"] such that [Sw TI = q+ k+½r or q+ k+½+ 1.
Hence, from Remark 2.4, the roots of the inequality v. x ~ pq in EP,, include the cuts 6(S) where S is of the following form: (1) S = S+u [1', q'] w T, where S+ is a subset of size s+ contained in an interval of [ 1, p] of size k + 1, T is a subset of [ 1", r"] of size t and t + s+ = k + ½r or k + ½r + 1.

Theorem 2 . 5 .
The inequality (2.1) defines a facet of the equicut polytope EPm for all m odd, m >~ n + 2 and, moreover, for m = n when k = 1.