On the Chvátal rank of linear relaxations of the stable set polytope

Authors


Abstract

We study the Chvátal rank of two linear relaxations of the stable set polytope, the edge constraint and the clique constraint stable set polytope. For some classes of graphs whose stable set polytope is given by 0/1-valued constraints only, we present either the exact value of the Chvátal rank or upper bounds (of the order of their largest cliques and stable sets), which improve the bounds previously known from the literature (of the order of the graph itself).

1. Introduction

For any polyhedron P, let PI denote the convex hull of all integer points in P. Chvátal (1973) (and implicitly Gomory, 1958) introduced a method to obtain approximations of PI outgoing from P as follows. If Σaixileqslant R: less-than-or-eq, slantb is a valid inequality for P and has integer coefficients only, then Σaixileqslant R: less-than-or-eq, slantb⌋ is a Chvátal–Gomory cut for P and valid for PI. Define P′ to be the set of points satisfying all Chvátal–Gomory cuts for P, and let P0=P and Pt+1=(Pt)′ for non-negative integers t. Obviously PIPtP holds for every t. An inequality Σaixileqslant R: less-than-or-eq, slantb is said to have Chvátal rank at most t if it is a valid inequality for the polytope Pt. Chvátal showed that for each polyhedron P there exists a finite tgeqslant R: gt-or-equal, slanted0 with Pt=PI; the smallest such t is the Chvátal rank of P.

In this paper, we consider the Chvátal rank of two different linear relaxations of the stable set polytope STAB(G) of a graph G, outgoing from the system of inequalities associated with all edges and with all cliques of G, respectively.

Given a graph G=(VE), a stable set in G is a set of mutually nonadjacent nodes. Accordingly, the stable set polytope STAB(G) is defined as the convex hull of the incidence vectors of all stable sets in G. Finding a stable set of maximum cardinality in G is an NP-hard problem, which can be stated as the following integer program:

image(1)

The optimal value of this problem is denoted by α(G), the stability number or independence number of G. A natural linear relaxation of (1) gives rise to the edge constraint stable set polytope:

image

A clique QG is a set of mutually adjacent nodes. It is easy to see that edges are special cliques, and that any stable set can contain at most one node of a clique. Hence, clique constraints

image

are valid for STAB(G) and can be used to strengthen the last relaxation, leading to the clique constraint stable set polytope

image

Further generalizations of clique constraints are 0/1-constraints associated with arbitrary induced subgraphs G′⊆G, the rank constraints

image

Using these, the rank constraint stable set polytope is defined by

image

Clearly,

image

holds for any graph G. One central question is to identify and characterize classes of graphs that satisfy some of these inclusions with equality. Here we take a slightly different approach and study for some prominent graph classes, how many applications of the Chvátal–Gomory procedure are required to arrive at STAB(G), starting from one of the relaxations mentioned above. More precisely, for inline image, define the Chvátal rank of G w.r.t. inline image as

image

We will consider the values of crE(G) and crQ(G) for several graph classes. The clique constraint stable set polytope has been widely studied due to its interesting polyhedral properties; for instance, QSTAB(G) is related to the stable set polytope of its complement inline image through anti-blocker theory (Fulkerson, 1971, 1972). From a practical point of view, however, it is more natural to consider ESTAB(G) as an initial stage for Chvátal–Gomory rounding, as QSTAB(G) might contain an exponentially large number of inequalities even for simple graph classes (see Section 2.2 for an example), whereas ESTAB(G) has only O(|V|2) for any graph. Moreover, the separation of clique inequalities is NP-hard in general (Grötschel et al., 1981), but trivial for edge inequalities.

STAB(G) and ESTAB(G) coincide if and only if G is bipartite (i.e., if V can be partitioned into two stable sets), and thus crE(G)=0 holds for any bipartite graph G. It is well known that ESTAB(G)′ has, besides edge constraints, only rank inequalities associated with all chordless odd cycles C2k+1G, the odd hole constraints

image

Chvátal called a graph G t-perfect if STAB(G) and ESTAB(G)′ coincide, i.e. if crE(G)=1 holds. More generally, a graph G is ti-perfect if STAB(G)=ESTAB(G)i, i.e. if crE(G)=i holds. Then bipartite (resp. t-perfect) graphs are exactly the t0-perfect (resp. t1-perfect) graphs.

STAB(G) and QSTAB(G) coincide if and only if G is perfect (Chvátal, 1975; Padberg, 1974), that is if for all induced subgraphs G′⊆G, the node set of G′ can be partitioned into as many stable sets as the size ω(G′) of a largest clique in G′. Thus, crQ(G)=0 holds if and only if G is perfect. According to a recent characterization achieved by Chudnovsky et al. (2006), perfect graphs are precisely those graphs without any odd holes C2k+1 with kgeqslant R: gt-or-equal, slanted2, or their complements, the odd antiholes inline image. Odd holes are t-perfect and, if G is either an odd hole or an odd antihole, then crQ(G)=1.

A graph G is said to be rank-perfect if STAB(G) equals RSTAB(G), see Wagler (2004a). By definition, all perfect, t-perfect and h-perfect graphs are rank-perfect (the latter are these graphs G where STAB(G) is given by clique and odd hole constraints). Further examples of rank-perfect graphs are odd antiholes (Padberg, 1974) and antiwebs (Wagler, 2004b).

Let [n] denote the integer set {0, … , n−1}. An antiwebinline image is a graph with node set [n] and edges ij if and only if inline image, where ngeqslant R: gt-or-equal, slanted2k+2. Antiwebs include all cliques inline image, odd antiholes inline image, and odd holes inline image. The clique constraints and rank constraints associated with induced (sub)antiwebs, together with non-negativity constraints, provide a complete linear description of the stable set polytope of antiwebs (Wagler, 2004b).

A graph G is a-perfect if STAB(G) is given by non-negativity inequalities and rank constraints associated with induced cliques and antiwebs in G. The class of a-perfect graphs is a subclass of rank-perfect graphs and a common superclass of h-perfect graphs and antiwebs. Thus, we have crR(G)=0 for all such graphs. In the following, we focus our attention on the Chvátal rank of ESTAB(G) and QSTAB(G) for some fundamental graph classes (cliques, odd holes, odd antiholes and antiwebs), as well as the implications for related subclasses of a-perfect graphs.

According to Chvátal et al. (1989), for any monotone polytope P in [0, 1]n, a valid inequality aTxleqslant R: less-than-or-eq, slantb of PI with integer coefficients can be obtained by applying the Chvátal–Gomory procedure at most inline image times. As ESTAB(G) and QSTAB(G) are monotone polytopes in the 0/1-cube, implying for rank-perfect graphs G that

image

holds, and hence their Chvátal rank is at most of order O(|G|). As the main result of this paper, we will provide an upper bound for the Chvátal rank of a-perfect graphs in terms of their clique and stability number (instead of their number of nodes).

2. Chvátal ranks of inequalities and graphs

In order to determine the Chvátal rank of STAB(G) w.r.t. some linear relaxation inline image, we need to know when the facet-defining inequalities for STAB(G) are valid for inline image. More precisely, for an inequality aTxleqslant R: less-than-or-eq, slantb being valid for STAB(G), we define by

image

its Chvátal rank w.r.t. the relaxation inline image. Clearly, we have that

image

In the following, we discuss for some classes of rank-perfect graphs their Chvátal rank w.r.t. QSTAB(G) and ESTAB(G) by determining the Chvátal rank of the facets of STAB(G).

We say that a subgraph G′⊆G induces a facet aTxleqslant R: less-than-or-eq, slantb of STAB(G) if

image

holds. For a family inline image of facet-inducing subgraphs of G, we say that a facet aTxleqslant R: less-than-or-eq, slantb of STAB(G) is generated byinline image if aTxleqslant R: less-than-or-eq, slantb can be obtained by applying (one round of) the Chvátal–Gomory procedure to inequalities induced by the subgraphs in inline image.

2.1. Cliques

Clique constraints are of particular interest as they are required to describe the stable set polytope of any graph (Padberg, 1974). We clearly have crQ(χQ, 1)=0 for every clique Q, and from Chvátal (1973) it is known that inline image. Our purpose is to determine the exact values for crE(χQ, 1) and crE(Kn).

Lemma 1.Let ngeqslant R: gt-or-equal, slanted3. A clique Kn and the corresponding clique constraint have Chvátal rank

image(2)

Proof. For qleqslant R: less-than-or-eq, slantn, let Qq be the family of all subcliques of Kn having size q. As any induced subgraph of Kn is a clique, we have inline image. Moreover, every node of Kn belongs to exactly inline image cliques in Qq. Hence, adding up all constraints corresponding to cliques from Qq yields

image

This is the required clique constraint if and only if we have inline image. The smallest integer satisfying this condition is inline image and thus

image(3)

We will prove (2) by induction on the value of n. The condition can be easily verified for n=3, as crE(K3)=1. Now assume (2) holds for all integers strictly smaller than n. Then it follows from (3) that

image

If n is even, then inline image, and the proof is completed. Otherwise, let inline image, which means inline image. Since n is odd, inline image holds strictly and hence inline image implies inline image. This finishes the proof, as inline image. ▪

As clique constraints suffice to describe the stable set polytope of perfect graphs, we have:

Corollary 2.A perfect graph G withinline imagehasinline image.

For any h-perfect graph G, the stable set polytope is given by clique and odd hole constraints. Odd hole constraints can be generated from edges by Chvátal (1973), and to generate a constraint x(Q)leqslant R: less-than-or-eq, slant1 associated with a clique QG, only subcliques of Q can be used; thus, we obtain:

Corollary 3.Every imperfect h-perfect graph G withinline imagehas crQ(G)=1 andinline image.

2.2. Odd antiholes

For every odd antihole inline image with kgeqslant R: gt-or-equal, slanted2 the only nontrivial facets of inline image are rank constraints associated with maximal cliques and the graph itself (Padberg, 1974). The full rank constraint inline image can be obtained by applying the Chvátal–Gomory procedure to the maximum clique constraints, i.e. inline image holds. However, the number of inclusion-wise maximal cliques contained in inline image grows exponentially with respect to k, as the next lemma shows.

Lemma 4.The odd antiholeinline imagecontains exactly

image

inclusion-wise maximal cliques of size l, withinline image.

Proof. Assume inline image and ijE if and only if inline image. Let inline image be a maximal clique in inline image, with inline image and inline image. Observe that Q may be uniquely represented by the vector

image

where inline image. In the following, we call i the start node and d0, … , dl−1 the distance components.

Since Q is a clique, every distance component must be larger than one. Moreover, if drgeqslant R: gt-or-equal, slanted4 holds for some distance component, then inline image is a clique, and hence Q is not maximal. Thus, we have inline image, for all 0leqslant R: less-than-or-eq, slantjleqslant R: less-than-or-eq, slantl−1. The maximality of Q also implies that inline image, with i=2 only if dl−1=3. Conversely, it is straightforward to see that any such vector inline image defines a maximal clique.

Let a, b∈Z+ denote the number of distance components equal to 2 and 3, respectively. These numbers must satisfy the following system:

image

It follows that a=3l−(2k+1), b=(2k+1)−2l and hence 2lleqslant R: less-than-or-eq, slant2k+1leqslant R: less-than-or-eq, slant3l, or equivalently, inline image.

Let inline image be the set of indices corresponding to distance components with value 3, and observe that the pair (iIb) uniquely determines the vector inline image and hence the clique Q. There are inline image possible choices for Ib, and each of them can be completed to a valid pair by choosing i∈{0, 1}. For i=2, the restriction dl−1=3 has to be considered, which is equivalent to asking l−1∈Ib. There are then inline image possible choices for Ib. Adding together these quantities and substituting b=(2k+1)−2l, the assertion of the lemma follows. ▪

To obtain all facets of inline image starting from edge constraints, note that we clearly have

image

since inline image and inline image holds. The next lemma shows that also non-maximum cliques can suffice to generate the rank constraint associated with an odd antihole.

Lemma 5.For any odd antiholeinline image, we have

image

withinline image.

Proof. For inline image, let inline image be an inclusion-wise maximal clique of size inline image and consider the family inline image of maximal cliques defined by

image

with all sums taken modulo 2k+1. Observe that each node of inline image is contained in exactly inline image cliques from F. Hence, adding up the 2k+1 clique inequalities we obtain

image

It follows that F generates the odd antihole constraint if and only if inline image. The smallest integer satisfying this inequalities is inline image and therefore inline image.

Since inline image is completely described by the full rank facet and clique inequalities, it also follows from the last observation that

image

To complete the proof, note that for all kgeqslant R: gt-or-equal, slanted4 we have inline image. Thus, 2(q−2)>k−2 and inline image. The result then follows as a consequence of Lemma 1. Finally, if k∈{2, 3}, one can verify that k=q holds.▪

Remark 6. For some values of k, we have that inline image holds, since

image

In these cases, the exact value of inline image can be obtained from inline image. This is true, for instance, for inline image, and inline image. In fact, we conjecture that inline image holds in general.

We call a graph m-perfect if its stable set polytope is given by non-negativity inequalities and rank constraints associated with cliques and minimally imperfect subgraphs (i.e., odd holes and odd antiholes). As odd hole constraints can be generated from edges by Chvátal (1973) and rank constraints associated with cliques and odd antiholes from subcliques of appropriate size, we obtain as a consequence of (3) in Lemma 1 and Lemma 5:

Corollary 7.An imperfect m-perfect graph G withinline imageand largest odd antiholeinline imagehasinline imageandinline imagewithinline image.

2.3. Antiwebs

We now turn to stable set polytopes of antiwebs. An antiweb inline image is called prime if k+1 and n are relatively prime. Note that all cliques inline image, odd antiholes inline image, and odd holes inline image are examples of prime antiwebs. Trotter (1975) showed that the rank constraint associated with an antiweb is facet-defining if and only if the antiweb is prime. Following Wagler (2004b), the stable set polytope of an antiweb inline image is completely described by non-negativity inequalities and rank constraints associated with induced prime subantiwebs (including clique constraints and the full rank constraint if inline image is prime).

Our aim is to determine inline image and inline image for all antiwebs. We first note which antiwebs are t0-perfect, then we characterize the t1-perfect antiwebs, and present an algorithm that determines an upper bound of inline image for general antiwebs. Finally, we show how to adapt the procedure to obtain upper bounds for inline image.

The t0-perfect graphs are exactly the bipartite graphs, i.e., the perfect graphs with clique number 2. According to Trotter (1975), the perfect antiwebs are exactly

  • the cliques inline image,
  • the antiholes inline image with even n,
  • the matchings inline image for all kgeqslant R: gt-or-equal, slanted0.

(Recall that we excluded the stable sets inline image with inline image by definition.) Hence, by inline image, an antiweb inline image is t0-perfect if and only if inline image holds.

We next determine all t1-perfect antiwebs. Besides inline image and the odd holes inline image for all kgeqslant R: gt-or-equal, slanted1, which are all antiwebs that do not induce the full rank facet and contain edges, triangles, and odd holes as only facet-inducing subgraphs. In other words, they are antiwebs with clique number <4 whose only prime subantiwebs are inline image for kgeqslant R: gt-or-equal, slanted0.

Trotter (1975) determined necessary and sufficient conditions for an antiweb inline image to be contained as an induced subgraph in a larger antiweb inline image. Since these conditions will be used throughout the section, we state them here as a separate lemma.

Lemma 8. (Trotter, 1975).The antiwebinline imageis an induced subgraph ofinline imageif and only if we have n′<n, k′<k, and

image(4)

We next provide the characterization of all non-bipartite t-perfect antiwebs. The term Moebius ladders refers to the antiwebs inline image with kgeqslant R: gt-or-equal, slanted2.

Theorem 9.A non-bipartite antiwebinline imageis t-perfect if and only if it is

  • an odd holeinline imagewith kgeqslant R: gt-or-equal, slanted1,
  • a Moebius ladderinline imagewith odd kgeqslant R: gt-or-equal, slanted3, or
  • equal toinline image.

Proof. In order to characterize the t-perfect antiwebs, we shall determine all antiwebs having no other prime subantiwebs than triangles and odd holes. More precisely, for all antiwebs inline image different from those listed in the theorem, we either exhibit at least one of the following

  • the clique inline image with inline image,
  • the odd antihole inline image with inline image,
  • a Moebius ladder inline image with even kgeqslant R: gt-or-equal, slanted2, which has inline image as it is prime,

or show that inline image itself is prime. We first observe:

Claim 10.No antiweb with clique number at least 4 is t-perfect.

Hence, we restrict our attention to antiwebs with clique number 2 or 3. By inline image, which are exactly the antiwebs inline image with inline image.

Claim 11.The only t-perfect antiwebs with clique number 3 areinline image.

Proof. For fixed k, an antiweb inline image has clique number 3 if inline image. For k=0, this is just inline image which is obviously t-perfect. For k=1, we have to consider inline image and inline image. The first one equals inline image, is perfect, and has inline image by Lemma 1. The latter one is the odd antihole inline image and has inline image. For kgeqslant R: gt-or-equal, slanted2, we show that all these antiwebs but inline image contain either inline image or the even Moebius ladder inline image as an induced subgraph, or are prime.

From Lemma 8, inline image if and only if

image

holds. Observe that the right-most inequality is always satisfied, as we have inline image, for all kgeqslant R: gt-or-equal, slanted1. Hence, no antiweb inline image with inline image is t-perfect.

Furthermore, inline image contains inline image if and only if

image

holds. By inline image for all kgeqslant R: gt-or-equal, slanted2, no antiweb with nleqslant R: less-than-or-eq, slant4k can be t-perfect.

Combining the two observations, all possible candidates for t-perfect antiwebs must satisfy

image

It is straightforward to see that this inequality system has no solution for kgeqslant R: gt-or-equal, slanted5. For 2leqslant R: less-than-or-eq, slantkleqslant R: less-than-or-eq, slant4, the only antiwebs fulfilling this condition are inline image, inline image, inline image and inline image. Among them, inline image, inline image and inline image are prime and, therefore, not t-perfect. inline image does not induce the full rank facet and has K3 and C5 as only facet-inducing subgraphs; hence, inline image is indeed t-perfect. ▪

Claim 12.The only non-bipartite t-perfect antiwebs with clique number 2 are odd holesinline imagewith kgeqslant R: gt-or-equal, slanted1 and Moebius laddersinline imagewith odd kgeqslant R: gt-or-equal, slanted3.

Proof. For fixed k, an antiweb inline image has clique number 2 if inline image. As the antiwebs inline image are bipartite for all kgeqslant R: gt-or-equal, slanted0, and inline image odd holes for all kgeqslant R: gt-or-equal, slanted1, we start with the Moebius ladders inline image for kgeqslant R: gt-or-equal, slanted2.

Any Moebius ladder is 3-regular (as, by definition, node i has exactly inline image as neighbors). Hence, all their subantiwebs are either 1-regular matchings or 2-regular odd holes. Whether or not a Moebius ladder is t-perfect depends, therefore, on whether or not it induces the full rank facet. This is the case if k is even, but not if k is odd (as then both k+1 and n=2k+4 are even). Hence, all odd Moebius ladders are indeed t-perfect. On the other hand, it is well known that even Moebius ladders are t2-perfect.

It remains to show that all other antiwebs inline image with clique number 2 are not t-perfect (note that they satisfy inline image and kgeqslant R: gt-or-equal, slanted3). For that, we either exhibit an even Moebius ladder as an induced subgraph, or show that inline image itself is prime.

From Lemma 8, inline image if and only if inline image, inline image and

image

Since inline image, the last condition can be shown to be equivalent to

image(5)

Moreover, for the same reason inline image and inline image hold, and hence (5) suffices to imply inline image. There is an inline image satisfying this condition if and only if

image

With inline image, this is equivalent to

image(6)

Now we prove that, with the only two exceptions inline image and inline image, all antiwebs having kgeqslant R: gt-or-equal, slanted3 and inline image satisfy (6). Let inline image and observe that inline image and kgeqslant R: gt-or-equal, slanted3 imply rgeqslant R: gt-or-equal, slanted1. Moreover, from inline image it follows

image(7)

We distinguish the following three cases:

  • If rgeqslant R: gt-or-equal, slanted3, then inline image and from (7) we obtain
    image
    The last inequality holds by inline image, and (6) yields inline image for some inline image.
  • If r=2 then (7) implies
    image
    Observe that the right-hand side is smaller than 2 for any xgeqslant R: gt-or-equal, slanted1, and hence (6) is again satisfied. For x=0, the definition of r implies that
    image
    Hence (6) holds for k∈{5, 6}, but not for k=7, which yields the exception inline image.
  • If r=1 the inequality (7) leads to
    image
    and implies (6) for any xgeqslant R: gt-or-equal, slanted2. Considering again the definition of r, if x=0 the only possible values for k are 3 and 4, while if x=1 then k may take values between 3 and 5. Moreover, by manually checking the condition (6) in those cases, it is straightforward to see that it is satisfied for all values except x=0 and k=4, which leads to inline image as the second exception.

As both inline image and inline image are prime and different from odd holes, the assertion follows.

Combining all three claims finally verifies the assertion of the theorem. ▪

We now consider the Chvátal rank of not t-perfect antiwebs. As observed at the beginning of the section, for a general graph G we have

image

As for antiwebs all non-trivial facets are rank constraints associated with prime subantiwebs, we focus on the generation of such inequalities.

Let G:=(VE) be an induced subgraph in an antiweb inline image. For any i∈[n] we denote by Gi the subgraph of inline image induced by the set inline image, with all sums taken modulo n. Observe that by circular symmetry of the antiweb, the graphs G and Gi are isomorphic. In the following, F(G) will refer to the family containing the n isomorphic subgraphs of inline image defined by

image

The next result provides a necessary and sufficient condition for F(G) to generate the full rank inequality of inline image.

Lemma 13.Let G be any induced subgraph ofinline image. The family F(G) generates the full rank inequalityinline imageif and only if

image(8)

where n(G) is the number of nodes of G.

Proof. Observe that each node of inline image is contained in exactly n(G) graphs from F(G). Hence, adding the rank inequalities inline image corresponding to all graphs inline image, we obtain

image

Now F(G) generates the full rank inequality of inline image if and only if

image

The left inequality in the chain holds trivially for any subgraph inline image, as the full rank inequality of an antiweb is always tight. The right inequality is equivalent to (8). ▪

Combining the results from Lemma 8 and Lemma 13, it is possible to determine when the full rank inequality of an antiweb can be generated from subantiwebs.

Lemma 14.For any integers n>n′, n>k′, the antiwebinline imagecontains antiwebsinline imagethat generate the full rank inequality if and only if

image(9)
image(10)

Proof. Assume inline image for some n′<n and k′<n. Observe that (4) in Lemma 8 implies the following two inequalities:

image

On the other hand, substituting inline image in (8) from Lemma 13 we obtain the other condition

image

Conversely, it is straightforward to verify that (9) and (10) together imply the conditions for Lemma 8 and Lemma 13, and this completes the proof. ▪

A clique of size q is isomorphic to an antiweb inline image, as observed at the beginning of this subsection. Hence, substituting inline image and inline image in (9) we obtain that the full rank constraint of an antiweb inline image can be generated by such cliques if and only if

image

holds. Moreover, it is straightforward to check that inline image provides such a value for any inline image, and the next result follows.

Corollary 15.For k>0 andinline image,

image

Lemma 13 and Lemma 14 can be exploited algorithmically for computing an upper bound for the Chvátal rank of inline image.

Lemma 16.Algorithm 1 ends in a finite number of steps and returns an upper boundinline imagefor the Chvátal rank of the edge constraint stable set polytope of an antiwebinline image.

Proof. The main algorithm contains, besides the call of subprocedure FACET_RANK in line 32, only finite loops and simple assignments. Moreover, observe that in each new recursive call of the subprocedure, we have k″<k′ and hence the condition k′=0 in line 2 of Algorithm 2 becomes true after a finite number of calls, which guarantees that the recursion terminates.

Algorithm 1. A recursive algorithm for computing inline image
 Input:n, k
 Output:inline image
3: {Check for t0-perfect antiwebs}
 ifinline imagethen
  return 0
6: {Check for t-perfect antiwebs, use Theorem 9}
 if (inline image) or (inline image and k odd) or (inline image) then
  return 1
9: {Check for t2-perfect Moebius ladders}
 ifinline image and k even then
  return 2
12: {Check for cliques, use Lemma 1}
 ifk=0 then
  returninline image
15: {Check for antiholes, note: nleqslant R: less-than-or-eq, slant6 has already been processed}
 ifk=1 then
  {Even antiholes are perfect, use Corollary 2}
18: ifn is even then
   returninline image
  {For odd antiholes, use Lemma 5}
21: ifn is odd then
   inline image
   returninline image
24: {Otherwise, collect all prime subantiwebs}
 inline image
 forinline imagedo
27:  forinline imagedo
   ifk′+1 and n′ are relatively prime then
    inline image
30: {Compute maximal Chvátal-rank of a facet}
 r+←0
 forinline imagedo
33: r′←FACET_RANK inline image
  ifr′>rthen
   rr
36: returnr

First, the algorithm detects if inline image is either t0-perfect (i.e., a matching inline image), non-bipartite t-perfect (from Theorem 9), or an even Moebius ladder (which are known to be t2-perfect), and returns the corresponding value of inline image. The exact Chvátal rank is also computed for when inline image is a clique, using Lemma 1. Then antiholes with ngeqslant R: gt-or-equal, slanted7 are considered, as antiholes with nleqslant R: less-than-or-eq, slant6 belong to one of the classes mentioned above. Even antiholes are perfect, and Corollary 2 provides an exact value for inline image. For odd antiholes, Lemma 5 is applied for obtaining the upper bound inline image, which is returned in line 23.

In all other cases, the algorithm first constructs a set L containing all facet-defining subantiwebs inline image, including possibly inline image itself. For each of these antiwebs, an upper bound inline image for the Chvátal rank of the corresponding rank constraint is calculated, and the algorithm then returns the maximum of these values.

Algorithm 2. Subprocedure FACET_RANK
 {Finish recursion if inline image is a clique}
 ifk′=0 then
3: returninline image
 r←∞
 {Check all proper subantiwebs}
6: forinline imagedo
  forinline imagedo
   {Determine if inline image can generate full rank facet}
9:  ifinline imagethen
   r′←FACET_RANK inline image
   ifr′<rthen
12:    rr
 returninline image

Computation of inline image is done in the subprocedure FACET_RANK. In each call, all proper subantiwebs inline image that can generate the facet are examined using (8) from Lemma 13, and upper bounds for the Chvátal ranks of the corresponding rank inequalities are determined via recursive calls. The procedure keeps track of the minimum r of these values, and finally returns r+1. The recursion ends whenever the procedure is called with k′=0, in which case inline image is a clique and Lemma 1 is used again to obtain inline image.▪

Note that the algorithm can be easily modified to compute an upper bound for inline image and inline image as follows:

  • the return values in line 14 of the main algorithm and in line 3 of the subprocedure are both set to zero, as inline image holds trivially for all cliques; and
  • the return value for odd antiholes is set to 1 in line 23, and for even antiholes is set to 0 in line 19.

Remark that inline image holds for all t0-perfect antiwebs, for all t-perfect antiwebs, and for all t2-perfect Moebius ladders as all of them, except for inline image, have inline image. Moreover, inline image, since inline image implies that this antiweb is not perfect.

Table 1 shows the values of inline image obtained by our algorithm for inline image and inline image, and Table 2 reports the corresponding upper bounds for inline image.

Table 1. 
Upper bounds for inline image, for kleqslant R: less-than-or-eq, slant32 and ngeqslant R: gt-or-equal, slanted2k+2
n\k01234567891011121314151617181920212223242526272829303132
31                                
420                               
521                               
6310                              
7321                              
83220                             
93311                             
1042210                            
1143221                            
12432220                           
13433221                           
144332210                          
154433221                          
1643322220                         
1744332221                         
18533332210                        
19543332221                        
205443323220                       
215433322221                       
2254433322210                      
2354433323221                      
24544343322220                     
25544433323221                     
265443333232210                    
275544333322221                    
2854443333232220                   
2955443433232221                   
30544434332332210                  
31554443333232221                  
325444434332322220                 
335544434333332221                 
3464444343332322210                
3565444443332332221                
36654443433333322220               
37655444343332322221               
386554443433323222210              
396544443433323322221              
4065544444433333222220             
4165544444433323322221             
42655444444333333232210            
43655444434333333322221            
446554444343333233222220           
456555444444333333322221           
4665544444443333333222210          
4765554444443333233232221          
48655444444443332332222220         
49655544443443333333232221         
506555444444433333333222210        
516655444444433333333222221        
5265554444444333333332322220       
5366554444444433333333222221       
54655554444444333333332322210      
55665544444344343333333322221      
566555544444444333333332322220     
576655544444444333323332322221     
5865555444444443333333332222210    
5966555444444443433333332322221    
60655554444444433333333333222220   
61665554444444443333333332322221   
626555544444444433333333323222210  
636655544544444434333333333322221  
6465555444444444433333333323222220 
6566555544444444433333333323222221 
66755555444444444343333333323222210
Table 2. 
Upper bounds for inline image, for kleqslant R: less-than-or-eq, slant32 and ngeqslant R: gt-or-equal, slanted2k+2
n\k01234567891011121314151617181920212223242526272829303132
30                                
400                               
501                               
6000                              
7011                              
80020                             
90111                             
1000110                            
1101121                            
12001220                           
13011121                           
140011210                          
150112221                          
1600112220                         
1701111221                         
18001122210                        
19011122221                        
200011223220                       
210111112221                       
2200111222210                      
2301111223221                      
24001122222220                     
25011112223221                     
260011112232210                    
270111112222221                    
2800111122232220                   
2901111222232221                   
30001112222332210                  
31011111122232221                  
320011112222322220                 
330111112222332221                 
3400111122222322210                
3501111222222332221                
36001111212223322220               
37011111122222322221               
380011111222223222210              
390111111222223322221              
4000111122222233222220             
4101111122222223322221             
42001111222222233232210            
43011111112222233322221            
440011111122222233222220           
450111111222222233322221           
4600111112222222333222210          
4701111112222222233232221          
48001111222222222332222220         
49011111121222222333232221         
500011111122222223333222210        
510111111122222222333222221        
5200111111222222223332322220       
5301111111222222223333222221       
54001111112222222223332322210      
55011111122122232223333322221      
560011111222222222233332322220     
570111111112222222223332322221     
5800111111122222222233332222210    
5901111111122222322233332322221    
60001111111222222222333333222220   
61011111112222222222233332322221   
620011111122222222222333323222210  
630111111222222223222333333322221  
6400111111212222222222333323222220 
6501111111112222222222333323222221 
66001111111122222232223333323222210

In general, the values of inline image tend to increase as n increases and to decrease as k increases. There are, however, exceptions to this rule, the smallest one being inline image. On the other hand, it is straightforward to verify that inline image coincides with the actual values of inline image for cliques, t0-perfect, and t-perfect antiwebs.

The same idea used in Algorithm 1 leads to the following general upper bound for inline image:

Corollary 17.For any antiwebinline image,

image

whereinline imageandinline imageare the cardinalities of the maximum stable set and the maximum clique ininline image, respectively.

Proof. Consider a prime subantiweb inline image whose corresponding facet-defining rank constraint achieves the maximal value for inline image. We know that this inequality can be generated by a family of cliques, or by a family of proper subantiwebs. Repeating this process for at most k′−1 times reveals a subantiweb inline image whose rank inequality can be generated by cliques. Moreover, these cliques have sizes no larger than inline image and hence,

image

The last inequality follows from the fact that every clique in inline image (resp. every stable set in inline image) is also a clique (resp. a stable set) in inline image. Applying now Lemma 1, the assertion of the corollary follows. ▪

Table 3 shows the values obtained from the last corollary for antiwebs inline image with inline image and inline image. These values coincide with the output of our algorithm for cliques and some odd antiholes, but they become considerably larger as k increases, which suggests that the bound can still be improved.

Table 3. 
Upper bounds obtained from Corollary 17
n\k01234567891011121314151617181920212223242526272829303132
31                                
421                               
521                               
6322                              
7322                              
83323                             
93333                             
1043334                            
1143334                            
12444445                           
13444445                           
144444456                          
154444556                          
1644455567                         
1744455567                         
18545556678                        
19545556678                        
205555666789                       
215555667789                       
22555566778910                      
23555566778910                      
2455566778891011                     
2555566778891011                     
265556677889101112                    
275556677899101112                    
28555667889910111213                   
29555667889910111213                   
30556677889101011121314                  
31556677889101011121314                  
3255667789910101112131415                 
3355667789910111112131415                 
346566778991011111213141516                
356566778991011111213141516                
366666788910101112121314151617               
376666788910101112121314151617               
38666678891010111212131415161718              
39666678891010111213131415161718              
4066677889101111121313141516171819             
4166677889101111121313141516171819             
426667789910111112131414151617181920            
436667789910111112131414151617181920            
44666778991011121213141415161718192021           
45666778991011121213141515161718192021           
4666677899101112121314151516171819202122          
4766677899101112121314151516171819202122          
4866677891010111213131415161617181920212223         
4966677891010111213131415161617181920212223         
506667889101011121313141516161718192021222324        
516667889101011121313141516171718192021222324        
52666788910101112131414151617171819202122232425       
53666788910101112131414151617171819202122232425       
5466778891011111213141415161718181920212223242526      
5566778891011111213141415161718181920212223242526      
566677889101111121314151516171818192021222324252627     
576677889101111121314151516171819192021222324252627     
58667788910111112131415151617181919202122232425262728    
59667788910111112131415151617181919202122232425262728    
6066778991011121213141516161718192020212223242526272829   
6166778991011121213141516161718192020212223242526272829   
626677899101112121314151616171819202021222324252627282930  
636677899101112121314151616171819202121222324252627282930  
64667789910111212131415161717181920212122232425262728293031 
65667789910111212131415161717181920212122232425262728293031 
6676778991011121313141516171718192021222223242526272829303132

Remark 18. Since for any a-perfect graph G the stable set polytope is given by non-negativity constraints and rank inequalities associated to induced cliques and antiwebs, we have as a direct consequence from the previous corollary

image

3. Conclusions

In this paper, we focus on the Chvátal rank of the linear relaxation ESTAB(G) of STAB(G) for several classes of graphs.

We first consider the rank of cliques and clique constraints, as they are required to describe the stable set polytope of any graph. We provide the exact value

image

for all ngeqslant R: gt-or-equal, slanted3. As a consequence, we further obtain the exact values of crE(G) whenever G is perfect or h-perfect, showing that crE(G) is of order inline image.

Next, we consider the rank of odd antiholes and show that

image

holds with inline image. As inline image, this again provides an upper bound for crE (G) of order inline image for odd antiholes and all m-perfect graphs G.

Finally, we turn to the general case of antiwebs inline image. We characterize the t-perfect antiwebs and provide a procedure to determine an upper bound for inline image, based on the generation of rank facets associated with antiwebs using prime subantiwebs only. It is open whether this procedure already determines the exact value of inline image. This would be the case if no other inequalities being valid for inline image than rank constraints associated with prime subantiwebs are required to generate the full rank constraint of inline image.

In any case, we have an upper bound

image

with inline image and α=k+1. We infer that crE(G) is at most of order inline image for all antiwebs and, thus, for all a-perfect graphs. Figure 1 shows the inclusion relations of the classes of a-perfect graphs that were considered so far.

Figure 1.

 Some classes of a-perfect graphs. The arrow showing that odd holes are a subclass of antiwebs has been omitted for clarity.

In addition, there are several further interesting graph classes known to be a-perfect. Wagler (2005) showed that complements of so-called fuzzy circular interval graphs are a-perfect.

Fuzzy circular interval graphs are defined as follows: given a collection inline image of intervals from a circle C, these are graphs whose nodes correspond to points xi on C and xixj is an edge if there is an interval in inline image containing xi and xj, where nodes corresponding to different endpoints of one interval may or may not be adjacent. Circular interval graphs are special cases without fuzziness (as also nodes corresponding to different endpoints of one interval are always adjacent).

A further subclass of fuzzy circular interval graphs are so-called concave-round graphs whose nodes can be circularly enumerated such that the closed neighborhood of each node forms an interval in the enumeration. Moreover, the class of concave-round graphs contains the class of proper circular-arc graphs.

Hence, a-perfect graphs constitute an interesting subclass of rank-perfect graphs G for which we improved the previously known bound on crE(G) of order O(|G|) from Chvátal et al. (1989) to an upper bound of order inline image.

Ancillary