Chapter 19. A ’Pseudopolynomial’ Algorithm for Sequencing Jobs to Minimize Total Tardiness
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The minimal integral separator
419
Proof. By (2) and property P for any non-stable set S,, x l E r a 2
i b + 1. If S, is stable
then either I C {1,2,3,. . . , t } or else I C {1,2, . . . , sj - 1,j } for some j. In either case
by (1) and (3) we must have that xCi,,aiG b.
Acknowledgement. Partial support through NRC (Grant A 8552) is gratefully
acknowledged.
Reference
[l] V. Chvital and P.L. Hammer, Aggregation of Inequalities in Integer Programming, Ann. Discrete
Math. 1 (1977) 145-162.
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Annals of Discrete Mathematics 1 (1977) 421-434
Publishing Company
0 North-Holland
ON THE COMPLEXITY OF SET PACKING POLYHEDRA*
Manfred W. PADBERG
Graduate School, Faculty of Business Administration, New York University, New York, N Y 10006,
U.S.A.
We review some of the more recent results concerning the facial structure of set packing
polyhedra. Utilizing the concept of a facet-producing graph we give a method that can be used
repeatedly to construct (arbitrarily) complex facet-producing graphs. A second method, edgedivision, is used to further enlarge the class of facet-defining subgraphs.
1. Introduction
We consider the set packing problem (SP)
max ck
AxSe
x i = O or 1 f o r j = 1 , ..., n
where A is a m X n matrix of zeros and ones, e T = (1,. . ., 1) is a vector of m ones
and c is a vector of n (arbitrary) rational components. This class of combinatorial
optimization problems has recently received much attention, both as a problem of
considerable practical interest -see e.g. [3,10,22] for recent survey articles for this
problem and its close relatives - as well as a combinatorial programming problem
that captures most of the difficulties and computational complexities that are
present in the general zero-one programming problem [2, 10, 11, 151.
In this paper we extend the class of “strongest” cutting planes or facets known
for problem (SP) and show that for set packing problems of sufficiently large size
arbitrarily “complex” valid inequalities can be constructed that, however, are facets
of the convex hull of (integer) sohtions to (SP), i.e. belong to the class of linear
inequalities that uniquely define the convex hull of solutions to (SP). Without
restriction of generality, we will assume throughout the paper that A does not have
any zero column or zero row. Denote by P = P(A, e ) the polyhedron given by the
feasible set of the linear programming problem associated with (SP), i.e.
P(A,e ) = {x € R“ I Ax
G
e, x
ZO}.
(1.1)
* Parts of this paper were presented in preliminary form at the NATO Advanced Study Institute
Symposium on Cornbinatorial Programming held in Versailles, France, September 1974.
42 1
M.W. Padberg
422
Furthermore, let PI = P l ( A ,e ) denote the set packing polyhedron, i.e. the convex
hull of integer points of P
~ , ( ~ , e ) = c o n v {ExP ( A , ~ ) integer}.
~x
By the theorem of Weyl[25], there exists a finite system of linear inequalities whose
solution set coincides with PI, i.e.
PI = { x E R " ( H x
h,x S O }
(1.3)
for some appropriate matrix H and vector h. Some research activity has recently
focused on identifying part (or all) of those linear inequalities that define PI, see [5,
16, 17, 18,20,21,23,24]. This interest is motivated in part by the desire to use linear
programming duality in proving optimality - with respect to the linear form cx of a given extreme point of PI.As every extreme point of PI is also an extreme point
of P and hence of any polyhedron P satisfying PI C P P, it is generally sufficient
to work with a partial - rather than a complete - linear characterization of PI.
More precisely, one is interested in finding part (or all) of the linear inequalities that
define facets of PI. Note that dim P = dimPI = n, i.e. both P and PI are fully
dimensional. As customary in the literature, we will call an inequality n-x G r0 a
facet of PI if (i) T X s m0 for all x E PI and (ii) there exist n affinely independent
vertices x ' of PI such that n-x' = nofor i = 1 , . . ., n. One readily verifies that each
inequality x, 2 0, j E N, is a (trivial) facet of PI, where N = {I,. . ., a } .
A construction that has proved useful in identifying facets of PI is the
intersection graph associated with the zero-one matrix A defining P. Denote by a,
the j t h column of the m x n matrix A . The intersection graph G = ( N ,E ) of A has
one node for every column of A, and one (undirected) edge for every pair of
nonorthogonal columns of A, i.e. (i, j ) E E iff a,u, 2 1. One verifies readily that the
weighted node packing problem (NP) on G for which the node weights equal c, for
j = 1 , . . ., n is equivalent to (SP), i.e. (NP) has the same solution set and set of
optimal solutions as the problem (SP). (The weighted node packing problem (NP)
on a finite, undirected, loopless graph G is the problem of finding a subset of
mutually non-adjacent vertices of G such that the total weight of the selected
subset is maximal, See [6, 10, 15, 181 for more detail on the stated equivalence.)
This observation is very useful as it permits one to restrict attention to node packing
problems in certain subgraphs of G when one tries to identify facets of PI.
2. Facet producing subgraphs of intersection graphs
Let n-x T,, be a non-trivial facet of PI. We can assume without loss of generality
that both rrJ,j E N, and r 0are integers. From the non-negativity of A, it follows
readily that 71; 2 0 for all j E N and .rr0>0. For suppose that N - = { j E N
T, < 0) # 0 for some non-trivial facet r x s noof PI. As r x s r 0is generated by n
affinely independent points of PI, there exists an X E PI such that TX = no and
I
423
Complexity of set packing polyhedra
2, = 1 for some j E N - . (For, if not, then by the assumed affine independence we
have that I N-(
= 1. It follows that r o= 0 and since all unit vectors of R" are
- feasible
that r x < r(,is a triuial facet of the form x, 2 0 ) . But the point given by
points,
x, = Z,, j_E N - N-, = 0, j E N - is contained in PI as A is non-negative and
hence, r: > r owhich is impossible. Consequently, every non-trivial facet r x s ro
satisfies r,2 0 for all j E N and r o > O .
Denote by S = S ( r ) the support of r , i.e. S = { j E N r,> O}. Let Gs = (S, E s )
be the induced subgraph of G with node set S and edge set Es C E and let
P s = P n { x E R" x, = 0 for all if?!!
S } . D u e to the non-negativity of A, one verifies
readily that the convex hull PS of integer points of P s satisfies P S =
PI n {x E R" x, = 0 for all j E S } . Furthermore, n-x S r oretains its property of
being a facet of PS. If one considers proper subgraphs of Gs, this property of
r x s r o may or may not be "inherited." In fact, if all components of T are
(non-negative) integers and r o= 1, then one readily verifies that for all subgraphs
(including those on single nodes) the property of r x rot o be a facet of the
resulting (lower-dimensional) packing polyhedron is retained. The next theorem
characterizes all facets of P, with integer r and r o =1, see [9, 181.
x
I
1
I
Theorem 1. The inequality c,,,x, S 1, where K C N,is a facet of PI if and only i f
K is the node set of a clique (maximal complete subgraph) of G.
Thus one knows all subgraphs of the intersection graph G of a zero-one matrix
A that give rise to facets r x rowith nonnegative integer r,,j = 1,. . ., n, and
r o= 1. If, however, r oa 2 and r odoes not divide all components of r,then there
v0
exists a smallest subgraph G' of G, G' not an isolated node, such that T X
looses its property of being a facet of the packing polytope associated with the
packing problem of any proper subgraph of G'. If r >O, i.e. if S = N, then, of
course, the (full) intersection graph G may have this property and thus G may be
itself strongly facet-producing [24].
Definition. A vertex-induced subgraph Gs = ( S , E s ) of G = (N, E ) with node set
S C N is facet-producing if there exists an inequality r x s 7r0 with nonnegative
integer components r, such that (i) r x s rois a facet of PS = PI n { x E R" x, = 0
for all j$Z S } and (ii) r x S m0 is not a facet for PT= PI n {x E R" x, = 0 for all
j e T } where T is any subset of S such that I T 1 = IS I - 1. A subgraph Gs of G is
called strongly fucet-producing if there exists an inequality r x c rosuch that (i)
holds and (ii) holds for all T C S satisfying I T 1 G 1 S 1 - 1. A subgraph Gs of G is
facet-defining if there exists an inequality r x s r,,such that (i) holds and (iii) such
that r, > O for j E S. (Shortly we will say that Gs defines the facet r x s ro.)
I
I
Remark 1. Every strongly facet-producing (sub-)graph is facet-producing. Every
facet-producing (sub-)graph is facet-defining. If Gs is facet-defining, then Gs is
connected.
M. W . Padberg
424
Proof. The first two parts being obvious, let rrx c rro be a facet defined by
Gs = ( S , Es), i.e. rr, > O for j E S, and suppose that Gs is not connected. Then
G, = ( S , E , ) can be written as Gs = G I U G, where G, = ( S , , E , )with S,# 0 for
i = 1 , 2 and S = S , U S,, S , f l Sz = 0 and Es = E l U EZ.Let P i be defined like P s
with S replaced by S,, and let rri = max {rrx x E P i } for i = 1,2. Since T X c
is
defined by Gs, it follows that rr; > 0 for i = 1,2. Define TTT; = rr, for j E S,, rrf = 0
for j e S,, and let x ’ E P ; be such that rrx‘ = a i for i = 1 ,2 . Since Gs is
disconnected, x ’ + x z E P: and hence, rrX + rri c T o . It follows that every x E PS
satisfying nx = rro satisfies rr’x = IT,!, for i = 1 , 2 and consequently, rrx =G
is not a
facet of PS.
By the discussion preceeding the definition, it is clear that every facet rrx s T o of
P, is either produced by the subgraph G, having S = S(T) or if not, that there
exists a subgraph GT of Gs with T C S such that the facet i i x =G m0 of PT is
produced by GT where 71, = rrl for j E T, i;, = 0 for jE T and PT is defined as
previously. The question is, of course, given i i x < no can we retrieve the facet
rrx n o of PI. The answer is positive and follows easily from the following theorem
which can be found in [16].
I
1
Theorem 2. Let P s = PI fl { x E R” x, = 0 for all j @ S} be the set packing polyhedron obtained from PI by setting all variables x,, j E N - S, equal to zero. If the
inequality
a,xl s a. is a facet of P?, then there exist integers &, 0 /?, ao,such
l is a facet of PI.
that Z J E S q x+I Z , E N - s ~ Jsx a.
c,,,
Generalizations of Theorem 2 for more general polyhedra encountered in
zero-one programming problems have been discussed in [l, 4,12,14,19,26,28].
The apparent conclusion from this result - in view of the notion of the
intersection graph discussed above - is that the problem of identifying part (or all)
of the facets of a set packing polyhedron P, is thus equivalent to the problem of
identifying all those subgraphs of the intersection graph G associated with a given
zero-one matrix A that are facet-producing in the sense defined above. (It should
be noted that the “facet-defining’’ property of (sub-)graphs is a considerable weaker
property as in this case we require solely that the corresponding facet has positive
coefficients. The choice of terminology may seem somewhat arbitrary, but the
positivity of all components of a facet furnished by a facet-defining (sub-)graph is
crucial in some of the arguments to follow.) O n e possible attack o n the problem of
finding a linear characterization of PI is thus t o “enumerate” all possible graphs
that are facet-producing, a truly difficult task as we will show in the next section.
3. Facet-producing graphs
Let G = (N, E ) be any finite undirected graph having no loops. Denote by Ac
the incidence matrix of all cliques of G (rows of A,) versus the nodes of G
Complexity of set packing polyhedra
425
1
(columns of Ac). Let N = (1,. . ., n } and let Pc = {x E R“ Acx S ec, x a 0) where
ec = (1,. . ., 1) is dimensioned compatibly with Ac. As before let PI denote the
convex hull of integer points of Pc,i.e. PI = conv{x E Pc x integer). We will not
note explicitly the dependence of t h e respective polyhedra upon the graph G which
may be taken as the intersection graph associated with some given zero-one matrix.
The term of a facet-producing (facet-defining) graph is used here analogously with
S = N. If x is a node (edge) of G, then by G - {x} we will denote the graph
obtained from G by deleting node x from G and all edges incident to x from G (by
deleting the edge x, but no node from G). Denote by G the complement of G, i.e.
G = (N, % ( N )- E ) where % ( N ) is the set of all edges on n nodes. Every clique in
G defines a stable (independent) node set (or node packing) in G and every
maximal stable node set in G defines a clique in G. Let Qc = {x E R“ Bcx s
dc, x 3 0) where Bc is the incidence matrix of all cliques in G and d z = (1,. . ., 1) is
dimensioned compatibly. Furthermore, let QI = conv{x E Qc x integer}. One
verifies readily that Qc is the anti-blocker of PI and that Pc is the anti-blocker of
QI, see [9, 201.
Before investigating special facet-producing graphs it is interesting to note the
following proposition which substantially reduces the search for facet-producing
graphs. Contrary to what one might expect intuitively, it i s not necessarily true that
the complement C? of a facet-producing graph G is again facet-producing. The
graph in Fig. 1 shows an example of a facet-producing graph G whose complement
G is not facet-producing. In fact, whereas the graph G of Fig. 1 produces the facet
x, s 4, its complement C? does not produce any facet in the sense of the above
definition; rather, every facet of the associated packing problem is obtained by
“lifting” the facets produced by some proper subgraph of G, as the clique-matrix of
G is of rank 9.
I
1
I
x:Z1
Fig. 1.
To state the next theorem we meet the following definition from linear algebra:
A square matrix M is said to be reducible if there exist permutation matrices P and
Q such that
N O
R ]
OM‘=[=
where N and R are square matrices and 0 is a zero-matrix. If no such permutation
matrices exist, then M is called irreducible.
Theorem 3.
Suppose that G defines the facet
TX
s r0 for PI such that
426
M . W . Padberg
I
max { r r x x E Pc}= r r f is assumed at a vertex X of Pc satisfying 0 < 2, < 1 for
j = 1,.. ., n. If the submatrix A 1of Acfor which AIX = e, is irreducible (and square),
then the complement graph
G is
strongly facet-producing.
Proof. Since every row of Ac defines a vertex of Q I , it follows from the
assumption that O < X j < 1 for all j = 1, .. ., n that the hyperplane X y = 1 is
generated by n linearly independent vertices of Q I . On the other hand, AcX ec
implies validity of X y S 1 for Q17 i.e. QI { y E R” Xy S l}.Hence, G defines the
facet Xy s 1 of Q,. Suppose that Xy 6 1 defines a facet for some k-dimensional
polyhedron Q, C QI satisfying k < n. Then there exist k linearly independent
vertices y ’ of 6,satisfying X y ‘ = 1. Since the submatrix A , of A, defining X is
square, it follows that upon appropriate reordering of the rows and columns of A 1 ,
A , can be brought into the form (3.1) contradicting the assumed irreducibility
of A , .
As an immediate consequence we have the corollary:
I
1
Corollary 3.1. If G defines a facet rrx rro of PI such that max{n-x x E Pc} = rrf
is assumed at a vertex X of Pc satisfying 0 < 2, < 1 for all j = 1,. . ., n, then defines
a facet of QI.
Chordless odd cycles (“holes”) as well as their complements (“anti-holes”) are
known to define facets. In the former case one readily verifies the hypothesis of
Theorem 3 to conclude that anti-holes as well as holes are strongly facet-producing.
More recently, L. Trotter [24] has introduced the notion of a “web” which properly
subsumes the aforementioned cases: A web, denoted W ( n ,k ) is a graph G = (N,
E)
such that
IN/= n 3 2 and for all i, j E N, ( i 7 j ) EE iff j =
i + k , i + k + 1,.. ., i + n - k , (where sums are taken modulo n ) , with 1 S k S [ n / 2 ] .
The web W ( n ,k ) is regular of degree n - 2k + 1, and has exactly n maximum node
packings of size k . The complement W ( n , k ) of a web W ( n ,k ) is regular of degree
2 ( k - 1 ) and has exactly n maximum cliques of size k . O n e verifies that W ( n ,1) is a
clique o n n nodes, and for integer s 3 2 , W ( 2 s + 1, s ) is an odd hole, while
W ( 2 s + 1 , 2 ) is an odd anti-hole. The following theorem is essentially from [ 2 4 ] ,see
the appendix.
Theorem 4. A web W ( n ,k ) strongly produces the facet C,”=,x, k i f and only if
k 2 2 and n and k are relatively prime. The complement W ( n , k ) of a facetproducing web W ( n ,k ) defines (strongly produces) the facet
x, s [ n / k ] ( i f and
only i f n = k [ n l k ] + 1).
c;=,
The next theorem due to V. Chvital [5] provides some graph-theoretical insights
into graphs that give rise to facets with zero-one coefficients. To this end, recall that
an edge e of a graph is called a-critical if a ( G - e ) = a ( G ) +1, where a ( G )
denotes the stability number of G, i.e. the maximum number of independent nodes
of G.
Complexity of set packing polyhedra
427
Theorem 5. Let G = ( V ,E ) be a graph ; let E * E be the set of its (Y -critical edges.
If G * = ( V , E * ) is connected, then G defines the facet &,x,
a(G).
It would be interesting to know whether all facets of set packing polyhedra
having zero-one coefficients and a positive right-hand side constant can be
described this way. (The question has been answered in the negative by Balas and
Zemel [4a]). V. Chvatal also discusses in his paper [5] several graph-theoretical
operations (such as the separation, join and sum of graphs) in terms of their
polyhedral counterparts. Though very interesting in their own right, we will not
review those results here. In particular, the two constructions given below are not
subsumed by the graphical constructions considered by V. Chvbtal.
We note next that graphs that satisfy the hypothesis of Theorem 5 need not be
facet-producing in the sense defined in Section 2. In fact, the graph of Fig. 2
provides a point in-case. The facet defined by the graph G of Fig. 2 is given by
x, 2 which, however, is produced by the odd cycle on nodes {1,2,3,4,5}. The
x, S 2, i.e. by applying
coefficient of x6 is obtained by “lifting” the facet
Theorem 2 .
x;=l
x:=l
11------7p-~
5
Fig. 2
W e next turn to a construction which permits one t o “build” arbitrarily complex
facet-producing graphs. Let G be any facet-defining graph with node set V =
(1,. . ., n } with n 3 2 and consider the graph G * obtained by joining the ith node of
G t o the ith node of the “claw” K1,”by an edge. The claw K,.,,- also referred to
as a “cherry” or “star”, see [13] - is the bipartite graph in which a single node is
joined by n edges to n mutually non-adjacent nodes. W e will give the node of K , , ,
that is joined to the ith node of G the number n + i for i = 1,. . ., n, whereas the
single node of Kl,,, that is not joined to any node of G, will be numbered 2n + 1.
(See Fig. 3 where the construction is carried out for a clique G = K4.) Denote by
V * = (1,. . ., 2n + 1) the node set of G * and by E * its edge-set. It turns out that G *
is facet-defining (this observation was also made by L. Woolsey [27]), and
moreover, that G * is strongly facet-producing.
Fig. 3.
Theorem 6. Let G = ( V , E ) be a graph on n 2 2 nodes and let T X S no be a
(non-trivial)facet defined by G. Denote by G * = ( V * ,E*)the graph obtained from
M . W. Padberg
428
G by joining every node of G to the pending notes of the claw K1,"a s indicated above.
Then G * strongly produces the facet
where x ( ' ) = ( x l , . . ., x,,), x(')= ( x " + ~.,. ., x2") and x ~ are
~ the
+ ~variables of the
node-packing problem on G * in the numbering defined above.
Proof. Let A, be the clique-matrix of G and denote by PI the set packing
polyhedron defined with respect to A,. The clique matrix of G * is given by A $:
A, 0 0
A$='[ I I 01
0 I e
(3.3)
where I is the n x n identity matrix, e is vector with n components equal to one,
and 0 are zero-matrices of appropriate dimension. Denote by PT the set packing
polyhedron defined with respect to A 2.. To establish validity of (3.2)for P : we note
that x, = 1 for some j E { n 1, , . ., 2 n ) implies that x ~ , , +=~0. Consequently, as
+ x(2) < e, every vertex of PT having x, = 1 for some j E { n + 1,. . ., 2 n ) satisfies
(3.2). On the other hand, since r x ( l ) < nofor every vertex of PT and no<
n;, it
follows that every vertex of PT satisfies (3.2). To establish that the inequality (3.2)
defines a facet of PT, let B denote any n X n nonsingular matrix whose rows
correspond to vertices of PI satisfying T X < n,, with equality. Then define matrix
B * as follows:
+
.
c;=,
0
(3.4)
where Z is the n x n identity matrix, E is the n X n matrix with all entries equal to
one, e is the vector with n components equal to one, and 0 are zero-matrices of
appropriate dimension. O n e verifies that the absolute value of the determinant of
B* is given by
c;=,
Hence, B * is non-singular since det Bf 0 and
r, > r,,> 0. On the other hand,
every row of B * corresponds to some vertex of PT satisfying (3.2) with equality.
Consequently, (3.2) defines a facet of PT. To prove that G * produces a facet in the
sense defined above, we show that no graph G * - { j } defines the facet (3.2) for
j = 1 , . . ., 2n + 1. Note first that from the positivity of rr it follows that every vertex
of PT satisfying (3.2) with equality satisfies x, + xn+, + x z n + ,3 1 for all j E V. Let
now j E V and consider G * - { j } . Every vertex of PT satisfying x, = 0 and (3.2)with
equality, necessarily satisfies x,+, + x ~ , , +=~1. Consequently, (3.2) does not define a
42 9
Complexity of set packing polyhedra
facet of P, = P: n{x E R * " + ' I x ,= 0) for j E V. Consider next vertices of P t
satisfying (3.2) with equality and x,+, = 0 for j E V. Since n 3 2 and rrx rro is
defined by G, it follows that the node j of G has a neighbor k ( j ) in G, i.e.
(j, k (j))E E for some k (j) # j , k (j) E V. Consequently, every vertex satisfying (3.2)
with equality and x,+, = 0, also satisfies the equation x n+k(,)+ x ~ = ~1. Hence
+ ~ (3.2)
does not define a facet of Pa+, for j E V. Finally, every vertex of P t satisfying (3.2)
with equality and xZ,,+'= 0 satisfies the equation x k x " + ~= 1 for all k E V, and
Consider next a subgraph
consequently, (3.2) does not define a facet of Pzn+l.
G' = (V', E ' ) of G * having I V'I s 2n - 1 nodes. If G' defines the facet given by
(3.2), then, as noted earlier, all vertices of P' = PT n { x E R2"+'x, =
0, j E V* - V'} satisfying ( 3 . 2 )with equality must satisfy x, + xn+, + x ~ 3 ~1 for+ all~
j E V, since a vertex of P' is also a vertex of P : . It follows that V' must contain the
V' implies n + i E V' for all
node numbered 2n + 1 and furthermore, that
i E V. Consequently, V C V' and V' n { n + 1 , . . ., 2 n } # 0. Let N ' C { n f 1 , . . ., 2 n }
be the nodes of G * that are not in G'. Then either there exists a node n + j E N'
such that the node j E V has a neighbor k (j) E V satisfying n + k (j)E V' or else,
G is disconnected. The latter contradicts Remark 1. Consequently, by the above
reasoning, we have N ' = 0, i.e. V = V'. This completes the proof of Theorem 6.
+
I
ie
Corollary 6.1. Let G, G * and rr be a s in Theorem 6 . If there exists a vertex X E Pc
such that max {rrx x E Pc}= rr2 is assumed at vertex 2 of Pc satisfying 0 < X, < 1
for j = 1, . . ., n, then the complement graph G * of G * defines a facet. Moreover, this
facet of the set packing polyhedron Q T associated with G * ( i n rational form ) is given
by
. + ( e - 2 ) . x ( * ) + f .x Z n i l s 1
(3.5)
1
where f
=
I
min {X, j = 1,. . ., n } .
Proof. Using the clique-matrix A T.as defined by (3.3) o n e verifies readily that the
coefficients of the inequality (3.5) define a vertex of PT.with all components strictly
between zero and one. Furthermore, the submatrix of A T.defining the vertex with
components (X,e - X, Z) is nonsingular. As the cliques in G *define vertices of 0 T,
Corollary 6.1 follows.
The second construction uses edge-division. Let G b e any facet-defining graph
with node set V = (1,. . ., n } and edge-set E. Let e = ( v , w ) E E and consider the
graph G * with nodes set V* = (1,. . ., n, n + 1 , n + 2) and edge-set
E * = ( E - { e } )U {(u, n
+ l),( n + 1 , n + 2), ( n + 2 , w ) } .
That is, G * is obtained from G by "inserting" two new nodes into an (existing)
edge of G. Let rrx rro be the facet defined by G. As usual, we will assume that rr is
a vector of positive integers. An edge e = (u, w ) E E will b e called rr-critical if
there exists an independent node set F in the graph G - { e } such that x , € F q > no
and
rr, = rro or
rr, = rro. Note that rr-criticality of an edge is entirely
analogous to the concept of (Y -criticality used above.
c,,,-,
c,,,-,