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On some weakly bipartite graphs
Volume 2, Number 5
OPERATIONS RESEARCH LETTERS
December 1983
ON S O M E WEAKLY B I P A R T I T E G R A P H S
Francisco BARAHONA lnstitut fiir Operations Research, Universiti~t Bonn, Fed. Rep. Germany Received May 1983 Revised September 1983
We characterize a class of weakly bipartite graphs. In this case, the max-cut problem can be solved by, finding a minimum two-commodity cut. Max.cut problem, polyhedral combinatorics, ellipsoid method
I. I n t r o d u c t i o n a n d n o t a t i o n
If G -- (V, E) is a graph, and W_c V, 8(W) denotes the set of edges with exactly one extremity in W, 8(W) is called a cut. The max-cut problem can be stated as follows; given a set of edge weights ce > 0 for all e ~ E, find a set W c_ V such that c(8(W)) - Y~{ ce : e ¢ 8(W)} is as large as possible. A maximum cut is a maximum bipartite subgraph of G. Since bipartite subgraphs do not have odd cycles one can ask which are the graphs such that the convex hull of incidence vectors of bipartite subgraphs is defined by the odd cycle inequalities. Gr6tschel and Pulleyblank [8] called those graphs weakly bipartite. Using this polyhedral characterization and the ellipsoid method for linear programming it is possible to solve the maxcut problem in polynomial time for this kind of graphs. Another characterization or a polynomial algorithm to recognize weakly bipartite graphs is not known. Clearly, bipartite graphs belong to this class, and planar graphs are also weakly bipartite. Our aim is to present a new family of weakly bipartite graphs: those graphs
Research supported by the Alexander yon Humboldt Stiftung. Present address: Dpto. De Matematicas, U. de Chile, Casilla 5272, Santiago 3, Chile.
such that there exist two vertices that cover all the odd cycles. If G - (V, E) is a graph the cardinality of V is called the order of G. If e E E is an edge with endnodes i and j we also write/j to denote the edge e. If H - (W, F ) is a graph with W _ V and F c_ E then H is called a subgrapb of G. If G - (V, E) is a graph and F _ E, then V(F) denotes the set of nodes of V which occur at least once as an eadnote of an edge in F. Similarly, for Wc_ V, E(W) denotes the set of all edges of G with both endnodes in W. For W c_ V, G \ W will denote the graph H = ( V \ W, E ( V \ W)). A graph G is called complete if every two different nodes of G are linked by an edge. The complete graph with n nodes is denoted by Kn. A graph is called bipartite if its node set can be partitioned into two nonempty, disjoint sets Vi and VI such that no two nodes in I/1 and no two nodes in I/2 are linked by an edge. We call V1, V2 a bipartition of V. If W_c V. Clearly, (W, 8(W)) is a maximal bipartite subgraph of G. We write 8(v) instead of 8({ v }) for v ~ V and call 8(o) the star of v. A path P in G - (V, E) is a sequence of edges e t , e2, . . . , e k such that e~ -- royt, e 2 -- v i v 2, . . . , ek--vk-~v~ and such that v ~ v j for i cj. The nodes v0 and v~ are the endnodes of P and we say that P links Vo and vk or goes from Vo to vs. The number k of edges of P is called the length of P. If
0167.6377/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North.Holland)
239
Volume2, Number 5
OPERATIONSRESEARCHLETTERS
P --- et, e 2, e k is a path linking Vo and vk and ek+t = VoVk ~ E, then the sequence et, e 2, • .., ek, ek+t is called a cycle of length k + 1. A cycle (path) is called odd if its length is odd, otherwise it is called even. Given F c: E, x F will denote the incidence vector of F. for x: E--,R, and F c_ E , x ( F ) will denote • { x(e): e ~ F }. The max-cut problem is NP-Complete, cf. [7], but it is polynomiaily solvable for planar graphs, cf. Hadlock [11], for graphs not contractible to K s, cf. Barahona [3], for weakly bipartite graphs, cf. Gr6tschel and Pulleyblank [8], and for graphs without long odd cycles, of. Grt~tschel and Nemhauser [9]. Moreover, the cut of maximum cardinality can be obtained in polynomial time for toroidal graphs, cf. Barahona [2]. The polytope ...,
subgraph of G } is called the bipartite subgraph polytope. Let Q(G) be the polytope defined by: C and odd cycle,
O ( x ( e ) ~ < 1, e 6 E .
There exist two vertices, pj and P2, such that (2.1) G \ { Pt, P2 } is bipartite If is clear that graphs in fi can be recognized in polynomial time. Let G - (V, E) be a graph in f~ and let Pl, P2 be the vertices mentioned in (2.1). In what follows we will construct a graph G such that the complement of a minimum two-commodity cut in G gives us a maximum bipartite subgraph of G. Let G ' (V', E') be the graph G \ { pt, P2 }, let U, W be a bipartition of V'. The graph ¢~- (V, E) is defined as follows:
if uv ~ E' then uv ~ E, if p~u ~ E, for u ¢ U, then s~u ~ E, 1
if p~w ~ E, for w ¢ W, then t~w ~ E,
i = 1, 2, i - 1, 2,
(1.1)
ifptp2 ~ E, then s~t 2 ~ E.
(1.2)
Let us call A the incidence matrix of all s~ to t~ paths, i - 1, 2. It follows from the two-commodity max-flow min-cut theorem, cf. Hu [12], and from the theory of blocking polyhedra, cf. Fulkerson [6], that the incidence matrix B of all cuts separating s~ from t,, i = 1, 2, is the blocking matrix of A. Hence, for w >_.0, the problem
x E
thus if G is weakly bipartite it permits to find a maximum cut. We will show a new class of weakly bipartite graphs, hence a class of graphs where the max-cut problem is polynomially solvable. Moreover, in this case this problem can be solved by finding a minimum two-commodity cut. In the last section we will study the clutter of odd cycles and its blocker. 240
Let us denote by fl the class of graphs G with the following property:
1
It is clear that Pn(G)c_ 9(G), and that each extreme point of Pa(G) is an extreme point of 9(G). Moreover, inequalities (1.1) and (1.2) define facets of Pa(G), cf. Barahona [1]. Gr6tschel and Pulleyblank [8] defined the class of Weakly Bipartite Graphs as the set of graphs G such that Ps(G)= Q(G). Moreover, they gave a polynomial algorithm to solve the separation problem for 9(G)" given a point x, either verify that it belongs to 9(G) or else find a hyperplane that separates it from 9(G). This algorithm combined with the ellipsoid method [10], permits to solve in polynomial time the problem max c x s.t.
2. A class of weakly bipartite graphs
V - V ' U { $ 1 , $ 2 , tl, t2},
•De(G ) - c o n v { x r ~ R El( V, F ) is a bipartite
x ( C ) ~
December 1983
min s.t.
wx
~ 1, (2.2) x>~0 has an integer valued optimal solution; moreover, if w is integer valued, the dual problem max
s.t.
Ax
yl yA <~w y>~0
(2.3)
has a half integral optimal solution. A vector y is said to be half integral if the vector 2y is integer valued. From the construction of G it is easy to see that every path from s~ to t~ has an odd number of edges, i = 1, 2. Hence A is the incident matrix of odd cycles of G. Let x* and y* be the optimal solutions of (2.2) and (2.3) respectively.
Volume 2, Number 5
OPERATIONS RESEARCH LETTERS
We claim that ,T(e)= 1 - x*(e), is an optimal solution of max wx s.t. x(C) ICI - 1, c an odd cycle in G,
O<~x(e)<~ l, e ~ E . (2.4) To see this, we shall show that (y*, z) is an optimal solution of the dual problem, where
z(e)=w(e)-~,(y,*" eisin the odd cycle C).
December 1983
A graph is called a bicycle p-wheel if G consists of a cycle of length p and two vertices that are adjacent to each other and to every vertex in the cycle. In [4] we have shown that if G = (1I, E) is a bicycle (2k + 1)-wheel, k >I 1, then the inequality x(E)~2(2k + 1) defines a facet of Pn(G). From this fact, and from Theorem (2.5) we obtain the following Remark 2.7. A bicycle p-wheel is weakly bipartite if and only if p is even.
Then Since K s is not weakly bipartite, property (2.1) cannot be replaced by
and
"There exist three vertices...".
(ICl- 1)7,* + z(E). Let us remark that if w is integer valued and y* is half integral then z is half integral. We can state the following Theorem 2.5. If G ~ 12 then g is weakly bipartite. Moreover, a maximum cut is the complement of the edge set corresponding to a minimum two-commodity cut in G. Thus the total amount of work to solve the problem is bounded by O(n3), where n is the order of G. If w(e)= 2 for e ~ E, problem (2.3) has an integer valued optimal solution. That implies the following Remark 2.6. Given G ~ fl, if each edge of G is replaced by two parallel edges in the new graph, the minimum number of edges that must be deleted to have a bipartite graph, equals the maximum number of edge-disjoint odd cycles. Figure 2.1 shows some graphs in ft.
Moreover, the word 'bipartite' cannot be replaced by 'planar'. The cut polytope 1,,.(G) of a graph G is the convex hull of incidence vectors of all edge sets of cuts of G. It is clear that Pc(G)_c Pa(G) and that each extreme point of Pc(G) is an extreme point of PB(G). For graphs in fl, even when Ps(G) can be easily characterized, an inequality system defining Pc(G) could be much more complicated. For instance, if G=(V, E) is the graph of figure 2.1(c), the inequality x( E \ { f }) - x ( f ) ~<6 defines a facet of Pc(G), cf. Barahona and Mahjoub [5].
3. On the blocker of odd cycles Let C be a clutter on a set E, and let A be the matrix with rows all incidence vectors of sets in C. If the pair of programs max s.t. min s.t.
P2
(a) Fig. 2.1.
(b)
('c)
yl ya w, y>~0,
(3.1)
wx Ax >_.1,
(3.2) x>_.0 both have integer optimizing vectors, for every integer vector w >10, the clutter is called mengerian, cf. Seymour [13]. Let G be a graph, let C(G) be the clutter of odd cycles, and let D(G) be the blocker of C(G). In [13], Seymour pointed out that the statement "if G is planar then D(G) is mengerian", implies the four-color theorem. This theorem is equivalent to the statement that for a loopless planar graph 241
Volume 2, Number 5
OPERATIONS RESEARCH LETTERS
A w
t i
(b)
(a)
Fig. 3.1.
there are three disjoint sets of edges intersecting all odd cycles, cf. Woodali [14]. But the graph G of Figure 3.1(a) is a planar graph such that D(G) is not mengerian. In fact, since G E f~, problem (3.1) is equivalent to the 'two-commodity cut packing problem' in the graph of Figure 3.1(b). For this graph the clutter of two-commodity cuts is not mengerian, cf. [13]. In what follows we will give a straightforward characterization of graphs such that C(G) and D(G) are mengerian. A signed graph is a pair (G, s), where G - ( V, E) is a graph, and s:E ~ { - 1, 1 } is called a sign function. Let us denote by $ the function $--- - 1 . A negative cycle C is a cycle such that n eGC
,(e)=-1.
For instance, negative cycles of (G, $) correspond to odd cycles in G. Let us denote by C(G.) the clutter of negative cycles of (G, s), and let D(G) be its blocker. C(G) and/)(G) are binary clutters. A signed graph (G, s) is said to be reducible to (G', s') if this last graph can be obtained from the former one by a sequence of the following operations: (i) deletion of an edge, (ii) contraction of an edge e with positive sign, (iii) changing the signs of the edges of a star ,~(v), t, G V.
I
I
edge with positive sign
I
I
Fig. 3.2.
242
edge with negative sign
December 1983
It is easy to see that ~'(G') and D(G') are minors of ¢~(G) and /)(G) respectively. Seymour [13] showed that the clutter Q6, defined by Q 6 - { {1, 3, 5}, {1, 4, 6}, {2, 3, 6}, {2, 4, 5} }, is. a minimal nonmengerian clutter. Moreover, he proved that a binary clutter is mengerian if and only if it has no Q6 minor. Hence, given a graph G, C(G) is mengerian if and only if (G, $) is not reducible to (/(4, $). And, D(G) is mengerian if and only if (G, $) is not reducible to the signed graph of Figure 3.2. References [1] F. Barahona, "On the complexity of max cut", Rapport de Recherche No. 186, Mathematiques Appliques et lnforma.. tique Universit6 Scientifique et M~dicale de Grenoble, France, February 1980. [2] F. Barahona, "Balancing.signed toroidal graphs in polynomial time", Depto. de Matematicas, Universidad de Chile, Santiago, Chile, June 1982. [3] F. Barahona, "The max cut problem on graphs not contractible to Ks", Report No. 82239-OR, lnstitut for Operations Research, UniversitM Bonn. To appear in Oper. Res. Lett. [4] F. Barahona, M. GrOtschel and A.R. Mahjoub, "Facets of the bipartite subgraph polytope", Research Report, institut fur Operations Research, Universittt Bonn (1983). [5] F. Barahona and A.R. Mahjoub, "On the cut polytope", Research Report, Institut for Operations Research, Universitfit Bonn (1983). [6] D.R. Fulkerson, "Blocking =nd antiblocking polyhedra", Math. Programming I, 168-194 (1971). [7] M.R. Garey and D.S. Johnson, Computers and lntractabiiit),: A Guide to the Theory of NP.Completeness, Freeman, San Francisco, 1979. [8] M. Gr6tscheland W.R. Pulleyblank, "Weakly bipartite graphs and the max-cut problem", Oper. Res. Lett. I, 23-27 (1981). [9] M. Grlitschel and G.L. Nemhauser, "A polynomial algorithm for the max-cut problem on graphs without long odd cycles", Report No. 8221-OR, Universititt Bonn (1982). [lOl M. Grittschel, L. Lov~sz and A. Sehrijver, "The ellipsoid method and its consequences in combinatorial optimization:', Combinatorica I, 169-197 1981 [11] F.O. Hadlock, "Finding a maximum cut o f a planar graph in polynomial time", SIAM J. Comput. 4, 221-225 (1975). [12] T.C. Hu, "Multi-commodity networks flows", Oper. Res. II, 344-360 (1963). (13l P.D. Seymour, "The matroids with the max-flow min-cut property", J. Combin. Theory Set. B 23, 189-222 (1977). [14] D.R. Woodall, "Property B and the four-color problem", in: Combinatorics, Proc. Combinatorial Conf., Oxford, 1972, pp. 322-340.
OPERATIONS RESEARCH LETTERS
December 1983
ON S O M E WEAKLY B I P A R T I T E G R A P H S
Francisco BARAHONA lnstitut fiir Operations Research, Universiti~t Bonn, Fed. Rep. Germany Received May 1983 Revised September 1983
We characterize a class of weakly bipartite graphs. In this case, the max-cut problem can be solved by, finding a minimum two-commodity cut. Max.cut problem, polyhedral combinatorics, ellipsoid method
I. I n t r o d u c t i o n a n d n o t a t i o n
If G -- (V, E) is a graph, and W_c V, 8(W) denotes the set of edges with exactly one extremity in W, 8(W) is called a cut. The max-cut problem can be stated as follows; given a set of edge weights ce > 0 for all e ~ E, find a set W c_ V such that c(8(W)) - Y~{ ce : e ¢ 8(W)} is as large as possible. A maximum cut is a maximum bipartite subgraph of G. Since bipartite subgraphs do not have odd cycles one can ask which are the graphs such that the convex hull of incidence vectors of bipartite subgraphs is defined by the odd cycle inequalities. Gr6tschel and Pulleyblank [8] called those graphs weakly bipartite. Using this polyhedral characterization and the ellipsoid method for linear programming it is possible to solve the maxcut problem in polynomial time for this kind of graphs. Another characterization or a polynomial algorithm to recognize weakly bipartite graphs is not known. Clearly, bipartite graphs belong to this class, and planar graphs are also weakly bipartite. Our aim is to present a new family of weakly bipartite graphs: those graphs
Research supported by the Alexander yon Humboldt Stiftung. Present address: Dpto. De Matematicas, U. de Chile, Casilla 5272, Santiago 3, Chile.
such that there exist two vertices that cover all the odd cycles. If G - (V, E) is a graph the cardinality of V is called the order of G. If e E E is an edge with endnodes i and j we also write/j to denote the edge e. If H - (W, F ) is a graph with W _ V and F c_ E then H is called a subgrapb of G. If G - (V, E) is a graph and F _ E, then V(F) denotes the set of nodes of V which occur at least once as an eadnote of an edge in F. Similarly, for Wc_ V, E(W) denotes the set of all edges of G with both endnodes in W. For W c_ V, G \ W will denote the graph H = ( V \ W, E ( V \ W)). A graph G is called complete if every two different nodes of G are linked by an edge. The complete graph with n nodes is denoted by Kn. A graph is called bipartite if its node set can be partitioned into two nonempty, disjoint sets Vi and VI such that no two nodes in I/1 and no two nodes in I/2 are linked by an edge. We call V1, V2 a bipartition of V. If W_c V. Clearly, (W, 8(W)) is a maximal bipartite subgraph of G. We write 8(v) instead of 8({ v }) for v ~ V and call 8(o) the star of v. A path P in G - (V, E) is a sequence of edges e t , e2, . . . , e k such that e~ -- royt, e 2 -- v i v 2, . . . , ek--vk-~v~ and such that v ~ v j for i cj. The nodes v0 and v~ are the endnodes of P and we say that P links Vo and vk or goes from Vo to vs. The number k of edges of P is called the length of P. If
0167.6377/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North.Holland)
239
Volume2, Number 5
OPERATIONSRESEARCHLETTERS
P --- et, e 2, e k is a path linking Vo and vk and ek+t = VoVk ~ E, then the sequence et, e 2, • .., ek, ek+t is called a cycle of length k + 1. A cycle (path) is called odd if its length is odd, otherwise it is called even. Given F c: E, x F will denote the incidence vector of F. for x: E--,R, and F c_ E , x ( F ) will denote • { x(e): e ~ F }. The max-cut problem is NP-Complete, cf. [7], but it is polynomiaily solvable for planar graphs, cf. Hadlock [11], for graphs not contractible to K s, cf. Barahona [3], for weakly bipartite graphs, cf. Gr6tschel and Pulleyblank [8], and for graphs without long odd cycles, of. Grt~tschel and Nemhauser [9]. Moreover, the cut of maximum cardinality can be obtained in polynomial time for toroidal graphs, cf. Barahona [2]. The polytope ...,
subgraph of G } is called the bipartite subgraph polytope. Let Q(G) be the polytope defined by: C and odd cycle,
O ( x ( e ) ~ < 1, e 6 E .
There exist two vertices, pj and P2, such that (2.1) G \ { Pt, P2 } is bipartite If is clear that graphs in fi can be recognized in polynomial time. Let G - (V, E) be a graph in f~ and let Pl, P2 be the vertices mentioned in (2.1). In what follows we will construct a graph G such that the complement of a minimum two-commodity cut in G gives us a maximum bipartite subgraph of G. Let G ' (V', E') be the graph G \ { pt, P2 }, let U, W be a bipartition of V'. The graph ¢~- (V, E) is defined as follows:
if uv ~ E' then uv ~ E, if p~u ~ E, for u ¢ U, then s~u ~ E, 1
if p~w ~ E, for w ¢ W, then t~w ~ E,
i = 1, 2, i - 1, 2,
(1.1)
ifptp2 ~ E, then s~t 2 ~ E.
(1.2)
Let us call A the incidence matrix of all s~ to t~ paths, i - 1, 2. It follows from the two-commodity max-flow min-cut theorem, cf. Hu [12], and from the theory of blocking polyhedra, cf. Fulkerson [6], that the incidence matrix B of all cuts separating s~ from t,, i = 1, 2, is the blocking matrix of A. Hence, for w >_.0, the problem
x E
thus if G is weakly bipartite it permits to find a maximum cut. We will show a new class of weakly bipartite graphs, hence a class of graphs where the max-cut problem is polynomially solvable. Moreover, in this case this problem can be solved by finding a minimum two-commodity cut. In the last section we will study the clutter of odd cycles and its blocker. 240
Let us denote by fl the class of graphs G with the following property:
1
It is clear that Pn(G)c_ 9(G), and that each extreme point of Pa(G) is an extreme point of 9(G). Moreover, inequalities (1.1) and (1.2) define facets of Pa(G), cf. Barahona [1]. Gr6tschel and Pulleyblank [8] defined the class of Weakly Bipartite Graphs as the set of graphs G such that Ps(G)= Q(G). Moreover, they gave a polynomial algorithm to solve the separation problem for 9(G)" given a point x, either verify that it belongs to 9(G) or else find a hyperplane that separates it from 9(G). This algorithm combined with the ellipsoid method [10], permits to solve in polynomial time the problem max c x s.t.
2. A class of weakly bipartite graphs
V - V ' U { $ 1 , $ 2 , tl, t2},
•De(G ) - c o n v { x r ~ R El( V, F ) is a bipartite
x ( C ) ~
December 1983
min s.t.
wx
~ 1, (2.2) x>~0 has an integer valued optimal solution; moreover, if w is integer valued, the dual problem max
s.t.
Ax
yl yA <~w y>~0
(2.3)
has a half integral optimal solution. A vector y is said to be half integral if the vector 2y is integer valued. From the construction of G it is easy to see that every path from s~ to t~ has an odd number of edges, i = 1, 2. Hence A is the incident matrix of odd cycles of G. Let x* and y* be the optimal solutions of (2.2) and (2.3) respectively.
Volume 2, Number 5
OPERATIONS RESEARCH LETTERS
We claim that ,T(e)= 1 - x*(e), is an optimal solution of max wx s.t. x(C) ICI - 1, c an odd cycle in G,
O<~x(e)<~ l, e ~ E . (2.4) To see this, we shall show that (y*, z) is an optimal solution of the dual problem, where
z(e)=w(e)-~,(y,*" eisin the odd cycle C).
December 1983
A graph is called a bicycle p-wheel if G consists of a cycle of length p and two vertices that are adjacent to each other and to every vertex in the cycle. In [4] we have shown that if G = (1I, E) is a bicycle (2k + 1)-wheel, k >I 1, then the inequality x(E)~2(2k + 1) defines a facet of Pn(G). From this fact, and from Theorem (2.5) we obtain the following Remark 2.7. A bicycle p-wheel is weakly bipartite if and only if p is even.
Then Since K s is not weakly bipartite, property (2.1) cannot be replaced by
and
"There exist three vertices...".
(ICl- 1)7,* + z(E). Let us remark that if w is integer valued and y* is half integral then z is half integral. We can state the following Theorem 2.5. If G ~ 12 then g is weakly bipartite. Moreover, a maximum cut is the complement of the edge set corresponding to a minimum two-commodity cut in G. Thus the total amount of work to solve the problem is bounded by O(n3), where n is the order of G. If w(e)= 2 for e ~ E, problem (2.3) has an integer valued optimal solution. That implies the following Remark 2.6. Given G ~ fl, if each edge of G is replaced by two parallel edges in the new graph, the minimum number of edges that must be deleted to have a bipartite graph, equals the maximum number of edge-disjoint odd cycles. Figure 2.1 shows some graphs in ft.
Moreover, the word 'bipartite' cannot be replaced by 'planar'. The cut polytope 1,,.(G) of a graph G is the convex hull of incidence vectors of all edge sets of cuts of G. It is clear that Pc(G)_c Pa(G) and that each extreme point of Pc(G) is an extreme point of PB(G). For graphs in fl, even when Ps(G) can be easily characterized, an inequality system defining Pc(G) could be much more complicated. For instance, if G=(V, E) is the graph of figure 2.1(c), the inequality x( E \ { f }) - x ( f ) ~<6 defines a facet of Pc(G), cf. Barahona and Mahjoub [5].
3. On the blocker of odd cycles Let C be a clutter on a set E, and let A be the matrix with rows all incidence vectors of sets in C. If the pair of programs max s.t. min s.t.
P2
(a) Fig. 2.1.
(b)
('c)
yl ya w, y>~0,
(3.1)
wx Ax >_.1,
(3.2) x>_.0 both have integer optimizing vectors, for every integer vector w >10, the clutter is called mengerian, cf. Seymour [13]. Let G be a graph, let C(G) be the clutter of odd cycles, and let D(G) be the blocker of C(G). In [13], Seymour pointed out that the statement "if G is planar then D(G) is mengerian", implies the four-color theorem. This theorem is equivalent to the statement that for a loopless planar graph 241
Volume 2, Number 5
OPERATIONS RESEARCH LETTERS
A w
t i
(b)
(a)
Fig. 3.1.
there are three disjoint sets of edges intersecting all odd cycles, cf. Woodali [14]. But the graph G of Figure 3.1(a) is a planar graph such that D(G) is not mengerian. In fact, since G E f~, problem (3.1) is equivalent to the 'two-commodity cut packing problem' in the graph of Figure 3.1(b). For this graph the clutter of two-commodity cuts is not mengerian, cf. [13]. In what follows we will give a straightforward characterization of graphs such that C(G) and D(G) are mengerian. A signed graph is a pair (G, s), where G - ( V, E) is a graph, and s:E ~ { - 1, 1 } is called a sign function. Let us denote by $ the function $--- - 1 . A negative cycle C is a cycle such that n eGC
,(e)=-1.
For instance, negative cycles of (G, $) correspond to odd cycles in G. Let us denote by C(G.) the clutter of negative cycles of (G, s), and let D(G) be its blocker. C(G) and/)(G) are binary clutters. A signed graph (G, s) is said to be reducible to (G', s') if this last graph can be obtained from the former one by a sequence of the following operations: (i) deletion of an edge, (ii) contraction of an edge e with positive sign, (iii) changing the signs of the edges of a star ,~(v), t, G V.
I
I
edge with positive sign
I
I
Fig. 3.2.
242
edge with negative sign
December 1983
It is easy to see that ~'(G') and D(G') are minors of ¢~(G) and /)(G) respectively. Seymour [13] showed that the clutter Q6, defined by Q 6 - { {1, 3, 5}, {1, 4, 6}, {2, 3, 6}, {2, 4, 5} }, is. a minimal nonmengerian clutter. Moreover, he proved that a binary clutter is mengerian if and only if it has no Q6 minor. Hence, given a graph G, C(G) is mengerian if and only if (G, $) is not reducible to (/(4, $). And, D(G) is mengerian if and only if (G, $) is not reducible to the signed graph of Figure 3.2. References [1] F. Barahona, "On the complexity of max cut", Rapport de Recherche No. 186, Mathematiques Appliques et lnforma.. tique Universit6 Scientifique et M~dicale de Grenoble, France, February 1980. [2] F. Barahona, "Balancing.signed toroidal graphs in polynomial time", Depto. de Matematicas, Universidad de Chile, Santiago, Chile, June 1982. [3] F. Barahona, "The max cut problem on graphs not contractible to Ks", Report No. 82239-OR, lnstitut for Operations Research, UniversitM Bonn. To appear in Oper. Res. Lett. [4] F. Barahona, M. GrOtschel and A.R. Mahjoub, "Facets of the bipartite subgraph polytope", Research Report, institut fur Operations Research, Universittt Bonn (1983). [5] F. Barahona and A.R. Mahjoub, "On the cut polytope", Research Report, Institut for Operations Research, Universitfit Bonn (1983). [6] D.R. Fulkerson, "Blocking =nd antiblocking polyhedra", Math. Programming I, 168-194 (1971). [7] M.R. Garey and D.S. Johnson, Computers and lntractabiiit),: A Guide to the Theory of NP.Completeness, Freeman, San Francisco, 1979. [8] M. Gr6tscheland W.R. Pulleyblank, "Weakly bipartite graphs and the max-cut problem", Oper. Res. Lett. I, 23-27 (1981). [9] M. Grlitschel and G.L. Nemhauser, "A polynomial algorithm for the max-cut problem on graphs without long odd cycles", Report No. 8221-OR, Universititt Bonn (1982). [lOl M. Grittschel, L. Lov~sz and A. Sehrijver, "The ellipsoid method and its consequences in combinatorial optimization:', Combinatorica I, 169-197 1981 [11] F.O. Hadlock, "Finding a maximum cut o f a planar graph in polynomial time", SIAM J. Comput. 4, 221-225 (1975). [12] T.C. Hu, "Multi-commodity networks flows", Oper. Res. II, 344-360 (1963). (13l P.D. Seymour, "The matroids with the max-flow min-cut property", J. Combin. Theory Set. B 23, 189-222 (1977). [14] D.R. Woodall, "Property B and the four-color problem", in: Combinatorics, Proc. Combinatorial Conf., Oxford, 1972, pp. 322-340.