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Separating from the dominant of the spanning tree polytope
Operations Research Letters 12 (1992)201-203 North-Holland
October 1992
Separating from the dominant of the spanning tree polytope Francisco Barahona IBM T.J. Watson Research Center, Yorktown Heights, N Y 10598, USA Received October 1991 Revised March 1992
We study the separation problem for the partition inequalities that define the dominant of the spanning tree polytope of a graph G = (V, f ) . We show that a most violated inequality can be found by solving at most I V ] maximum flow problems. Cunningham (1985) had solved this as a sequence of [El maximum flow problems. spanning tree polyhedron: separation problem
1, Introduction Given a graph G = (V, E ) the spanning tree polytope T(G) is the convex hull of incidence vectors of spanning trees of G, its dominant is the polyhedron P(G) = T(G) + ~ obtained by adding the nonnegative orthant. In this paper we study the problem of finding a hyperplane that separates a given vector £ from P(G) or prove that ~ e P(G). Cunningham (19851 studied this problem and reduced it to I E I maximum flow problems. We are going to show a reduction to I V I maximum flow problems. Consider a partition {V1. . . . . Vp} of V. We denote by 6(V 1. . . . . VR) the set of edges having their endnodes in different members of the partition. When the partition consists of two sets S and T \ S sometimes we use 6(S). Given x • 7~~: we denote Y]x(e) I e • S} by x(S). Fulkerson (19711 (see also Chopra (19891) proved that P(G) is defined by
x ( 6 ( V I ..... Vv)) > p -
1
the 2-connected subgraph polytope as follows. Let H be a graph and consider the convex hull of incidence vectors of 2-node-connected spanning subgraphs of H denoted by T N C P ( H ) . For any node u the restriction to H \ u of a 2-connected subgraph contains a spanning tree, then the partition inequalities associated with H \ u are valid for T N C P ( H ) . In some cases they define facets of this polytope, see Gr6tschel et al. (1989) and Barahona and Mahjoub (19911. So the problem is: given 2 • ~t: find a parti-
tion of V that minimizes .~ ( (~ ( V l ..... V p ) ) - p-[-1. Cunningham (1985) refers to this as the attack
problem, he also showed that the problem minimize £( 3( VI . . . . . Vp)) p-1 called strength problem, reduces to a sequence of I V I attack problems.
(1.1)
for every partition of V, x_>O.
Inequalities (1.1) are called partition inequafities. They also give rise to valid inequalities for Correspondence to: F. Barahona, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA.
2. The algorithm Jiinger extended ciate the variables
and Pulleyblank (1991) have given an formulation for P(G) as follows. Assovariables x with the edges and the y with the nodes. Given S _c V, we use
0167-6377/92/$(15.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
201
Volume 12, Number 4
OPERATIONS RESEARCH LETTERS
y(S) to denote E{y(u)l u ~ S}. The system below defines a polyhedron whose projection onto the variables x is P(G). The node r is an arbitrary element of V.
x(6(S))+y(S)>2 x(a(S))+y(S)>O, y(V) =0.
ifrf~S, S c V , ifrcS, ScV,
(2.1)
So given a vector ~ we want to find a vector such that (Y, ~) satisfies (2.1) or prove that does not exist. Let
f(S)=
{
2-~(6(S)), -Y(6(S)),
ifr~S, ifr~S,
y( V)
subject to
y ( S ) >_f(S)
for S_c V.
(2.2)
(2.3)
lu S}=l
for all u E V,
z>0. Given a vector y satisfying (2.2), a set S is called tight if y(S)=f(S). We need the following well known lemma. U n c r o s s i n g L e m m a . If S and T are tight and S n T 4=O, then S n T and S U T are also tight. Proof.
y(SUT)+y(SOT) =y(S) +y(T) +f(T) < f ( S U T) + f ( S n T ) .
=f(S)
[]
Let V = {v 1. . . . . v.}. We start with y ( v ) = 2 Vi, and decrease the value of each y(v i) until a set 202
for
i = l . . . . . n;
k~l;
Step 1. If v~ belongs to a set in 9- go to Step 3, otherwise set a = f ( S ) - p ( S ) = max{f(S) - Y(S) I v k e S},
{f}.
gu
Step 2. While there are two sets S and T in 9with S n T 4= 0 do
;q) u { r u s}. Step 3 Set k <---k + 1, if k < n go to Step 1, otherwise stop. The vector ~ family 3 - d e f i n e s for every S ~ 9 - . 2 s = 0 otherwise.
= E{
~Zsf( S)
subject to
E{
StepO. Set p(v i ) = 2
g+-
As shown by Edmonds (1970), the greedy algorithm solves this linear program. This algorithm produces also an optimal solution of the dual problem, as we shall see, this will give us a most violated partition inequality if there is any. The dual problem is maximize
becomes tight. We denote by 3 the family of tight sets. If we try to add S to 3- and there is a set T e 9- with S • T =~ O, then we replace S and T by S u T. This is also tight because of the uncrossing lemma. The algorithm is:
y ( v k ) +-
for S __ V. The function f ( ' ) is supermodular for intersecting pairs, i.e., f(S U T) + f(S (~ T) >_ f(S) +f(T) for sets S and T in 2 v with S n T=~ 0. We are going to solve minimize
October 1992
is built so it satisfies (2.2), the a partition of V and ~(S) = f ( S ) We set ~ s = l , if S ~ 9 - , and We have
(s)ls
= F_,{f(s)ls
g}
= £{f(S)$sISgV
}.
This proves that ~ and 2 are optimal solutions. If the value of the optimum is 0 then (.2, Y) satisfies (2.1). In this case we can pick any partition of V into V p . . . , Vp, add the inequalities in (2.1) associated with the sets {V/} and - y ( V ) >_ 0, we obtain a partition inequality. This shows that satisfies all the partition inequalities. Now assume that the value of the optimum is greater than 0. Let zT be a 0-1 vector that satisfies the equations of (2.3), the family W = {Slzs = 1} = {S l . . . . , Sp} gives a partition of the set V and
Y'.f(S)Y. s = 2( p - 1) - 2 ~ ( a ( S , . . . . . St,)), since 2 is an optimum of (2.3) it gives a most violated partition inequality.
OPERATIONS RESEARCH LETTERS
Volume 12, Number 4
October 1992
Fig. 1
It remains to show how to compute the number a in Step 1. Construct a directed graph D = ( N , A), where N = V U { s , t} and A = {(i, j), (j, i) 16 ~ E} u {(s, i), (i, t)li ~ V}. Define r/(i)=~(i)
fori~V,
References
define capacities
c(s, Uk)
c(i,t)=O, c(s,i)=O,
if B(i) < 0 , if ~ ( i ) > 0 ,
C(Uk, t) =0,
c(i, j) =c(j, i) =.~(i, j)
for ij~E.
If {s} U S induces a cut separating s from t that has capacity A, see Figure 1, then we have
+2(a(S))
=a + E
I n(t ) < 0}.
Therefore if /3 is the minimum capacity of a cut separating s from t, then the value a is 2-/3-
I am grateful to the referee and to the associate editor for their suggestions on the presentation of this paper.
i4:r,
rt(r ) = y ( r ) + 2,
c ( s , i ) = -~7(i), i4~uk, i ~ V , c(i,t)=~(i), i4:uk, i ~ V ,
Acknowledgements
~ {rt(t,) I rt(t, ) < 0 } .
F. Barahona and A.R. Mahjoub (1991), "On 2-connected subgraph polytopes", preprint. S. Chopra (1989), "On the spanning tree polyhedron", Oper. Res. Lett. 8, 25-29. W.H. Cunningham, (1985), "Optimal attack and reinforcement of a network", J. ACM 32, 549-561. M. Jfinger and W.R. Pulleyblank (1991), "New primal and dual matching heuristics", Report No. 91.105. Institut fflr Informatik, Universitiit zu K61n. J. Edmonds (1970), "Submodular functions, matroids and certain polyhedra", in: R.K. Guy, E. Milner and N. Sauers (eds.), Combinatorial Structures and Their Applications, Gordon and Breach, New York, 69-87. D.R. Fulkerson (1971), "Blocking and anti-blocking pairs of polyhedra", Math. Programming 1, 127-136. M. Gr6tschel, C.L. Monma and M. Stoer (1989), "Facets for polyhedra arising in the design of communication networks with low connectivity constraints", Report No. 187, Institut ffir Mathematik, Universit~t Augsburg.
203
October 1992
Separating from the dominant of the spanning tree polytope Francisco Barahona IBM T.J. Watson Research Center, Yorktown Heights, N Y 10598, USA Received October 1991 Revised March 1992
We study the separation problem for the partition inequalities that define the dominant of the spanning tree polytope of a graph G = (V, f ) . We show that a most violated inequality can be found by solving at most I V ] maximum flow problems. Cunningham (1985) had solved this as a sequence of [El maximum flow problems. spanning tree polyhedron: separation problem
1, Introduction Given a graph G = (V, E ) the spanning tree polytope T(G) is the convex hull of incidence vectors of spanning trees of G, its dominant is the polyhedron P(G) = T(G) + ~ obtained by adding the nonnegative orthant. In this paper we study the problem of finding a hyperplane that separates a given vector £ from P(G) or prove that ~ e P(G). Cunningham (19851 studied this problem and reduced it to I E I maximum flow problems. We are going to show a reduction to I V I maximum flow problems. Consider a partition {V1. . . . . Vp} of V. We denote by 6(V 1. . . . . VR) the set of edges having their endnodes in different members of the partition. When the partition consists of two sets S and T \ S sometimes we use 6(S). Given x • 7~~: we denote Y]x(e) I e • S} by x(S). Fulkerson (19711 (see also Chopra (19891) proved that P(G) is defined by
x ( 6 ( V I ..... Vv)) > p -
1
the 2-connected subgraph polytope as follows. Let H be a graph and consider the convex hull of incidence vectors of 2-node-connected spanning subgraphs of H denoted by T N C P ( H ) . For any node u the restriction to H \ u of a 2-connected subgraph contains a spanning tree, then the partition inequalities associated with H \ u are valid for T N C P ( H ) . In some cases they define facets of this polytope, see Gr6tschel et al. (1989) and Barahona and Mahjoub (19911. So the problem is: given 2 • ~t: find a parti-
tion of V that minimizes .~ ( (~ ( V l ..... V p ) ) - p-[-1. Cunningham (1985) refers to this as the attack
problem, he also showed that the problem minimize £( 3( VI . . . . . Vp)) p-1 called strength problem, reduces to a sequence of I V I attack problems.
(1.1)
for every partition of V, x_>O.
Inequalities (1.1) are called partition inequafities. They also give rise to valid inequalities for Correspondence to: F. Barahona, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA.
2. The algorithm Jiinger extended ciate the variables
and Pulleyblank (1991) have given an formulation for P(G) as follows. Assovariables x with the edges and the y with the nodes. Given S _c V, we use
0167-6377/92/$(15.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
201
Volume 12, Number 4
OPERATIONS RESEARCH LETTERS
y(S) to denote E{y(u)l u ~ S}. The system below defines a polyhedron whose projection onto the variables x is P(G). The node r is an arbitrary element of V.
x(6(S))+y(S)>2 x(a(S))+y(S)>O, y(V) =0.
ifrf~S, S c V , ifrcS, ScV,
(2.1)
So given a vector ~ we want to find a vector such that (Y, ~) satisfies (2.1) or prove that does not exist. Let
f(S)=
{
2-~(6(S)), -Y(6(S)),
ifr~S, ifr~S,
y( V)
subject to
y ( S ) >_f(S)
for S_c V.
(2.2)
(2.3)
lu S}=l
for all u E V,
z>0. Given a vector y satisfying (2.2), a set S is called tight if y(S)=f(S). We need the following well known lemma. U n c r o s s i n g L e m m a . If S and T are tight and S n T 4=O, then S n T and S U T are also tight. Proof.
y(SUT)+y(SOT) =y(S) +y(T) +f(T) < f ( S U T) + f ( S n T ) .
=f(S)
[]
Let V = {v 1. . . . . v.}. We start with y ( v ) = 2 Vi, and decrease the value of each y(v i) until a set 202
for
i = l . . . . . n;
k~l;
Step 1. If v~ belongs to a set in 9- go to Step 3, otherwise set a = f ( S ) - p ( S ) = max{f(S) - Y(S) I v k e S},
{f}.
gu
Step 2. While there are two sets S and T in 9with S n T 4= 0 do
;q) u { r u s}. Step 3 Set k <---k + 1, if k < n go to Step 1, otherwise stop. The vector ~ family 3 - d e f i n e s for every S ~ 9 - . 2 s = 0 otherwise.
= E{
~Zsf( S)
subject to
E{
StepO. Set p(v i ) = 2
g+-
As shown by Edmonds (1970), the greedy algorithm solves this linear program. This algorithm produces also an optimal solution of the dual problem, as we shall see, this will give us a most violated partition inequality if there is any. The dual problem is maximize
becomes tight. We denote by 3 the family of tight sets. If we try to add S to 3- and there is a set T e 9- with S • T =~ O, then we replace S and T by S u T. This is also tight because of the uncrossing lemma. The algorithm is:
y ( v k ) +-
for S __ V. The function f ( ' ) is supermodular for intersecting pairs, i.e., f(S U T) + f(S (~ T) >_ f(S) +f(T) for sets S and T in 2 v with S n T=~ 0. We are going to solve minimize
October 1992
is built so it satisfies (2.2), the a partition of V and ~(S) = f ( S ) We set ~ s = l , if S ~ 9 - , and We have
(s)ls
= F_,{f(s)ls
g}
= £{f(S)$sISgV
}.
This proves that ~ and 2 are optimal solutions. If the value of the optimum is 0 then (.2, Y) satisfies (2.1). In this case we can pick any partition of V into V p . . . , Vp, add the inequalities in (2.1) associated with the sets {V/} and - y ( V ) >_ 0, we obtain a partition inequality. This shows that satisfies all the partition inequalities. Now assume that the value of the optimum is greater than 0. Let zT be a 0-1 vector that satisfies the equations of (2.3), the family W = {Slzs = 1} = {S l . . . . , Sp} gives a partition of the set V and
Y'.f(S)Y. s = 2( p - 1) - 2 ~ ( a ( S , . . . . . St,)), since 2 is an optimum of (2.3) it gives a most violated partition inequality.
OPERATIONS RESEARCH LETTERS
Volume 12, Number 4
October 1992
Fig. 1
It remains to show how to compute the number a in Step 1. Construct a directed graph D = ( N , A), where N = V U { s , t} and A = {(i, j), (j, i) 16 ~ E} u {(s, i), (i, t)li ~ V}. Define r/(i)=~(i)
fori~V,
References
define capacities
c(s, Uk)
c(i,t)=O, c(s,i)=O,
if B(i) < 0 , if ~ ( i ) > 0 ,
C(Uk, t) =0,
c(i, j) =c(j, i) =.~(i, j)
for ij~E.
If {s} U S induces a cut separating s from t that has capacity A, see Figure 1, then we have
+2(a(S))
=a + E
I n(t ) < 0}.
Therefore if /3 is the minimum capacity of a cut separating s from t, then the value a is 2-/3-
I am grateful to the referee and to the associate editor for their suggestions on the presentation of this paper.
i4:r,
rt(r ) = y ( r ) + 2,
c ( s , i ) = -~7(i), i4~uk, i ~ V , c(i,t)=~(i), i4:uk, i ~ V ,
Acknowledgements
~ {rt(t,) I rt(t, ) < 0 } .
F. Barahona and A.R. Mahjoub (1991), "On 2-connected subgraph polytopes", preprint. S. Chopra (1989), "On the spanning tree polyhedron", Oper. Res. Lett. 8, 25-29. W.H. Cunningham, (1985), "Optimal attack and reinforcement of a network", J. ACM 32, 549-561. M. Jfinger and W.R. Pulleyblank (1991), "New primal and dual matching heuristics", Report No. 91.105. Institut fflr Informatik, Universitiit zu K61n. J. Edmonds (1970), "Submodular functions, matroids and certain polyhedra", in: R.K. Guy, E. Milner and N. Sauers (eds.), Combinatorial Structures and Their Applications, Gordon and Breach, New York, 69-87. D.R. Fulkerson (1971), "Blocking and anti-blocking pairs of polyhedra", Math. Programming 1, 127-136. M. Gr6tschel, C.L. Monma and M. Stoer (1989), "Facets for polyhedra arising in the design of communication networks with low connectivity constraints", Report No. 187, Institut ffir Mathematik, Universit~t Augsburg.
203