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Combinatorial Optimization:

Packing and Covering

G´erard Cornu´ejols

Carnegie Mellon University

July 2000

1

Preface

The integer programming models known as set packing and set cov-

ering have a wide range of applications, such as pattern recognition,

plant location and airline crew scheduling. Sometimes, due to the spe-

cial structure of the constraint matrix, the natural linear programming

relaxation yields an optimal solution that is integer, thus solving the

problem. Sometimes, both the linear programming relaxation and its

dual have integer optimal solutions. Under which conditions do such

integrality properties hold? This question is of both theoretical and

practical interest. Min-max theorems, polyhedral combinatorics and

graph theory all come together in this rich area of discrete mathemat-

ics. In addition to min-max and polyhedral results, some of the deepest

results in this area come in two ﬂavors: “excluded minor” results and

“decomposition” results. In these notes, we present several of these

beautiful results. Three chapters cover min-max and polyhedral re-

sults. The next four cover excluded minor results. In the last three, we

present decomposition results. We hope that these notes will encourage

research on the many intriguing open questions that still remain. In

particular, we state 18 conjectures. For each of these conjectures, we

oﬀer $5000 as an incentive for the ﬁrst correct solution or refutation

before December 2020.

2

Contents

1 Clutters 7

1.1 MFMC Property and Idealness . . . . . . . . . . . . . . 9

1.2 Blocker . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 st-Cuts and st-Paths . . . . . . . . . . . . . . . . 15

1.3.2 Two-Commodity Flows . . . . . . . . . . . . . . . 17

1.3.3 r-Cuts and r-Arborescences . . . . . . . . . . . . 18

1.3.4 Dicuts and Dijoins . . . . . . . . . . . . . . . . . 19

1.3.5 T -Cuts and T -Joins . . . . . . . . . . . . . . . . . 20

1.3.6 Odd Cycles in Graphs . . . . . . . . . . . . . . . 21

1.3.7 Edge Coloring of Graphs . . . . . . . . . . . . . . 21

1.3.8 Feedback Vertex Set . . . . . . . . . . . . . . . . 22

1.4 Deletion, Contraction and Minor . . . . . . . . . . . . . 23

2 T -Cuts and T -Joins 27

2.1 Proof of the Edmonds-Johnson Theorem . . . . . . . . . 28

2.2 Packing T -Cuts . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Theorems of Seymour and Lov´asz . . . . . . . . . 30

2.2.2 More Min Max Results . . . . . . . . . . . . . . . 33

2.3 Packing T -Joins . . . . . . . . . . . . . . . . . . . . . . . 34

3 Perfect Graphs and Matrices 35

3.1 The Perfect Graph Theorem . . . . . . . . . . . . . . . . 36

3.2 Perfect Matrices . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Antiblocker . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Minimally Imperfect Graphs . . . . . . . . . . . . . . . . 42

3

4 CONTENTS

4 Ideal Matrices 49

4.1 Minimally Nonideal Matrices . . . . . . . . . . . . . . . . 49

4.1.1 Proof of Lehman’s Theorem . . . . . . . . . . . . 52

4.1.2 Examples of mni Clutters . . . . . . . . . . . . . 56

4.2 Ideal Minimally Nonpacking Clutters . . . . . . . . . . . 59

4.3 Clutters such that τ

2

(C) < τ

1

(C) . . . . . . . . . . . . . . 61

5 Odd Cycles in Graphs 63

5.1 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Signed Graphs . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Proof Outline of Guenin’s Theorem . . . . . . . . . . . . 66

5.4 Binary Clutters . . . . . . . . . . . . . . . . . . . . . . . 69

5.4.1 Seymour’s Conjecture . . . . . . . . . . . . . . . 71

5.4.2 Seymour’s MFMC Theorem . . . . . . . . . . . . 72

6 0, ±1 Matrices and Integral Polyhedra 73

6.1 Totally Unimodular Matrices . . . . . . . . . . . . . . . . 74

6.2 Balanced Matrices . . . . . . . . . . . . . . . . . . . . . 80

6.2.1 Integral Polytopes . . . . . . . . . . . . . . . . . 81

6.2.2 Total Dual Integrality . . . . . . . . . . . . . . . 83

6.2.3 A Bicoloring Theorem . . . . . . . . . . . . . . . 86

6.3 Perfect and Ideal 0, ±1 Matrices . . . . . . . . . . . . . . 88

6.3.1 Relation to Perfect and Ideal 0,1 Matrices . . . . 88

6.3.2 Propositional Logic . . . . . . . . . . . . . . . . . 90

6.3.3 Bigraphs and Perfect 0, ±1 Matrices . . . . . . . 92

7 Signing 0,1 Matrices to be TU or Balanced 95

7.1 Camion’s Signing Algorithm . . . . . . . . . . . . . . . . 95

7.2 Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.3 Tutte’s Theorem . . . . . . . . . . . . . . . . . . . . . . 99

7.4 Truemper’s Theorem . . . . . . . . . . . . . . . . . . . . 101

7.4.1 Proof of Truemper’s Theorem . . . . . . . . . . . 103

7.4.2 Another Proof of Tutte’s Theorem . . . . . . . . 105

8 Decomposition by k-Sum 107

8.1 Binary Matroids . . . . . . . . . . . . . . . . . . . . . . . 107

8.2 Decomposition of Regular Matroids . . . . . . . . . . . . 111

CONTENTS 5

8.2.1 Proof Outline . . . . . . . . . . . . . . . . . . . . 114

8.2.2 Recognition Algorithm . . . . . . . . . . . . . . . 116

8.3 Binary Clutters . . . . . . . . . . . . . . . . . . . . . . . 116

8.3.1 Relation to Binary Matroids . . . . . . . . . . . . 116

8.3.2 The MFMC Property . . . . . . . . . . . . . . . . 121

8.3.3 Idealness . . . . . . . . . . . . . . . . . . . . . . . 121

9 Decomposition of Balanced Matrices 125

9.1 Recognition Algorithm for Balanced 0,1 Matrices . . . . 126

9.1.1 Smallest Unbalanced Holes . . . . . . . . . . . . . 126

9.1.2 Recognition Algorithm . . . . . . . . . . . . . . . 129

9.2 Proof Outline of the Decomposition Theorem . . . . . . 137

9.2.1 Even Wheels . . . . . . . . . . . . . . . . . . . . 137

9.2.2 A Conjecture . . . . . . . . . . . . . . . . . . . . 140

9.3 Balanced 0, ±1 Matrices . . . . . . . . . . . . . . . . . . 141

9.3.1 Decomposition Theorem . . . . . . . . . . . . . . 142

9.3.2 Recognition Algorithm . . . . . . . . . . . . . . . 144

10 Decomposition of Perfect Graphs 147

10.1 Basic Classes . . . . . . . . . . . . . . . . . . . . . . . . 147

10.2 Perfection-Preserving Compositions . . . . . . . . . . . . 148

10.3 Meyniel Graphs . . . . . . . . . . . . . . . . . . . . . . . 154

10.3.1 D-structures . . . . . . . . . . . . . . . . . . . . . 156

10.3.2 M-structures . . . . . . . . . . . . . . . . . . . . . 158

10.3.3 Expanded Holes . . . . . . . . . . . . . . . . . . . 160

10.3.4 The Main Theorem . . . . . . . . . . . . . . . . . 162

Acknowledgements: Many thanks to Jon Lee, Kristina Vuˇskovi´c

and Carl Lee for encouraging me to write this monograph and to

Fran¸cois Margot, Beth Novick, Doug West, Michele Conforti, Thomas

Zaslavsky and Andre Kundgen for their comments on the ﬁrst draft.

These notes are based on a series of ten lectures delivered at a regional

6 CONTENTS

CBMS-NSF Conference at the University of Kentucky in May 1999.

The work was supported in part by NSF grant DMI-9802773 and ONR

grant N00014-97-1-0196.

Chapter 1

Clutters

A clutter C is a pair (V, E), where V is a ﬁnite set and E is a family of

subsets of V none of which is included in another. The elements of V

are the vertices of C and those of E are the edges. For example, a simple

graph (V, E) (no multiple edges or loops) is a clutter. We refer to West

[208] for deﬁnitions in graph theory. In a clutter, a matching is a set of

pairwise disjoint edges. A transversal is a set of vertices that intersects

all the edges. A clutter is said to pack if the maximum cardinality

of a matching equals the minimum cardinality of a transversal. This

terminology is due to Seymour 1977. Many min-max theorems in graph

theory can be rephrased by saying that a clutter packs. We give three

examples. The ﬁrst is K¨onig’s theorem.

Theorem 1.1 (K¨onig [130]) In a bipartite graph, the maximum cardi-

nality of a matching equals the minimum cardinality of a transversal.

As a second example, let s and t be distinct nodes of a graph G.

Menger’s theorem states that the maximum number of pairwise edge-

disjoint st-paths in G equals the minimum number of edges in an st-cut

(see West [208] Theorem 4.2.18). Let C

1

be the clutter whose vertices

are the edges of G and whose edges are the st-paths of G (Following

West’s terminology [208], paths and cycles have no repeated nodes).

We call C

1

the clutter of st-paths. Its transversals are the st-cuts. Thus

Menger’s theorem states that the clutter of st-paths packs.

Interestingly, some diﬃcult results and famous conjectures can be

rephrased by saying that certain clutters pack. As a third example,

7

8 CHAPTER 1. CLUTTERS

consider the four color theorem [2] stating that every planar graph is

4-colorable. Tait [190] showed that this theorem is equivalent to the fol-

lowing statement: Every simple 2-connected cubic planar graph G is 3-

edge-colorable (see West [208] Theorem 7.3.3). Let C

2

= (V (C

2

), E(C

2

))

be the clutter whose vertices are the maximal matchings of G and whose

edges are indexed by the edges of G, with S

e

∈ E(C

2

) if and only if

S

e

= {M ∈ V (C

2

) : e ∈ M}. In these notes, a maximal set in a given

family refers to an inclusion-maximal set, whereas a maximum set refers

to a set of maximum cardinality. We make the same distinction between

minimal and minimum. We leave it as an exercise to check that, in a

cubic graph, C

2

packs if and only if G is 3-edge-colorable. Therefore,

the four color theorem is equivalent to stating that C

2

packs for sim-

ple 2-connected cubic planar graphs. The smallest simple 2-connected

cubic graph that is not 3-edge-colorable is the Petersen graph (see Fig-

ure 1.1). Tutte [203] conjectured that every simple 2-connected cubic

graph that is not 3-edge-colorable (i.e. C

2

does not pack) is contractible

to the Petersen graph. (Graph G is contractible to graph H if H can be

obtained from G by a sequence of edge contractions and edge deletions.

Contracting edge e = uv is the operation of replacing u and v by a

single node whose incident edges are the edges other than e that were

incident to u or v. Deleting e is the operation of removing e from the

graph.) Since the Petersen graph is not planar, the four color theorem

is a special case of Tutte’s conjecture. Tutte’s conjecture was proved

recently by Robertson, Sanders, Seymour and Thomas ([165], [169],

[170]). A more general conjecture of Conforti and Johnson [55] is still

open (see Section 1.3.5). This indicates that a full understanding of the

clutters that pack must be extremely diﬃcult. More restricted notions

are amenable to beautiful theories, while still containing rich classes

of examples. In this chapter, we introduce several such concepts and

examples.

Exercise 1.2 In a cubic graph G, show that C

2

packs if and only if G

is 3-edge-colorable.

1.1. MFMC PROPERTY AND IDEALNESS 9

Figure 1.1: The Petersen graph.

1.1 MFMC Property and Idealness

We deﬁne a clutter C to be a family E(C) of subsets of a ﬁnite ground

set V (C) with the property that S

1

6⊆ S

2

for all distinct S

1

, S

2

∈ E(C).

V (C) is called the set of vertices and E(C) the set of edges of C. A

clutter is trivial if it has no edge or if it has the empty set as unique

edge. Clutters are also called Sperner families in the literature.

Given a nontrivial clutter C, we deﬁne M(C) to be a 0,1 matrix

whose columns are indexed by V (C), whose rows are indexed by E(C)

and where m

ij

= 1 if and only if the vertex corresponding to column j

belongs to the edge corresponding to row i. In other words, the rows of

M(C) are the characteristic vectors of the sets in E(C). Note that the

deﬁnition of M(C) is unique up to permutation of rows and permutation

of columns. Furthermore, M(C) contains no dominating row, since C

is a clutter (A vector r ∈ F is said to be dominating if there exists

v ∈ F distinct from r such that r ≥ v). A 0,1 matrix containing no

dominating rows is called a clutter matrix. Given any 0,1 clutter matrix

M, let C(M) denote the clutter such that M(C(M)) = M.

Let M 6= 0 be a 0,1 clutter matrix and consider the following pair

of dual linear programs.

min{wx : x ≥ 0, Mx ≥ 1} (1.1)

10 CHAPTER 1. CLUTTERS

= max{y1 : y ≥ 0, yM ≤ w} (1.2)

Here x and 1 are column vectors while w and y are row vectors. 1

denotes a vector all of whose components are equal to 1.

Deﬁnition 1.3 Clutter C(M) packs if both (1.1) and (1.2) have opti-

mal solution vectors x and y that are integral when w = 1.

Deﬁnition 1.4 Clutter C(M) has the packing property if both (1.1)

and (1.2) have optimal solution vectors x and y that are integral for all

vectors w with components equal to 0, 1 or +∞.

Deﬁnition 1.5 Clutter C(M) has the Max Flow Min Cut property (or

MFMC property) if both (1.1) and (1.2) have optimal solution vectors

x and y that are integral for all nonnegative integral vectors w.

Clearly, the MFMC property for a clutter implies the packing prop-

erty which itself implies that the clutter packs. Conforti and Cornu´ejols

[41] conjectured that, in fact, the MFMC property and the packing

property are identical. This conjecture is still open.

Conjecture 1.6 A clutter has the MFMC property if and only if it has

the packing property.

Deﬁnition 1.7 Clutter C(M) is ideal if (1.1) has an optimal solution

vector x that is integral for all w ≥ 0.

The notion of idealness is also known as the width-length property

(Lehman [133]), the weak Max Flow Min Cut property (Seymour [183])

or the Q

+

-MFMC property (Schrijver [172]). It is easy to show that

the MFMC property implies idealness. Indeed, if (1.1) has an optimal

solution vector x for all nonnegative integral vectors w, then (1.1) has

an optimal solution x for all nonnegative rational vectors w and, since

the rationals are dense in the reals, for all w ≥ 0. In fact, the packing

property implies idealness.

Theorem 1.8 If a clutter has the packing property, then it is ideal.

This follows from a result of Lehman [133] that we will prove in Chap-

ter 4 (see Theorem 4.1 and Exercise 4.8).

1.1. MFMC PROPERTY AND IDEALNESS 11

IDEAL

CLUTTERS THAT PACK

PACKING PROPERTY

MAX FLOW MIN CUT PROPERTY

C

4

2

Q

2+

6

Q

6

+

C

3

2

C

3

Figure 1.2: Classes of clutters.

12 CHAPTER 1. CLUTTERS

Exercise 1.9 Let Q

6

=

1 1 0 1 0 0

1 0 1 0 1 0

0 1 1 0 0 1

0 0 0 1 1 1

, C

2

3

=

1 1 0

0 1 1

1 0 1

and C

2

4

=

1 1 0 0

0 1 1 0

0 0 1 1

1 0 0 1

. For an m ×n 0,1 matrix M, let M

+

denote

the m ×(n +1) matrix obtained from M by adding the column vector 1.

Find which of the clutters C(Q

6

), C(Q

+

6

), C(C

2

3

), C(C

2+

3

), C(C

2

4

) pack,

which have the packing property, which have the MFMC property and

which are ideal. See Figure 1.2 for a hint.

Clearly, C(M) is ideal if and only if P = {x ≥ 0 : Mx ≥ 1}

is an integral polyhedron, that is, P has only integral extreme points.

Equivalently, C is ideal if and only if

x(S) ≥ 1 for all S ∈ E(C)

x ≥ 0

is an integral polyhedron, where x(S) denotes

P

i∈S

x

i

.

A linear system Ax ≥ b is Totally Dual Integral (TDI) if the linear

program min{wx : Ax ≥ b} has an integral optimal dual solution y

for every integral w for which the linear program has a ﬁnite optimum.

Edmonds and Giles [81] showed that, if Ax ≥ b is TDI and b is integral,

then P = {x : Ax ≥ b} is an integral polyhedron. The interested

reader can ﬁnd the proof of the Edmonds-Giles theorem in Schrijver

[173] pages 310–311, or Nemhauser and Wolsey [146] pages 536–537. It

follows that C(M) has the MFMC property if and only if (1.2) has an

optimal integral solution y for all nonnegative integral vectors w.

Deﬁnition 1.10 Let k be a positive integer. Clutter C(M) has the

1/k -MFMC property if it is ideal and, for all nonnegative integral

vectors w, the linear program (1.2) has an optimal solution vector y

such that ky is integral.

When k = 1, this deﬁnition reduces to the MFMC property. If

C(M) has the 1/k-MFMC property, then it also has the 1/q-MFMC

property for every integer q that is a multiple of k.

1.2. BLOCKER 13

For convenience, we say that trivial clutters have all the above prop-

erties: MFMC, ideal, etc.

The min-max equation (1.1)=(1.2) has a close max-min relative

max{wx : x ≥ 0, Mx ≤ 1}

= min{y1 : y ≥ 0, yM ≥ w}

discussed in Chapter 3.

1.2 Blocker

The blocker b(C) of a clutter C is the clutter with V (C) as vertex set

and the minimal transversals of C as edge set. That is, E(b(C)) consists

of the minimal members of {B ⊆ V (C) : |B ∩ A| ≥ 1 for all A ∈ E(C)}.

In other words, the rows of M(b(C)) are the minimal 0,1 vectors x

T

such that x belongs to the polyhedron {x ≥ 0 : M(C)x ≥ 1}.

Example 1.11 Let G be a graph and s, t be distinct nodes of G. If C

is the clutter of st-paths, then b(C) is the clutter of minimal st-cuts.

Exercise 1.12 Show that the blocker of a trivial clutter is a trivial

clutter.

Edmonds and Fulkerson [80] observed that b(b(C)) = C. Before

proving this property, we make the following remark.

Remark 1.13 Let H and K be two clutters deﬁned on the same vertex

set. If

(i) every edge of H contains an edge of K and

(ii) every edge of K contains an edge of H,

then H = K.

Exercise 1.14 Prove Remark 1.13.

Theorem 1.15 If C is a clutter, then b(b(C)) = C.

14 CHAPTER 1. CLUTTERS

Proof: Let A be an edge of C. The deﬁnition of b(C) implies that

|A ∩ B| ≥ 1, for every edge B of b(C). So A is a transversal of b(C), i.e.

A contains an edge of b(b(C)).

Now let A b e an edge of b(b(C)). We claim that A contains an edge

of C. Suppose otherwise. Then V (C) − A is a transversal of C and

therefore it contains an edge B of b(C). But then A∩B = ∅ contradicts

the fact that A is an edge of b(b(C)). So the claim holds.

Now the theorem follows from Remark 1.13. 2

Two 0,1 matrices of the form M(C) and M(b(C)) are said to form a

blocking pair. The next theorem is an important result due to Lehman

[132]. It states that, for a blocking pair A, B of 0,1 matrices, the poly-

hedron P deﬁned by

Ax ≥ 1 (1.3)

x ≥ 0 (1.4)

is integral if and only if the polyhedron Q deﬁned by

Bx ≥ 1 (1.5)

x ≥ 0 (1.6)

is integral. The proof of this result uses the following remark.

Remark 1.16

(i) The rows of B are exactly the 0,1 extreme points of P .

(ii) If an extreme point x of P satisﬁes x

T

≥ λ

T

B where λ

i

≥ 0 and

P

λ

i

= 1, then x is a 0,1 extreme point of P .

Proof: (i) follows from the fact that the rows of B are the minimal 0,1

vectors in P .

To prove (ii), note that x is an extreme point of P

I

= {χ : χ

T

≥

λ

T

B where λ

i

≥ 0 and

P

λ

i

= 1} for otherwise x would be a convex

combination of distinct x

1

, x

2

∈ P

I

and, since P

I

⊆ P , this would

contradict the assumption that x is an extreme point of P . Now (ii)

follows by observing that the extreme points of P

I

are exactly the rows

of B. 2

1.3. EXAMPLES 15

Theorem 1.17 (Lehman [132]) A clutter is ideal if and only if its

blocker is.

Proof: By Theorem 1.15, it suﬃces to show that if P deﬁned by (1.3)-

(1.4) is integral, then Q deﬁned by (1.5)-(1.6) is also integral.

Let a be an arbitrary extreme point of Q. By (1.5), Ba ≥ 1, i.e.

a

T

x ≥ 1 is satisﬁed by every x such that x

T

is a row of B. Since P

is an integral polyhedron, it follows from Remark 1.16(i) that a

T

x ≥ 1

is satisﬁed by all the extreme points of P . By (1.6), a ≥ 0. Therefore

a

T

x ≥ 1 is satisﬁed by all points in P . Furthermore, a

T

x = 1 for some

x ∈ P . Now, by linear programming duality, we have

1 = min{a

T

x : x ∈ P } = max{λ

T

1 : λ

T

A ≤ a

T

, λ ≥ 0}.

Therefore, by Remark 1.16(ii) applied to Q, a is a 0,1 extreme point of

Q. 2

Exercise 1.18 Let Q

6

denote the 4 × 6 incidence matrix of triangles

versus edges of K

4

. Describe the blocker of C(Q

6

). Is it ideal? Does it

pack? Does it have the MFMC property? Compare with the properties

of Q

6

found in Exercise 1.9.

1.3 Examples

1.3.1 st-Cuts and st-Paths

Consider a digraph (N, A) with s, t ∈ N. Let C be the clutter where

V (C) = A and E(C) is the family of st-paths.

For any arc capacities w ∈ Z

A

+

, the Ford-Fulkerson theorem [84]

states that (1.1) and (1.2) both have optimal solutions that are integral:

(1.1) is the min cut problem and (1.2) is the max ﬂow problem (a ﬂow

y is a multiset of st-paths such that each arc a ∈ A belongs to at most

w

a

st-paths of y. A max ﬂow is a ﬂow containing the maximum number

of st-paths). Using the terminology introduced in Deﬁnition 1.5, the

Ford-Fulkerson theorem states that the clutter C of st-paths has the

MFMC property.

16 CHAPTER 1. CLUTTERS

Theorem 1.19 (Ford-Fulkerson [84]) The clutter C of st-paths has the

MFMC property.

This result implies that C is ideal and therefore the polyhedron

{x ∈ R

A

+

: x(P ) ≥ 1 for all st-paths P }

is integral. Its extreme points are the minimal st-cuts. In the remain-

der, it will be convenient to refer to minimal st-cuts simply as st-cuts.

As a consequence of Lehman’s theorem (Theorem 1.17), the clutter

of st-cuts is also ideal, i.e. the polyhedron

{x ∈ R

A

+

: x(C) ≥ 1 for all st-cuts C}

is integral. So, minimizing a nonnegative linear function over this poly-

hedron solves the shortest st-path problem. We leave it as an exercise

to show that the clutter of st-cuts has the MFMC property.

Exercise 1.20 Show that the clutter of st-cuts packs by using graph

theoretic arguments. Then show that the clutter of st-cuts has the

MFMC property.

In a network, the product of the minimum number of edges in an st-

path by the minimum number of edges in an st-cut is at most equal to

the total number of edges in the network. This width-length inequality

can be generalized to any nonnegative edge lengths `

e

and widths w

e

:

the minimum length of an st-path times the minimum width of an

st-cut is at most equal to the scalar product `

T

w. This width-length

inequality was observed by Moore and Shannon [145] and Duﬃn [78].

A length and a width can be deﬁned for any clutter and its blocker.

Interestingly, Lehman [132] showed that the width-length inequality

can be used as a characterization of idealness.

Theorem 1.21 (Width-length inequality, Lehman [132]) For a clutter

C and its blocker b(C), the following statements are equivalent.

• C and b(C) are ideal;

• min{w(C) : C ∈ E(C)} × min{`(D) : D ∈ E(b(C))} ≤ w

T

` for

all `, w ∈ R

n

+

.

1.3. EXAMPLES 17

Proof: Let A = M(C) and B = M(b(C)) be the blocking pair of 0,1

matrices associated with C and b(C) respectively.

First we show that if C and b(C) are ideal then, for all `, w ∈ R

n

+

,

αβ ≤ w

T

` where α ≡ min{w(C) : C ∈ E(C)} and β ≡ min{`(D) :

D ∈ E(b(C))}.

If α = 0 or β = 0, then this clearly holds.

If α > 0 and β > 0, we can assume w.l.o.g. that α = β = 1

by scaling ` and w. So Aw ≥ 1, i.e. w belongs to the polyhedron

P ≡ {x ≥ 0, Ax ≥ 1}. Therefore w is greater than or equal to a

convex combination of the extreme points of P , which are the rows of

B by Remark 1.16(i) since P is an integral polyhedron. It follows that

w

T

≥ λ

T

B where λ ≥ 0 and

P

i

λ

i

= 1. Similarly, one shows that

`

T

≥ µ

T

A where µ ≥ 0 and

P

i

µ

i

= 1. Since BA

T

≥ J, where J

denotes the matrix of all 1’s, it follows that

w

T

` ≥ λ

T

BA

T

µ ≥ λ

T

Jµ = 1 = αβ

Now we prove the converse. Let C be a nontrivial clutter and let

w be any extreme point of P ≡ {x ≥ 0 : Ax ≥ 1}. Since Aw ≥ 1,

it follows that min{w(C) : C ∈ E(C)} ≥ 1. For any point z in

Q ≡ { z ≥ 0 : Bz ≥ 1}, we also have min{z(D) : D ∈ E(b(C))} ≥ 1.

Using the hypothesis, it follows that w

T

z ≥ 1 is satisﬁed by all points

z in Q. Furthermore, equality holds for at least one z ∈ Q. Now, by

linear programming duality,

1 = min{w

T

z : z ∈ Q} = max{µ

T

1 : µ

T

B ≤ w

T

, µ ≥ 0}.

It follows from Remark 1.16(ii) that w is a 0,1 extreme point of P .

Therefore, C is ideal. By Theorem 1.17, b(C) is also ideal. 2

1.3.2 Two-Commodity Flows

Let G be an undirected graph and let {s

1

, t

1

} and {s

2

, t

2

} be two pairs

of nodes of G with s

1

6= t

1

and s

2

6= t

2

. A two-commodity cut is a

minimal set of edges separating each of the pairs {s

1

, t

1

} and {s

2

, t

2

}.

A two-commodity path is an s

1

t

1

-path or an s

2

t

2

-path.

For any edge capacities w ∈ R

E(G)

+

, Hu [126] showed that a minimum

capacity two-commodity cut can be obtained by solving the linear pro-

18 CHAPTER 1. CLUTTERS

gram (1.1), where M is the incidence matrix of two-commodity paths

versus edges.

Theorem 1.22 (Hu [126]) The clutter of two-commodity paths is ideal.

This theorem states that the polyhedron

x(P ) ≥ 1 for all two-commodity paths P

x

e

≥ 0 for all e ∈ E(G)

is integral. Using Lehman’s theorem (Theorem 1.17), the clutter of

two-commodity cuts is ideal.

The clutters of two-commodity paths and of two-commodity cuts

do not pack, but both have the 1/2-MFMC property (Hu [126] and

Seymour [184], respectively).

For more than two commodities, the clutter of multicommodity

paths is not always ideal but conditions on the graph G and the source-

sink pairs {s

1

, t

1

}, . . . , {s

k

, t

k

} have been obtained under which it is

ideal. See Papernov [159], Okamura and Seymour [151], Lomonosov

[135] and Frank [85] for examples.

1.3.3 r-Cuts and r-Arborescences

Consider a connected digraph (N, A) with r ∈ N and nonnegative

integer arc lengths `

a

for a ∈ A. An r-arborescence is a minimal arc

set that contains an r v-dipath for every v ∈ N. It follows that an

r-arborescence contains |N| − 1 arcs forming a spanning tree and each

node of N −{r} is entered by exactly one arc. The minimal transversals

of the clutter of r-arborescences are called r-cuts.

Theorem 1.23 (Fulkerson [91]) The clutter of r-cuts has the MFMC

property, i.e. the minimum length of an r-arborescence is equal to the

maximum number of r-cuts such that each a ∈ A is contained in at

most `

a

of them.

In other words, both sides of the linear programming duality relation

min {

P

a∈A

`

a

x

a

: x(C) ≥ 1 for all r-cuts C

1.3. EXAMPLES 19

x

a

≥ 0}

= max {

P

C r -cut

y

C

:

X

C3a

y

C

≤ `

a

for all a ∈ A

y

C

≥ 0}

have integral optimal solutions x and y.

Theorem 1.24 (Edmonds [79]) The clutter of r-arborescences has the

MFMC property.

In other words, both sides of the linear programming duality relation

min {

P

a∈A

`

a

x

a

: x(B) ≥ 1 for all r-arborescences B

x

a

≥ 0}

= max {

P

B r-arborescence

y

B

:

X

B3a

y

B

≤ `

a

for all a ∈ A

y

B

≥ 0}

have integral optimal solutions x and y. The fact that the minimization

problem has an integral optimal solution

x

follows from Theorems 1.17

and 1.23, but the fact that the dual also does cannot be deduced from

these theorems.

1.3.4 Dicuts and Dijoins

Let (N, A) be a digraph. An arc set C ⊆ A is called a dicut if there

exists a nonempty node set S ⊂ N such that (S, N − S) = C and

(N − S, S) = ∅ where (S

1

, S

2

) denotes the set of arcs ij with i ∈ S

1

and

j ∈ S

2

, and C is minimal with this property. A dijoin is a minimal arc

set intersecting every dicut.

Theorem 1.25 (Lucchesi-Younger [141]) The clutter of dicuts has the

MFMC property.

By Lehman’s theorem, it follows that the clutter of dijoins is ideal.

However, Schrijver [171] showed by an example that the clutter of di-

joins does not always have the MFMC property. Two additional exam-

ples are given in [63].

Conjecture 1.26 (Woodall [210]) The clutter of dijoins packs.

20 CHAPTER 1. CLUTTERS

1.3.5 T -Cuts and T -Joins

Let G be an undirected graph and T ⊆ V (G) a node set of even cardi-

nality. Such a pair (G, T ) is called a graft. An edge set J ⊆ E(G) is a

T -join if it induces an acyclic graph where the odd nodes coincide with

T . The minimal transversals of the clutter of T -joins are called T -cuts.

For disjoint node sets S

1

, S

2

, let (S

1

, S

2

) denote the set of edges with

one endnode in S

1

and the other in S

2

. T -cuts are edge sets of the form

(S, V (G) − S) where |T ∩ S| is odd.

When T = {s, t}, the T -joins are the st-paths of G and the T -cuts

are the st-cuts.

When T = V (G), the T -joins of size |V (G)|/2 are the perfect match-

ings of G.

Theorem 1.27 (Edmonds-Johnson [82]) The clutter of T -cuts is ideal.

Hence, the polyhedron

x(C) ≥ 1 for all T -cuts C

x

e

≥ 0 for all e ∈ E(G)

is integral.

The Edmonds-Johnson theorem together with Lehman’s theorem

(Theorem 1.17) implies that the clutter of T -joins is also ideal. Hence

the polyhedron

x(J) ≥ 1 for all T -joins J

x

e

≥ 0 for all e ∈ E(G)

is integral.

The clutter of T-cuts does not pack, but it has the 1/2-MFMC

property (Seymour [188]). The clutter of T -joins does not have the

1/2-MFMC property (there is an example requiring multiplication by

4 to get an integer dual), but it may have the 1/4-MFMC property

(open problem). Seymour [185] showed that the 1/4-MFMC property

would follow from a conjecture of Fulkerson, namely Conjecture 1.32

mentioned later in this chapter. Another intriguing conjecture is the

following. In a graph G, a postman set is a T -join where T coincides

with the odd degree nodes of G.

1.3. EXAMPLES 21

Conjecture 1.28 (Conforti and Johnson [55]) The clutter of postman

sets packs in graphs not contractible to the Petersen graph.

If true, this implies the four color theorem (see Exercise 1.31)!

We discuss T -cuts and T -joins in greater detail in Chapter 2.

1.3.6 Odd Cycles in Graphs

Let G be an undirected graph and C the clutter of odd cycles, i.e.

V (C) = E(G) and E(C) is the family of odd cycles in G (viewed as

edge sets). Seymour [183] characterized exactly when the clutter of

odd cycles has the MFMC property and Guenin [110] characterized

exactly when it is ideal. These results are described in Chapter 5 in

the more general context of signed graphs.

1.3.7 Edge Coloring of Graphs

In a simple graph G, consider the clutter C whose vertices are the

maximal matchings of G and whose edges are indexed by the edges of

G, with S

e

∈ E(C) being the set of maximal matchings that contain

edge e. The problem

χ

0

(G) = min{1x : M(C)x ≥ 1, x ≥ 0 integral} (1.7)

is that of ﬁnding the edge-chromatic number of G. The dual problem

max{y1 : yM(C) ≤ 1, y ≥ 0 integral}

is that of ﬁnding a maximum number of edges in G no two of which

belong to the same matching. This equals the maximum degree ∆(G),

except in the trivial case where ∆(G) = 2 and G contains a triangle.

By Vizing’s theorem [205], χ

0

(G) and ∆(G) diﬀer by at most one. It

is NP-complete to decide whether χ

0

(G) = ∆(G) or ∆(G) + 1 (Holyer

[123]). For the Petersen graph, χ

0

(G) = 4 > ∆(G) = 3. The following

conjecture of Tutte [203] was proved recently.

Theorem 1.29 (Robertson, Sanders, Seymour, Thomas [165], [169],

[170]) Every 2-connected cubic graph that is not contractible to the Pe-

tersen graph is 3-edge-colorable.

22 CHAPTER 1. CLUTTERS

The Petersen graph is nonplanar. So, by specializing the above

theorem to planar graphs, we get the following corollary.

Theorem 1.30 (Appel and Haken [2]. See also [164]) Every 2-connected

cubic planar graph is 3-edge-colorable.

This is equivalent to the famous 4-color theorem for planar maps,

as shown by Tait [190] over a century ago. In fact, Tait mistakenly

believed that he had settled the 4-color conjecture b ecause he thought

that every 2-connected cubic planar graph is Hamiltonian (which would

imply Theorem 1.30). Tutte [200] found a counterexample over sixty

years later!

Exercise 1.31 Show that Conjecture 1.28 implies Theorem 1.29 and

therefore the 4-color theorem.

Let χ

0

2

(G) = min{1x : M(C)x ≥ 1, x ≥ 0, 2x integral}. (1.8)

For the Petersen graph, it is easy to check that χ

0

2

(G) = 3.

Conjecture 1.32 (Fulkerson [89]) For every 2-connected cubic graph,

χ

0

2

(G) = 3.

Seymour [185] showed that Fulkerson’s conjecture holds if one re-

laxes the condition x ≥ 0 in (1.8).

1.3.8 Feedback Vertex Set

Given a digraph D = (V, A), a vertex set S ⊆ V is called a feedback

vertex set if V (C) ∩ S 6= ∅ for every directed cycle C. Let C denote the

clutter with V (C) = V and E(C) the family of minimal directed cycles

(viewed as sets of vertices). Then a feedback vertex set is a transversal

of C. Guenin and Thomas [113] characterize the digraphs D for which

C packs for every subdigraph H of D. Cai, Deng and Zang [23] and [24]

consider the feedback vertex set problem in tournaments and bipartite

tournaments respectively. A tournament is an orientation of a complete

1.4. DELETION, CONTRACTION AND MINOR 23

graph. A bipartite tournament is an orientation of a complete bipartite

graph. In the ﬁrst case, C consists of the directed triangles in D and,

in the second case, C consists of the directed squares (check this). Cai,

Deng and Zang [23], [24] characterize the tournaments and bipartite

tournaments D for which C has the MFMC property.

Recently, Ding and Zang [77] solved a similar problem on undirected

graphs. They characterized in terms of forbidden subgraphs the graphs

G for which the clutter C of cycles has the MFMC property. Here

V (C) ≡ V (G) and E(C) is the family of cycles of G viewed as sets of

vertices.

1.4 Deletion, Contraction and Minor

Let C be a clutter. For j ∈ V (C), the contraction C/j and deletion

C \ j are clutters deﬁned as follows: both have V (C) − {j} as vertex

set, E(C/j) is the set of minimal members of {S − {j} : S ∈ E(C)}

and E(C \ j) = {S ∈ E(C) : j 6∈ S}.

Exercise 1.33 Given an undirected graph G, consider the clutter C

whose vertices are the edges of G and whose edges are the cycles of

G (viewed as edge sets). Describe C\j and C/j. Relate to the graph-

theoretic notions of edge deletion and edge contraction in G.

Contractions and deletions of distinct vertices of C can be performed

sequentially, and it is easy to show that the result does not depend on

the order.

Proposition 1.34 For a clutter C and distinct vertices j

1

, j

2

,

(i) (C\j

1

)\j

2

= (C\j

2

)\j

1

(ii) (C/j

1

)/j

2

= (C/j

2

)/j

1

(iii) (C\j

1

)/j

2

= (C/j

2

)\j

1

Proof: Use the deﬁnitions of contraction and deletion! 2

24 CHAPTER 1. CLUTTERS

Contracting j ∈ V (C) corresponds to setting x

j

= 0 in the set

covering constraints Mx ≥ 1 of (1.1) since column j is removed from

M as well as the resulting dominating rows. Deleting j corresponds to

setting x

j

= 1 since column j is removed from M as well as all the rows

with a 1 in column j.

A clutter D obtained from C by a sequence of deletions and con-

tractions is a minor of C. For disjoint subsets V

1

and V

2

of V (C), we let

C/V

1

\V

2

denote the minor obtained from C by contracting all vertices

in V

1

and deleting all vertices in V

2

. If V

1

6= ∅ or V

2

6= ∅, the minor is

proper.

Proposition 1.35 For a clutter C and U ⊂ V (C),

(i) b(C\U) = b(C)/U

(ii) b(C/U) = b(C)\U

Proof: Use the deﬁnitions of contraction, deletion and blocker! 2

Proposition 1.36 (Seymour [183]) If a clutter C has the MFMC prop-

erty, then so do all its minors.

Proof: Trivial clutters have the MFMC property. So let C

0

=

C/V

1

\V

2

be a nontrivial minor of C. It suﬃces to show that, for every

w

0

∈ Z

V ( C

0

)

+

,

max{y1 : y ≥ 0, yM (C

0

) ≤ w

0

} (1.9)

has an integral optimal solution. Since b(C

0

) is nontrivial (Exercise 1.12),

τ = min{w

0

(B

0

) : B

0

∈ E(b(C

0

))} is well deﬁned. Deﬁne w by

w

j

= w

0

j

for j ∈ V (C

0

)

w

j

= τ for j ∈ V

1

w

j

= 0 for j ∈ V

2

.

If B is an edge of b(C) and B ∩V

1

6= ∅, then w(B) ≥ τ . On the other

hand, if B ∩ V

1

= ∅, then, by Proposition 1.35, b(C

0

) = b(C) \ V

1

/V

2

and therefore B contains an edge B

0

of b(C

0

) and w(B) ≥ w

0

(B

0

) ≥ τ.

1.4. DELETION, CONTRACTION AND MINOR 25

Furthermore, there exists B with w(B) = τ. Since C has the MFMC

property, it follows that

max{y1 : y ≥ 0, yM (C) ≤ w}

has an integral optimal solution y

∗

with function value τ. y

∗

can be

used to construct a solution y

∗∗

of (1.9) with function value τ as follows.

Start with y

∗∗

= 0. Let A be an edge of C such that y

∗

A

> 0. The fact

that w

j

= 0 for j ∈ V

2

implies that A ∩ V

2

= ∅. Hence A contains an

edge A

0

of C

0

. Increase y

∗∗

A

0

by y

∗

A

. Repeat for each A such that y

∗

A

> 0.

2

Similarly, one may prove the following result.

Proposition 1.37 If a clutter is ideal, then so are all its minors.

Exercise 1.38 Prove Proposition 1.37.

Corollary 1.39 Let M be a 0,1 matrix. The following are equivalent.

• The polyhedron {x ≥ 0 : Mx ≥ 1} is integral.

• The polytope {0 ≤ x ≤ 1 : Mx ≥ 1} is integral.

Propositions 1.36 and 1.37 suggest the following concepts.

Deﬁnition 1.40 A clutter is minimally non MFMC if it does not have

the MFMC property but all its proper minors do.

A clutter is minimally nonideal if it is not ideal but all its proper

minors are.

A clutter is minimally nonpacking if it does not pack but all its

proper minors do.

Properties of these clutters are investigated in Chapters 4 and 5.

26 CHAPTER 1. CLUTTERS

Chapter 2

T -Cuts and T -Joins

Consider a connected graph G with nonnegative edge weights w

e

, for

e ∈ E(G). The Chinese Postman Problem consists in ﬁnding a min-

imum weight closed walk going through each edge at least once (the

edges of the graph represent streets where mail must be delivered and

w

e

is the length of the street). Equivalently, the postman must ﬁnd a

minimum weight set of edges J ⊆ E(G) such that J ∪ E(G) induces an

Eulerian graph, i.e. J induces a graph the odd degree nodes of which

coincide with the odd degree nodes of G. Since w ≥ 0, we can assume

w.l.o.g. that J is acyclic. Such an edge set J is called a postman set .

The problem is generalized as follows. Let G be a graph and T a

node set of G of even cardinality. An edge set J of G is called a T -join

if it induces an acyclic graph the odd degree nodes of which coincide

with T . For disjoint node sets S

1

, S

2

, let (S

1

, S

2

) denote the set of edges

with one endnode in S

1

and the other in S

2

. A T -cut is a minimal edge

set of the form (S, V (G) − S) where S is a set of nodes with |T ∩ S|

odd. Clearly every T -cut intersects every T -join.

Edmonds and Johnson [82] considered the problem of ﬁnding a min-

imum weight T -join. One way to solve this problem is to reduce it to

the perfect matching problem in a complete graph K

p

, where p = |T |.

Namely, compute the lengths of shortest paths in G between all pairs of

nodes in T , use these values as edge weights in K

p

and ﬁnd a minimum

weight perfect matching in K

p

. The union of the corresponding paths

in G is a minimum weight T -join. There is another way to solve the

minimum weight T -join problem: Edmonds and Johnson gave a direct

27

28 CHAPTER 2. T -CUTS AND T -JOINS

primal-dual algorithm and, as a by-product, obtained that the clutter

of T -cuts is ideal.

Theorem 2.1 (Edmonds-Johnson [82])

The polyhedron

x(C) ≥ 1 for all T − cuts C (2.1)

x

e

≥ 0 for all e ∈ E(G). (2.2)

is integral.

In the next section, we give a non-algorithmic proof of this theorem

suggested by Pulleyblank [162].

The clutter of T -cuts does not pack, but Seymour [188] showed that

it has the 1/2-MFMC property. In Section 2.2, we prove Seymour’s

result, following a short argument of Seb¨o [174] and Conforti [39].

As we have seen in Chapter 1, clutters come in pairs: To each

clutter C, we can associate its blocker b(C) whose edges are the minimal

transversals of C. Lehman [132] showed that C is ideal if and only if

b(C) is ideal (Theorem 1.17). The Edmonds-Johnson theorem together

with Lehman’s theorem implies that the clutter of T -joins is also ideal,

i.e. the polyhedron

x(J) ≥ 1 for all T − joins J

x

e

≥ 0 for all e ∈ E(G).

is integral. The clutter of T -joins does not pack in general. In Sec-

tion 2.3, we present two special cases where it does.

2.1 Proof of the Edmonds-Johnson Theo-

rem

First, we prove the following lemma. For v ∈ V (G), let δ(v) denote the

set of edges incident with v. A star is a tree where one node is adjacent

to all the other nodes.

2.1. PROOF OF THE EDMONDS-JOHNSON THEOREM 29

Lemma 2.2 Let ˜x be an extreme point of the polyhedron

x(δ(v)) ≥ 1 for all v ∈ T (2.3)

x

e

≥ 0 for all e ∈ E(G). (2.4)

The connected components of the graph

˜

G induced by the edges such

that ˜x

e

> 0 are either

(i) odd cycles with nodes in T and edges ˜x

e

= 1/2, or

(ii) stars with nodes in T , except possibly the center, and edges ˜x

e

= 1.

Proof: Every connected component C of

˜

G is either a tree or con-

tains a unique cycle, since the number of edges in C is at most the

number of inequalities (2.3) that hold with equality.

Assume ﬁrst that C contains a unique cycle. Then (2.3) holds with

equality for all nodes of C, which are therefore in T . Now C is a cycle

since, otherwise, C has a pendant edge e with ˜x

e

= 1 and therefore

C is disconnected, a contradiction. If C is an even cycle, then by

alternately increasing and decreasing ˜x around the cycle by a small ²

(−² respectively), ˜x can be written as a convex combination of two

points satisfying (2.3) and (2.4). So (i) must hold.

Assume now that C is a tree. Then (2.3) holds with equality for at

least |V (C)| − 1 nodes of C. In particular, it holds with equality for at

least one node of degree one. Since C is connected, this implies that C

is a star and (ii) holds. 2

Proof Theorem 2.1: In order to prove the theorem, it suﬃces to

show that every extreme point ˜x of the polyhedron (2.1)–(2.2) is the

incidence vector of a T -join. We proceed by induction on the number

of nodes of G.

Suppose ﬁrst that ˜x is an extreme point of the polyhedron (2.3)–

(2.4). Consider a connected component of the graph

˜

G induced by the

edges such that ˜x

e

> 0 and let S be its node set. Since ˜x(S, V (G)−S) =

0, it follows from (2.1) that S contains an even number of nodes of T .

By Lemma 2.2,

˜

G contains no odd cycle, showing that ˜x is an integral

vector. Furthermore, ˜x is the incidence vector of a T -join since, by

Lemma 2.2 again, the component of

˜

G induced by S is a star and

30 CHAPTER 2. T -CUTS AND T -JOINS

|S ∩ T | even implies that the center is in T if and only if the star has

an odd number of edges.

Assume now that ˜x is not an extreme point of the polyhedron (2.3)–

(2.4). Then there is some T -cut C = (V

1

, V

2

) with |V

1

| ≥ 2 and |V

2

| ≥ 2

such that

˜x(C) = 1 .

Let G

1

= (V

1

∪ {v

2

}, E

1

) be the graph obtained from G by contracting

V

2

to a single node v

2

. Similarly, G

2

= (V

2

∪ {v

1

}, E

2

) is the graph

obtained from G by contracting V

1

to a single node v

1

. The new nodes

v

1

, v

2

belong to T . For i = 1, 2, let ˜x

i

be the restriction of ˜x to E

i

.

Since every T -cut of G

i

is also a T -cut of G, it follows by induction

that ˜x

i

is greater than or equal to a convex combination of incidence

vectors of T -joins of G

i

. Let T

i

be this set of T -joins. Each T -join in

T

i

has exactly one edge incident with v

i

. Since ˜x

1

and ˜x

2

coincide on

the edges of C, it follows that the T -joins of T

1

can be combined with

those of T

2

to form T -joins of G and that ˜x is greater than or equal to

a convex combination of incidence vectors of T -joins of G. Since ˜x is

an extreme point, it is the incidence vector of a T -join. 2

We have just proved that the clutter of T -cuts is ideal. It does

not have the MFMC property in general graphs. However Seymour

proved that it does in bipartite graphs. Seymour also showed that, in a

general graph, if the edge weights w

e

are integral and their sum is even

in every cycle, then the dual variables can be chosen to be integral in

an optimum solution. We prove these results in the next section.

2.2 Packing T -Cuts

2.2.1 Theorems of Seymour and Lov´asz

The purpose of this section is to prove the following theorems.

Theorem 2.3 (Seymour [188]) In a bipartite graph, the clutter of T -

cuts packs, i.e. the minimum cardinality of a T -join equals the maxi-

mum number of disjoint T -cuts.

2.2. PACKING T -CUTS 31

Theorem 2.4 (Lov´asz [138]) In a graph, the clutter of T -cuts has the

1/2-MFMC property.

Exercise 2.5 Consider the complete graph K

4

on four nodes and let

|T | = 4. Show that there are exactly seven T -joins. Describe the T -cuts.

Do you see any relation with Q

6

(see Exercise 1.18 for the deﬁnition of

Q

6

)?

We give a proof of Theorem 2.3 based on ideas of Seb¨o [174] and

Conforti [39].

Given a graph G and a T -join J, let G

J

be the weighted graph

obtained by assigning weights −1 to the edges of J and +1 to all the

other edges.

Remark 2.6

(i) J is a minimum T -join if and only if G

J

has no negative cycle.

(ii) If J is a minimum T -join and C is a 0-weight cycle in G

J

, then

J∆C is a minimum T -join.

Proof of Theorem 2.3: The result is trivial for |T | = 0, so assume

|T | ≥ 2. The proof is by induction on the number of nodes of the

bipartite graph G. Let J be a minimum T-join chosen so that its

longest path Q ⊆ J is longest possible among all minimum T -joins.

Since J is acyclic, the endnodes of Q have degree 1 in J, so both are

in T . Let u be an endnode of Q and let x be the neighbor of u in Q.

Since J is minimum, G

J

has no negative cycle. We claim that every 0-

weight cycle of G

J

that contains node u also contains edge ux. Suppose

otherwise. If C contains some other node of Q, then Q ∪ C contains

a negative cycle (check this!), a contradiction to J being a minimum

T -join. If u is the unique node of Q in C, then J∆C is a minimum

T -join (by Remark 2.6) with a longer path than Q, a contradiction to

our choice of J. So the claim holds and, since G is bipartite,

(*) every cycle that contains node u but not edge ux has weight at

least 2 in G

J

.

Let U be the node set comprising u and its neighbors in G. Let

G

∗

be the bipartite graph obtained from G by contracting U into a

single node u

∗

. If |U ∩ T | is even, set T

∗

= T \ U and if |U ∩ T |, set

32 CHAPTER 2. T -CUTS AND T -JOINS

T

∗

= (T \ U) ∪ {u

∗

}. Let J

∗

= J ∩ E(G

∗

). Then J

∗

is a T

∗

-join of G

∗

and, by (*) the graph G

∗

J

∗

has no negative cycle. So J

∗

is a minimum

T

∗

-join of G

∗

by Remark 2.6.

Now, by induction, G

∗

has |J

∗

| disjoint T

∗

-cuts. Since δ(v) is a

T -cut of G disjoint from them, G has |J

∗

| + 1 = |J| disjoint T -cuts.

Since G can have at most |J| disjoint T -cuts, the theorem holds. 2

This result implies the following theorem of Lov´asz [138].

Theorem 2.7 In a graph G, the minimum cardinality of a T -join is

equal to one half of the maximum cardinality of a set of T -cuts such

that no edge belongs to more than two T -cuts in the set.

Proof: Subdivide each edge of G by a new node and apply Theo-

rem 2.3. 2

Exercise 2.8 Prove Theorem 2.4 using Theorem 2.7.

Another useful consequence of Theorem 2.3 is the following result

of Seymour [188].

Deﬁnition 2.9 Let G be a graph and let w ∈ Z

E(G)

+

. The even cycle

property holds if, in every cycle of G, the sum of the weights is even.

Theorem 2.10 (Seymour [188]) Assume graph G and weight vector

w ∈ Z

E(G)

+

satisfy the even cycle property. Then the minimum weight

of a T -join equals the maximum number of T -cuts such that no edge e

belongs to more than w

e

of these T -cuts.

Proof: If w

e

> 0, subdivide edge e into w

e

edges, each of weight

1. If w

e

= 0, contract edge e and consider the resulting node as being

in T if exactly one of the endnodes of e in G belongs to T . Now the

theorem follows from Theorem 2.3. 2

The next result follows from a diﬃcult theorem of Seymour on bi-

nary clutters with the MFMC property (Theorem 5.30). A graph G

can be T -contracted to K

4

if its node set V can be partitioned into

{V

1

, V

2

, V

3

, V

4

} so that each V

i

induces a connected graph containing an

odd number of nodes of T and, for each i 6= j, there is an edge with

endnodes in V

i

and V

j

.

2.2. PACKING T -CUTS 33

Theorem 2.11 In a graph that cannot be T -contracted to K

4

, the clut-

ter of T -cuts has the MFMC property.

2.2.2 More Min Max Results

Theorem 2.4 can be strengthened as follows. For any node set X,

denote by q

T

(X) the number of connected components of G \ X that

contain an odd number of nodes of T .

Theorem 2.12 (Frank, Seb¨o, Tardos [86]) In a graph G, the mini-

mum cardinality of a T -join is equal to

1

2

max{

P

i

q

T

(V

i

)}, where the

maximum is taken over all partitions {V

1

, . . . , V

`

} of V .

This theorem can be used to prove Tutte’s theorem on perfect

matchings.

Theorem 2.13 (Tutte [201]) A graph contains no perfect matching if

and only if there exists a node set X such that G \ X contains at least

|X| + 1 components of odd cardinality.

Proof: (Frank and Szigeti [87]) Apply Theorem 2.12 with the choice

T = V . Note that in this case, q

T

(X) is the number of components of

odd cardinality in G \ X. If there is no perfect matching, the minimum

cardinality of a T -join is larger than

1

2

|V |. By Theorem 2.12, there is

a partition {V

1

, . . . , V

`

} of V such that

1

2

P

i

q

T

(V

i

) >

1

2

|V |. Therefore,

there must be a subscript i such that q

T

(V

i

) > |V

i

|, that is, the number

of components of G \ V

i

with odd cardinality is larger than |V

i

|, as

required. 2

Seb¨o proved yet another min-max theorem concerning T -joins. A

multicut is an edge set whose removal disconnects G into two or more

connected components. If each of these connected components contains

an odd number of nodes of T , the multicut is called a T -border. Clearly,

the number k of connected