integer and combinatorial optimization: perfect...

Integer and Combinatorial Optimization:Perfect Graphs

John E. Mitchell

Department of Mathematical SciencesRPI, Troy, NY 12180 USA

March 2019

Mitchell Perfect Graphs 1 / 14

Perfect graphs

See Nemhauser and Wolsey, section III.1.5, for more information.

DefinitionA graph G = (V ,E) is perfect if �(G) = !(G) for every inducedsubgraph G, where

�(G) = chromatic number of G= least number of colors needed to color the vertices

so no two adjacent vertices have the same color

!(G) = size of maximum clique of G

Note that every graph has �(G) � !(G), since every vertex in thelargest clique requires a different color.


"⇐±÷..G

r a

X CGI(3w ( a ) = 3

Examples

Bipartite graphs

ExampleBipartite graphs are perfect:

1

2

3

4

5

�(G) = !(G) = 2


Examples

Complements of bipartite graphsExampleComplements of bipartite graphs are perfect:

1

2

3

4

5

�(G) = !(G) = 3


G . I

÷ . . ÷⇐@1/61=161--2. x ( f )ea t#4

Examples

Odd holes

ExampleOdd holes are not perfect:

1

2

3

4

5

67

�(G) = 3, !(G) = 2


Examples

Odd anitholesExampleOdd antiholes are not perfect:

1

2

3

4

5

67

�(G) = 4, !(G) = 3Mitchell Perfect Graphs 6 / 14

B

R B

R G

4191=4 off.-z? G

Complement•

,i . o f f1 . 1 . .

alsoa cycleoflength 5 .

Packing LP

Complements of perfect graphs

A graph is perfect if and only if its complement is perfect.

A clique in a graph corresponds to a node packing in its complement.

A coloring in a graph corresponds to a clique cover in its complement;that is, a collection of cliques so that each vertex is in (at least) oneclique.


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"÷::÷÷÷÷÷÷÷÷÷÷÷÷O'

''÷"

Packing LP

A packing LP

Consider the packing linear program:

maxx2Rn cT xsubject to Ax e

x � 0

where e is the vector of ones, and every entry in A is either 0 or 1. Weconstruct a graph:

Each column of A gives a node in a graph.Each row of A gives a clique in the graph. So if aij = aik = 1 thenvertices j and k are adjacent.


Packing LP

A packing LP

Consider the packing linear program:

maxx2Rn cT xsubject to Ax e

x � 0

where e is the vector of ones, and every entry in A is either 0 or 1. Weconstruct a graph:



{Equivalentt o

g ¥ a node packingproblem.

Packing LP

Constructing a graph from a matrix


For example:

A =

2

664

1 1 1 0 01 0 0 0 10 0 1 1 00 0 0 1 1

3

775

r s t u vr

st

u

v�(G) = !(G) = 3�(G) = !(G) = 2


Packing LP

Constructing a graph from a matrix


For example:

A =

2

664

1 1 1 0 01 0 0 0 10 0 1 1 00 0 0 1 1

3

775

r s t u vr

st

u

v�(G) = !(G) = 3�(G) = !(G) = 2


Axel,!}xrtxstxc.clx , t X v E la -t x u E lx u t x u s I

⑥ Complement:

✓ t o09/40 £ 100µ @ @

parking: Eau}0151=2 =X(E)

cliquecover:clique: {s,a }

{ n s . t } ,E u n }

Packing LP

Node packings

Binary solutions to the packing LP correspond to node packings in theoriginal graph and to cliques in the complement of the graph:

in a clique C in the complement, none of the vertices are adjacent inthe original graph, so there is no clique in the original graph thatcontains two of the vertices in C.


Packing LP

Clique coversBinary dual solutions to the packing LP with c = e correspond tocoverings by cliques in the original graph and to colorings in thecomplement of the graph.

The dual problem is

minx2Rm eT ysubject to AT y � e

y � 0

which is a covering problem.

Every vertex must be covered by a color.

If two vertices are not adjacent they must have different colors; thiscorresponds to the two columns not appearing in the same clique inthe original graph.


* A

a:c::::"

Packing LP

Two theorems

Theorem (Chvatal)The LP has an integral optimal solution for every nonnegative integralvector c if and only if the underlying graph is perfect.

Theorem(Perfect graph conjecture of Berge (1961). Proved byChudnovsky, Robertson, Seymour, and Thomas (2003).) A graph isperfect if and only if it contains no odd holes and no odd antiholes.


Packing LP

The Petersen graphExample

r

s

t

u

v

w

p

q

k

l

�(G) = 5. E.g. clique cover {r ,w}, {s, p}, {t , q}, {u, k}, {v , l}

!(G) = 4. E.g. packing {r , t , l , k}


0 0 00 0

@

6 ¥t olook for packing'.Have 2 cycle, of lengle 5 .

Su a t m o s t 4 verticesi n a packing.

Packing LP

Another exampleExampleA graph that is not perfect because an induced subgraph is not perfect:

r

st

u

vw p

q k

Here we have �(G) = !(G) = 3. However, if we restrict attention to thesubset {r , s, t , u, v} of the vertices then this subgraph is an odd hole,so this is not perfect. In the packing LP, we get a binary solution withc = e, but not with c = (1, 1, 1, 1, 1, 0, 0, 0, 0), with the “ones”corresponding to vertices {r , s, t , u, v}: set xr = xs = xt = xu = xv = 1

2 .Mitchell Perfect Graphs 14 / 14

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