Perfect Graphs

1
Perfect Graphs

• Lecture 23 Apr 17

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Hard Optimization Problems

• Independent set
• Clique
• Colouring
• Clique cover

coding
scheduling
Hard to approximate within a factor of
3
Graph Product
Given G1(V1,E1) and G2(V2,E2), their product
G1xG2 is the graph whose vertex set is V1xV2 and
the edge set is ((u1,v1),(u2,v2)) u1u2
and (v1,v2) in E2 or v1v2 and (u1,u2) in
E1 or (u1,u2) in E1 and (v1,v2) in E2.
Claim There is a clique of size k in G if and
only if there is a clique of size k2 in GxG.
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Constant Factor Hardness
Assuming clique is hard to approximate within a
factor of (1?), then it is also hard to
approximate within any constant factor.
Idea take graph product.
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Perfect Graph
Graphs in which these hard problems are nice.

• Independent set
• Clique
• Colouring
• Clique cover

Easy equalities
Easy inequalities
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Perfect Graph
In what graphs does the equalitiy
holds?
A graph is perfect if for every induced subgraph
H of G,
A graph is co-perfect if for every induced
subgraph H of G,
What graphs are perfect?
Bipartite graphs
What graphs are co-perfect?
Bipartite graphs
What graphs are not perfect?
Odd cycles
What graphs are not co-perfect?
Odd cycles
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Line Graph of Bipartite Graph
Vertices correspond to edges of a bipartite
graph, and two vertices have an edge if and only
if the corresponding edges share an endpoint.
Line graph of bipartite graphs are perfect.
Line graph of bipartite graphs are co-perfect.
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Interval Graph
Vertices correspond to intervals, and two
vertices have an edge if and only if the
corresponding intervals overlap.
Interval graphs are perfect.
Interval graphs are co-perfect.
Many applications.
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Chordal Graph
Also known as triangulated graphs.

• A graph is chordal if every cycle of length 4
  has a chord.
• A graph is chordal if it is the intersection
  graph of subtrees of a tree.
• A graph is chordal if it has an ordering such
  that for each vertex
• the neighbours in front form a clique.

Chordal graphs are perfect.
Chordal graphs are co-perfect.
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Conjectures
Perfect graph conjecture A graph is perfect if
and only if it is co-perfect.
Lovasz 1970
Strong perfect graph conjecture Berge 1960 A
graph is perfect if and only if it does not
contain odd cycles and the complement of odd
cycles as induced subgraphs.
Chudnovsky, Robinson, Seymour, Thomas 2003
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LP-Perfect
In which graphs do the LP always have integral
solutions for independent set?
This LP is integral only if the input graph is
bipartite.
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LP-Perfect
In which graphs do the LP always have integral
solutions for independent set?
for each clique C
No known polynomial time separation oracle.
A graph is LP-perfect if and only if the above
linear program is integral.
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Perfect Graph Theorem

• The following are equivalent
• A graph is perfect.
• A graph is LP-perfect.
• A graph is co-perfect.

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Duplication Lemma
Let G be a graph and v be a vertex. Let G be
the graph obtained from G by adding a new vertex
v and connecting it to v and the neighbours of
v.
Lemma. If G is perfect, then G is perfect.
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Proof of Duplication Lemma
It is enough to prove
If v is in a maximum clique, then both sides plus
1.
Consider an optimal colouring of G. Let v be
coloured red.
Consider G-Rv, with maximum clique at most
w(G)-1.
Colour G-Rv using w(G)-1 colours.
So colour G using w(G) colours as R-vv is an
independent set.
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Perfect LP-perfect
for each clique C
Compute an optimal solution of the LP. Let qx be
integral.
Duplicate each vertex by qx(v) times to obtain a
graph G.
G is perfect and has clique size exactly q and
total cost at most qLP.
Decompose G into q independent set, one must
have cost at most LP.
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LP-perfect co-Perfect
for each clique C
for each vertex v
Consider a clique C with positive value in an
optimal dual solution.
This clique C must intersect every maximum
independent set.
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co-Perfect Perfect
Take the complement of G and apply perfect
co-perfect!
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Strong Perfect Graph Theorem
http//users.encs.concordia.ca/chvatal/perfect/sp
gt.html
A polynomial time algorithm to recognize perfect
graphs.
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Whats Next
Shannon coding Lovasz Theta function Solve the
clique LP using SDP Colouring 3-colourable graphs