Advanced Topics in Graph Algorithms
Advanced Topics in Graph Algorithms
Advanced Topics in Graph Algorithms
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<strong>Advanced</strong> <strong>Topics</strong> <strong>in</strong> <strong>Graph</strong><br />
<strong>Algorithms</strong><br />
Fedor V. Fom<strong>in</strong><br />
Paderborn 2002<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Lecture 1<br />
What is this course about?<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
What this course is about?<br />
• <strong>Graph</strong>s<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
What this course is about?<br />
• <strong>Graph</strong>s<br />
• Problems on graphs<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
What this course is about?<br />
• <strong>Graph</strong>s<br />
• Problems on graphs<br />
• <strong>Algorithms</strong> to solve these problems<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
<strong>Graph</strong>s<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
<strong>Graph</strong>s<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Problems<br />
• <strong>Graph</strong> color<strong>in</strong>g<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Maximum clique size<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Independent set<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
What properties of a graph can<br />
be useful to solve the problem?<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Basics. <strong>Graph</strong> Theory<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Path P
Cycle C<br />
If<br />
is a path, the graph obta<strong>in</strong>ed from P by add<strong>in</strong>g<br />
is cycle
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Some special graphs<br />
• Complete graph K n<br />
• Trees<br />
• Bipartite graph<br />
• Complete bipartite graph<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Clique<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
<strong>Algorithms</strong><br />
• We say that an algorithm for problem<br />
Π runs <strong>in</strong> time O(f(n)) if for some<br />
constant c there exists an implementation of A<br />
which term<strong>in</strong>ates after at most O(f(n))<br />
computational steps for all<br />
<strong>in</strong>stances of size n.<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
L<strong>in</strong>ear time algorithm<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
How to represent a graph<br />
• Adjacency list<br />
• Adjacency matrix<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example of l<strong>in</strong>ear problem:<br />
convert<strong>in</strong>g the adjacency list of a graph<br />
<strong>in</strong>to sorted adjacency lists.<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Buck sort<strong>in</strong>g<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Buck sort<strong>in</strong>g<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Buck sort<strong>in</strong>g<br />
• Buck sort<strong>in</strong>g takes<br />
O(n+d i ) time<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Algorithm<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Theorem<br />
• Algorithm Sort<strong>in</strong>g the Adjacency Lists of<br />
<strong>Graph</strong>s runs <strong>in</strong> O(m+n) time.<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Proof:<br />
{<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Another examples<br />
• Test<strong>in</strong>g planarity, connectivity,<br />
biconnectivity<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
NP complete problems<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Part 1. Perfect graphs<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Clique and IS<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Colour<strong>in</strong>g<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Def<strong>in</strong>ition of perfect graphs<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Examples of perfect graphs<br />
• Bipartite graphs<br />
• Cliques<br />
• Odd cycle --- NOT<br />
• More examples to appear later<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example: Integer programm<strong>in</strong>g<br />
Def<strong>in</strong>e<br />
and<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example: Integer programm<strong>in</strong>g<br />
# of IS conta<strong>in</strong>ig is<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example<br />
The dual<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example<br />
If graph is perfect<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Example: F<strong>in</strong>al remark<br />
For l<strong>in</strong>ear programm<strong>in</strong>g (solution is not<br />
necessarily <strong>in</strong>teger) the value of an optimal<br />
solution of the primal is equal to the value of<br />
an optimal solution of the dual. Stat<strong>in</strong>g graph<br />
problems as l<strong>in</strong>ear programm<strong>in</strong>g problem<br />
leads to an <strong>in</strong>terest<strong>in</strong>g field: fractional graph<br />
theory.<br />
In fractional graph theory it is possible that the<br />
fractional chromatic number is not <strong>in</strong>teger but<br />
it should be equal to fractional clique number!<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Chapter I. CHORDAL GRAPHS<br />
CHORDAL GRAPHS PLAY VERY<br />
IMPORTANT ROLE IN OUR COURSE.<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Def<strong>in</strong>ition of chordal graph<br />
• A chord of a cycle C is an edge not <strong>in</strong> C that<br />
has endpo<strong>in</strong>ts <strong>in</strong> C<br />
• A chordless cycle <strong>in</strong> G is a cycle of length<br />
more than three that has no chord.<br />
• A graph G is chordal if it does not conta<strong>in</strong> a<br />
chordless cycle.<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Simlicial vertex: def<strong>in</strong>ition<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Hereditary property<br />
• Exercise:<br />
Prove that chordality is hereditary<br />
property. (In other words, every<br />
<strong>in</strong>duced subgraph of a chordal graph is<br />
chordal.)<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
We shall prove that<br />
every chordal graph has a simplicial vertex. S<strong>in</strong>ce<br />
chordality is hereditary property, this implies the<br />
follow<strong>in</strong>g naive algorithm of test<strong>in</strong>g chordality of a<br />
graph G:<br />
o While there are simplicial vertices do<br />
o F<strong>in</strong>d a simplicial vertex and remove it<br />
o If the result<strong>in</strong>g graph is empty then G is chordal<br />
o Else G is not chordal<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
Runn<strong>in</strong>g time of the naive algorithm<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
PE-order<strong>in</strong>g def<strong>in</strong>ition<br />
Vertex order<strong>in</strong>g of a graph G=(V,E) is a<br />
PE-order<strong>in</strong>g (perfect elim<strong>in</strong>ation order<strong>in</strong>g)<br />
if for every<br />
vertex v i is simplicial <strong>in</strong> the<br />
graph <strong>in</strong>duced by vertex set<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
M<strong>in</strong>imal separators<br />
• A subset S of vertices of a connected graph G is called an a,bseparator<br />
for non adjacent vertices a and b <strong>in</strong> V(G)\ S if a and b<br />
are <strong>in</strong> different connected component of the subgraph of G<br />
<strong>in</strong>duced by V(G)\ S.<br />
• If no proper subset of an a,b-separator S separates a and b <strong>in</strong><br />
this way, then S is called a m<strong>in</strong>imal a,b-separator .<br />
• A subset S is referred to as a m<strong>in</strong>imal separator, if there exist<br />
non adjacent vertices a and b for which S is a m<strong>in</strong>imal a,bseparator.<br />
• Notice that a m<strong>in</strong>imal separator can be strictly conta<strong>in</strong>ed <strong>in</strong><br />
another m<strong>in</strong>imal separator.<br />
<strong>Advanced</strong> <strong>Topics</strong> <strong>Graph</strong> <strong>Algorithms</strong>
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