Probabilistic Proof and Graph Properties

1
Probabilistic Proof and Graph Properties

• Andrew Huss
• EC 517
• 5/2/08

2
Agenda

• The Probabilistic Method
• Graph Theory Basics
• Some Simple Proofs
• Existence of Independent Sets
• Results

3
The Probabilistic Method

• What is it?
• A tool that can be used to prove the existence of
  objects with desired properties
• Why use it?
• Providing an explicit construction may be
  difficult
• Whats the main idea?
• If property A holds with non-zero probability in
  an appropriately-defined probability space, then
  at least one object with property A must exist in
  reality (or be constructable)

4
The Probabilistic Method (cont)

• Toy example
• If the expected payoff of a lottery ticket is
  10, then the lottery must involve at least one
  ticket worth at least 10
• We could set up this proof as
• Probability Space Uniform distribution over all
  lottery tickets
• Object of Interest Ticket worth at least 10
• Its impossible for everyone to be above average
• Its also impossible for everyone to be below
  average

5
The Probabilistic Method (cont)

• Where have we seen this before?
• Shannons Channel Coding Theorem For any rate
  less than capacity, there exist block channel
  codes that can achieve arbitrarily small Perr
• Proof Outline
• Probability Space Uniform distribution over a
  set of binary codebooks with a fixed rate
• Object of Interest Single codebook with
  arbitrarily small Perr
• With very simple decoding assumptions, we showed
  that the expected Perr over all codes could be
  made arbitrarily small (lt2e)

6
The Probabilistic Method

• Other Uses
• The probabilistic method has been a very
  successful approach in a number of fields
• Most prominently, combinatorics and theoretical
  computer science have benefited
• Method pioneered by Erdos
• Our Focus
• Application to a specific class of problems in
  graph theory

7
Graph Theory

• A Graph G is defined as a set of n vertices (V)
  connected by m edges (E)

2
3
1
4
5
8
Graph Theory (cont)

• A Hypergraph H is defined as a set of n vertices
  (V) connected by m edges (E)

2
3
1
4
5
9
Graph Theory (cont)

• Definitions
• A hypergraph is Uniform if all edges connect the
  same number of vertices
• Note a basic graph in which each edge connects
  two vertices is a 2-Uniform hypergraph
• A Complete graph on n vertices (denoted Cn) is a
  maximally connected graph (all possible edges are
  present)
• Note usually subject to uniformity i.e. a
  Complete 3-Uniform graph contains all possible
  3-edges
• Induced Subgraph on vertex set V
• Remove all vertices not in V
• Remove all edges touching vertices not in V
• Complement of H
• Switch edges and non-edges
• A Clique of H is a subset of vertices of H s.t.
  the induced subgraph is complete
• An Independent Set is a subset of vertices of H
  s.t. the induced subgraph has no edges
• An independent set of H is a clique of the
  complement of H

10
A Simple Proof

• Probability Space All edge 2-colorings
• Object of Interest coloring s.t. no k-clique is
  monochromatic

11
A Simple Proof (cont)

• Let Ai be the event the clique i is monochromatic
• Probability of choosing a coloring s.t. at least
  one k-clique is monochromatic
• Probability of choosing a coloring s.t. no
  k-clique is monochromatic
• The desired probability is strictly gt0 iff

Desired object guaranteed to exist if
12
Independent Sets

• Consider a 2-Uniform hypergraph (graph) G(V,E)
  with n vertices and m edges
• Probability Space All induced subgraphs S
• Object of interest S from which we can
  construct an independent set (twist!)

13
Independent Sets
14
Independent Sets

• So, there exists at least one S that has at least
  n2/4m more vertices than edges
• Select one vertex from each edge and remove it
  from S (thus removing the edge as well)
• The new graph (call it S) has at least n2/4m
  vertices and no edges and is therefore an
  independent set

15
Independent Sets

• Notice that this proof has also provided a
  easy-to-implement algorithm for finding a large
  independent set
• Delete each vertex from G with probability
  1-n/(2m)
• For each remaining edge, remove it and one of its
  adjacent vertices
• The expected cardinality of the remaining
  independent set is n2/4m

16
Independent Sets

• Can we obtain a stronger result?
• Lets try a different probability space
• Probability Space uniform over all orderings of
  vertices
• Object of Interest Independent Set
• v ? I if in the ordering lt, v lt all other
  vertices with which it shares an edge

I v ? V v,w ? E ? v lt w
17
Independent Sets
All permutations of v and its neighbors are
equally likely, and v has dv neighbors
18
Independent Sets

• I is an independent set since if v1,v2 ? I and
  v1, v2 ? E, then v1ltv2 and v2ltv1, a
  contradiction
• The result (for 2-uniform hypergraphs) is known
  as Turans theorem
• It has been extended (Griggs 1983)

19
Independent Sets

• The probabilistic method can be used to extend
  the same line of thinking to t-uniform
  hypergraphs with t gt 2
• Probability Space uniform over all orderings of
  vertices
• Object of Interest Independent Set
• v should not be last in any of its edges under a
  given ordering lt

I v ? V v,w1,,wt-1 ? E ? v lt wi for
some i
20
Other Interesting Results

• 2-Point concentration theorem (Bollabas Erdos
  1976)
• Independent Set bounds for uniform hypergraphs
  (Caro Tuza 1991)

21
Comparison of Results
p 0.5
Bounds calculated for random graphs with edge
probability p 0.5
22
Comparison of Results
p 0.1
Bounds calculated for random graphs with edge
probability p 0.1
23
Conclusion

• The Probabilistic Method is a powerful tool for
  proving existence of objects we dont want to
  construct explicitly
• Knowing that something exists can be a big
  advantage
• Consider pruning a graph by removing every vertex
  that cannot be part of a large independent set
• If we have good bounds, can improve algorithms

24
References

• Y. Caro and Z. Tuza. Improved lower bounds on
  k-independence. J. Graph Theory, 15, pp. 99-107,
  1991
• Griggs, J.R. Lower bounds on the independence
  number in terms of the degrees, J. Combinatorial
  Theory (B) 34, 22-39, 1983.
• B. Bollobas and P. Erdos, Cliques in Random
  Graphs, Mathematics Proceedings of the Cambridge
  Philosophical Society 80, 419-427, 1976
• N. Alon, J. Spencer, The Probabilistic Method,
  1992
• B. Bollobas, Random Graphs, 1985
• M. Mitzenmacher, E. Upfal, Probability and
  Computing, 2005