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Theta Function
Lecture 24: Apr 18
Error Detection Code
Given a noisy channel, and a finite alphabet V,
and certain pairs that can be confounded,
the goal is to select as many words of length k
as possible so that no two can be confounded.
Let G be the graph. Then for k=1 it is the independent set problem.
What about for general k?
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)) : u1=u2 and (v1,v2) in E2
or v1=v2 and (u1,u2) in E1
or (u1,u2) in E1 and (v1,v2) in E2.
The problem is now to find a maximum independent set in Gk.
Shannon Capacity
The Shannon capacity is defined to be
Consider G = C4
Consider all the code words using “a” and “c”
On the other hand, each codeword forbids 2k codewords, and so
So Shannon capacity is 2 if G = C4
Shannon Capacity
The Shannon capacity is defined to be
What about C5? Obviously
Consider {(0,0),(1,2),(2,4),(3,1),(4,3)}.
It is an independent set of size 5 in C52
So
Can we do better?
So Shannon capacity is at least √5 if G = C5
Lovasz
Geometry
Vertex vs Vector
Independent set of vertices vs Orthogonal set of vectors
Let the handle be e1. Let all be unit vectors.
Let S be an independent set. The corresponding vectors form an orthogonal set.
Suppose we can find a drawing so that each projection to the handle has length x.
Each term is >= x2
So |S| <= 1/x2 A geometric upper bound for maximum independent set!
Orthogonal Representation
To give the best upper bound, find a drawing with the maximum projection.
Umbrella
Each term is >= x2
So |S| <= 1/x2 A geometric upper bound for maximum independent set!
For C5,
So |S| <= √5
Higher Dimension
For C5,
Use v1,v2,v3,v4,v5 as building block.
For the vector corresponds to (i,j) would be
Tensor product:
So independent set in the power corresponds to orthogonal set of these vectors.
Tight Analysis
For C5,
Use v1,v2,v3,v4,v5 as building block.
Tensor product:
This term becomes
So |S| <= 1/x2
In general Shannon capacity is at most √5 for C5
To give the best upper bound, find a drawing with the maximum projection.
Lovasz Theta Function
This can be computed using SDP for any graph!
over all orthogonal representation {v1,…,vn}.
Solving Clique LP
for each clique C
Let’s write a better LP using Lovasz idea.
Theta LP
for each c and ONR {vi}
Each independent set would satisfy this LP, because:
Theta LP
for each c and ONR {vi}
This LP is stronger than the clique LP, because:
Given any clique C, set vi=1 if i is in C; otherwise set vi=0 if i is not in C.
Then
The Sandwich Theorem
Each independent set is a feasible solution for Theta-LP, so
For clique LP,
its optimal value <= minimum clique cover .
Many Faces of Theta
Theta <= Theta-1
Easy computation.
Theta is maximum fractional independent set.
Theta-1 is the umbrella upper bound.
Theta-1 <= Theta-2
From Theta-2, use those vectors vi plus a vector c orthogonal to all vi.
Consider ui = (c + vi)/√t
This will show Theta-1 is at most t.
Theta-2 is minimum vector clique cover.
Theta-2 <= Theta-3
The most important step
Duality of SDP.
Theta-3 is maximum vector independent set.
Theta-3 <= Theta-4
Use wi in Theta-3.
Set
This will show Theta-3 is at most Theta-4.
Theta-4 is another form of maximum vector independent set.
Theta-4 <= Theta
Set
This is a feasible solution of Theta.
Idea: use the projection to get fractional solution.
SDP
for every ij not in E(G)
This is a vector program, and can be solved in polynomial time!
How to construct an independent set? Blackbox construction!
How to construct a clique cover? Compute the dual solution of clique LP.
Colouring a 3-Colourable Graph
Each vertex of the same colour corresponds to the same vector above.
for all ij in E
Solve this SDP and turn it into a colouring using colours.
Colouring a 3-Colourable Graph
Observation: adjacent vertices are far apart.
Idea: Take a random vector.
Find a “large” independent set close to it.
Use one colour for that set and repeat.
Pick g=(g1,g2,…,gn), each gi is independently drawn from a Normal distribution.
Random vector
Finding a Large Independent Set
If t is large, not enough vertices; if t is small, may have many edges.
First compute
By symmetry, assume v=(1,0,0,…,0).
Then
How Many Edges?
What is the probability that v has a neighbour in Vg(t)?
If this probability is <= 1/2, then we can keep >= half the vertices in Vg(t)?
Analysis
By symmetry,
assume v=(1,0,0,…,0)
u=(-1/2,√3/2,0,0,…,0)
Both >= t
Since g2 is normally distributed,
How Many Edges?
What is the probability that v has a neighbour in Vg(t)?
If this probability is <= 1/2, then we can keep half the vertices in Vg(t)?
Set t to find an independent set
of size
Summary
Idea: Take a random vector.
Find a “large” independent set close to it.
Use one colour for that set and repeat.
“large” means:
So we repeat for iterations.
Kneser Graph
KG(n,k) has a vertex for each k-element subset of a ground set of size n,
two vertices have an edge if and only if the corresponding subsets are disjoint.
Colouring <= n – 2k + 2
e.g. when n=3k-1, no triangle, but need k+1 colors.
Lovasz, topological method, colouring = n-2k+2.
Vector colouring is 3!
Kneser conjecture: minimum colouring = n – 2k + 2.
Open Problems
•A combinatorial algorithm to compute
maximum independent set in perfect graphs?
•Just a better rounding algorithm?
•Class of graphs with bounded Theta gap?
Remarks
Thanks!