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!