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 G k.

Shannon Capacity The Shannon capacity is defined to be Consider G = C 4 Consider all the code words using “a” and “c” On the other hand, each codeword forbids 2 k codewords, and so So Shannon capacity is 2 if G = C 4

Shannon Capacity The Shannon capacity is defined to be What about C 5 ? Obviously Consider {(0,0),(1,2),(2,4),(3,1),(4,3)}. It is an independent set of size 5 in C 5 2 So Can we do better? So Shannon capacity is at least √5 if G = C 5 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 >= x 2 So |S| <= 1/x 2 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 >= x 2 So |S| <= 1/x 2 A geometric upper bound for maximum independent set! For C 5, So |S| <= √5

Higher Dimension For C 5, 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 C 5, Use v1,v2,v3,v4,v5 as building block. Tensor product: This term becomes So |S| <= 1/x 2 In general Shannon capacity is at most √5 for C 5

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 V g (t)? If this probability is = half the vertices in V g (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 V g (t)? If this probability is <= 1/2, then we can keep half the vertices in V g (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 foriterations.

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!