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1. Theta Function Lecture 24: Apr 18

2. 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?

3. 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.

4. 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

5. 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 So Shannon capacity is at least √5 if G = C5 Can we do better? Lovasz

6. 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.

7. Orthogonal Representation Suppose we can find a drawing so that each projection to the handle has length x. Each term is >= x2 A geometric upper bound for maximum independent set! So |S| <= 1/x2

8. Umbrella To give the best upper bound, find a drawing with the maximum projection. For C5, So |S| <= √5 Each term is >= x2 A geometric upper bound for maximum independent set! So |S| <= 1/x2

9. Higher Dimension For C5, Use v1,v2,v3,v4,v5 as building block. Tensor product: For the vector corresponds to (i,j) would be So independent set in the power corresponds to orthogonal set of these vectors.

10. Tight Analysis For C5, Use v1,v2,v3,v4,v5 as building block. Tensor product: This term becomes So |S| <= 1/x2 Shannon capacity is at most √5 for C5 In general

11. Lovasz Theta Function To give the best upper bound, find a drawing with the maximum projection. over all orthogonal representation {v1,…,vn}. This can be computed using SDP for any graph!

12. Solving Clique LP for each clique C Let’s write a better LP using Lovasz idea.

13. Theta LP for each c and ONR {vi} Each independent set would satisfy this LP, because:

14. 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

15. The Sandwich Theorem Each independent set is a feasible solution for Theta-LP, so For clique LP, its optimal value <= minimum clique cover .

16. Many Faces of Theta

17. Theta <= Theta-1 Theta is maximum fractional independent set. Theta-1 is the umbrella upper bound. Easy computation.

18. Theta-1 <= Theta-2 Theta-2 is minimum vector clique cover. 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.

19. Theta-2 <= Theta-3 Theta-3 is maximum vector independent set. The most important step Duality of SDP.

20. Theta-3 <= Theta-4 Theta-4 is another form of maximum vector independent set. Use wi in Theta-3. Set This will show Theta-3 is at most Theta-4.

21. Theta-4 <= Theta Idea: use the projection to get fractional solution. Set This is a feasible solution of Theta.

22. 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.

23. Colouring a 3-Colourable Graph for all ij in E Each vertex of the same colour corresponds to the same vector above. Solve this SDP and turn it into a colouring using colours.

24. 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. Random vector Pick g=(g1,g2,…,gn), each gi is independently drawn from a Normal distribution.

25. 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

26. 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)?

27. Analysis Both >= t By symmetry, assume v=(1,0,0,…,0) u=(-1/2,√3/2,0,0,…,0) Since g2 is normally distributed,

28. 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

29. 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.

30. 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 Vector colouring is 3! Kneser conjecture: minimum colouring = n – 2k + 2. e.g. when n=3k-1, no triangle, but need k+1 colors. Lovasz, topological method, colouring = n-2k+2.

31. 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?

32. Remarks Thanks!