The Sandwich Theorem

This report contains expository notes about a function ti(G) that is popularly known as the Lovasz number of a graph G. There are many ways to define G(G), and the surprising variety of different characterizations indicates in itself that ti(G) should be interesting. But the most interesting property of 8(G) is probably the fact that it can be computed efficiently, although it lies “sandwiched” between other classic graph numbers whose computation is NP-hard. I have tried to make these notes self-contained so that they might serve as an elementary introduction to the growing literature on Lovasz’s fascinating function. The Sandwich Theorem DEK notes last revised 6 December 1993 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convex labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The theta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative definitions of ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization via eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A complementary characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary facts about cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definite proof of a semidefinite fact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Another characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The final link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The main converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Another look at TH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonzero weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The direct sum ofgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The direct cosum of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A direct product of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A direct coproduct of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Odd cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comments on the previous example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regular graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consequence for eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further examples of symmetric graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A bound on ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compatible matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiblockers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perfect graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A characterization of perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Another definition of 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Facets ofTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonal labelings in a perfect graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The smallest non-perfect graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perplexing questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 4 5 6 6 8 8 9 11 12 13 14 14 15 16 17 17 19 19 20 21 22 24 27 27 28 30 30 31 32 35 36 37 39 40 42 43 45 The Sandwich Theorem It is NP-complete to compute w(G), the size of the largest clique in a graph G, and it is NP-complete to compute x(G), the minimum number of colors needed to color the vertices of G. But Grotschel, Lovasz, and Schrijver proved [5] that we can compute in polynomial time a real number that is “sandwiched” between these hard-to-compute integers: w(G) I g(c) L x(G). ( > * Lovasz [ 131 called this a “sandwich theorem.” The book [7] develops further facts about the function 6(G) and shows that it possesses many interesting properties. Therefore I think it’s worthwhile to study ti(G) closely, in hopes of getting acquainted with it and finding faster ways to compute it. Caution: The function called d(G) in [13] is called G(c) in [7] and [12]. I am following the latter convention because it is more likely to be adopted by other researchers-[71 is a classic book that contains complete proofs, while [13] is simply an extended abstract. In these notes I am mostly following [ 71 and [la] with minor simplifications and a few additions. I mention several natural problems that I was not able to solve immediately although I expect (and fondly hope) that they will be resolved before I get to writing this portion of my forthcoming book on Combinatorial Algorithms. I’m grateful to many people-especially to Martin Grotschel and L&z16 Lovasz--for their comments on my first drafts of this material. These notes are in numbered sections, and there is at most one Lemma, Theorem, Corollary, or Example in each section. Thus, “Lemma 2” will mean “the lemma in section 2” . 0. Preliminaries. Let’s begin slowly by defining some notational conventions and by stating some basic things that will be assumed without proof. All vectors in these notes will be regarded as column vectors, indexed either by the vertices of a graph or by integers. The notation J: 2 y, when II: and y are vectors, will mean that xV > yV for all v. If A is a matrix, A, will denote column v, and A,,,, will be the element in row u of column v. The zero vector and the zero matrix and zero itself will all be denoted by 0. We will use several properties of matrices and vectors of real numbers that are familiar to everyone who works with linear algebra but not to everyone who studies graph theory, so it seems wise to list them here: (i) The dot product of (column) vectors a and b is