1. Topological nonexistence results in complexity theory and combinatorics László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com

2. Lower bounds on complexity of algorithms non-existence of efficient algorithms very difficult! - define measure of complexity of instance - it is high on appropriate instances - it is low on instances where algorithm works efficiently topology provides such measures!

3. Decision trees Boolean function depth Y N Y N YN Y N N Y

4. Example: tournament diagnostic  : tournament property (connected, no source,...) invariant under isomorphism Access to tournament: does i defeat j ? How many questions (in the worst case) to decide if property  holds? Tournament: complete oriented graph on n nodes

5. “no source”: 2n-3 questions suffice Example: tournament diagnostic (1) Knock-out tournament: read n-1 variables (2) Test winner against those knocked out by someone else: read  n-2 variables  2 n-3

6. Example: graph diagnostic  : graph property (connected, planar, no isolated node,...) invariant under isomorphism Access to graph: are nodes i and j connected? How many questions (in the worst case) to decide if property  holds? “no isolated node”: questions are needed!

7. Every non-constant monotone graph property is evasive. ? Anderraa-Rozenberg-Karp Conjecture: Lenstra et al; Rivest and Vuillemin; Kahn – Saks – Sturtevant Forman True if n is a prime, prime power,  is cyclic,,... Every non-constant monotone weakly symmetric Boolean function is evasive. ? Invariant under a transitive permutation group  on the variables

8. f monotone: simplicial complex f non-evasive  K f contractible Key Lemma:

9. f non-evasive  K f contractible f weakly symmetric   acts on K f f monotone  K f can be constructed G has a fixed point f =constant

10. Application: monotone graph properties Monotone non-trivial graph property, # of nodes prime power  evasive Monotone non-trivial graph property  decision tree depth  ( n 2 )

11. More complicated decision trees: comparisons decision tree node: Given are there 2 equal? n log n Given are they all equal? n Given are there k equal? n log (n/k) Björner-L-Yao

12. Chromatic number and topology Conj. Kneser ( t =2 ), Erdős-Gyárfás ( t >2 ) Proved L 78 ( t =2 ), Alon-Frankl-L 86 ( t >2 ) t=2

13. Kneser’s graphs (Petersen graph) 35 45 34 12 1 2 34 5

14. easy general lower bound on chromatic number?

15. set of homorphisms from G to H

16. set of colorations of G with r colors set of independent node sets in G set of walks in G path of length n “hard-core” models in statistical mechanics

17. set of homorphisms from G to H graph disconnected: “qualitative log-range interaction” Brightwell-Winkler

18. set of homorphisms from G to H graph convex cell complex cell 

19. k -connected  L 78 connected  Brightwell - Winkler 01  Kneser’s conjecture (n-2k-1) -connected

20. neighborhood complex of graph G (n-2k-1) -connected

21. neighborhood complex of graph G (n-2k-1) -connected

22. neighborhood complex of graph G (n-2k-1) -connected

23. P : convex polytope in d dim neighborhood complex of graph G (n-2k-1) -connected G(P) : connect vertices on each facet with opposite vertex (vertices) Combinatorial Borsuk-Ulam Bajmóczy-Bárány

24. (n-2k-1) -connected homotopy equivalence Nerve Lemma: is contractible or empty 

25. (n-2k-1) -connected k 2n-k more Nerve Lemma, or...

26. Crosscut Theorem Mather Contractible Carrier Lemma Quillen k-connected Nerve Lemma Björner-Korte-L Rank selection, shellability... Combinatorial theory of homotopy equivalence? Ziegler-Zivaljević Topology’s gain?

27. Decision trees Boolean function Y N Y N Y N depth size

28. 1. Evasiveness 2. Chromatic number 4. Linear decision trees 3. Communication complexity