Topological nonexistence results
in complexity theory and combinatorics
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
Lower bounds on complexity of algorithmsnon-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!
Decision trees
:{0,1} {0,1}:nf D Boolean function
depth
1 2 2 3 3 4( ) ( ) ( )x x x x x x‚ ƒ ‚ ƒ ‚
2 1?x "Y N
3 1?x 3 1?x "
Y N
0f "1f "
1f " 0f "
Y N
1 1?x 4 1?x Y N NY
1f " 0f
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
{0,1} (1 )ijx i j n
12 13 1,( , ,..., ) : Booleanfunctionn nx x x
“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
2n-3
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!2
n
Every non-constant monotone graph property is evasive. ?
Anderraa-Rozenberg-Karp Conjecture:
Lenstra et al; Rivest and Vuillemin;Kahn – Saks – SturtevantForman
( )2
nD f
True if n is a prime, prime power, is cyclic,
,...(2)nS
Every non-constant monotone weakly symmetric Boolean function is evasive. ?
Invariant under a transitive permutation group on the variables
f monotone: ff K simplicial complex
{1,..., }V n {0,1}S VS V
( ) 0.SfK fS
1 2 2 3 3 4( ) ( ) ( )x x x x x x‚ ƒ ‚ ƒ ‚
f non-evasive Kf contractible
Key Lemma:
f non-evasive Kf contractible
f weakly symmetric acts on Kf
f monotone Kf can be constructed
has afixed point
( )f fV K Kf =constant
Application: monotone graph properties
Monotone non-trivial graph property, # of nodes prime power
evasive
Monotone non-trivial graph property decision tree depth (n2)
More complicated decision trees: comparisons
?i jx x<decision tree node:
Given are there 2 equal?
1,... ,nx x Î ¡n log n
Given are they all equal?
1,... ,nx x Î ¡n
Given are there k equal?
1,... ,nx x Î ¡n log (n/k)
Björner-L-Yao
Chromatic number and topology
( 1) 1Let .
If all -subsets of an -set are -colored,
then disjoint sets with the same color.
n t r tk
k n r
t
= - + -
$
Conj. Kneser (t =2), Erdős-Gyárfás (t >2)
Proved L 78 (t =2), Alon-Frankl-L 86 (t >2)
t=2
Kneser’s graphs
|, |finite setS S n
: | | [ ,( ) , ( ]) :{ } { }k kn nV K EA S A k A B A BK
25 :K
(Petersen graph)
3545
34
121
2
34
5
( ) 2 2knK n kc = - +
( ) : chromatic number ofG Gc
( ) 2 2knK n kc £ - +
easy
( ) 2 2knK n kc ³ - +
general lower boundon chromatic number?
( , ) :G HHom set of homorphisms from G to H
( , ) :G HHom set of homorphisms from G to H
( , ) :rG KHom set of colorations of G with r colors
( , ) :GHom set of independent node sets in G
( , ) :nP GHom set of walks in G
path of length n
“hard-core” models in statistical mechanics
( , ) :G HHom set of homorphisms from G to H
graph ( ( , ))
#{ : ( ) ( )} 1
E G H
x x x
Hom
disconnected: “qualitative log-range interaction”
Brightwell-Winkler
( , ) :G HHom set of homorphisms from G to H
graph
convex cell complex
cell 0 1 1 2
2( , )K HHom k-connected ( ) 3.G kc ³ +
L 78
2 1( , )nC H+Hom connected ( ) 4.Gc ³
Brightwell - Winkler 01
Kneser’s conjecture
2( , )knK KHom (n-2k-1)-connected
( ( ) ) ( )
( ) nodes in have common neighbor
V G V G
X G X
=
Î Û
N
N
neighborhood complex of graph G( ):GN
2( , )knK KHom (n-2k-1)-connected
neighborhood complex of graph G( ):GN
2( , )knK KHom (n-2k-1)-connected
neighborhood complex of graph G( ):GN
2( , )knK KHom (n-2k-1)-connected
P: convex polytope in d dim
neighborhood complex of graph G( ):GN
2( , )knK KHom (n-2k-1)-connected
1( ( )) dG P S -»N
G(P): connect vertices on each facet with opposite vertex (vertices)
Combinatorial Borsuk-Ulam
Bajmóczy-Bárány
2( , )knK KHom (n-2k-1)-connected
2( , ) ( )K G GHom N homotopy equivalence
Nerve Lemma:
1 ,n= È ÈK K ... K
1 ri iÇ ÇK ... K is contractible or empty
1nerve( ,..., ),n»K K K
( ) :knKN
2( , )knK KHom (n-2k-1)-connected
k
2n-k
more Nerve Lemma, or...
Crosscut Theorem MatherContractible Carrier Lemma Quillenk-connected Nerve Lemma Björner-Korte-LRank selection, shellability...
Combinatorial theory of homotopy equivalence?
Ziegler-Zivaljević
Topology’s gain?
Decision trees:{0,1} {0,1}:nf D Boolean function
1 1?x "Y N
2 1?x " 3 1?x "
Y N Y N
0f " 1f " 1f " 0f "
depth
size
1 2 1 3( ) ( )x x x x‚ ƒ ‚
( )D f minimum depth
1. Evasiveness
2. Chromatic number
4. Linear decision trees
3. Communication complexity
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