Topological nonexistence results

in complexity theory and combinatorics

László Lovász

Microsoft Research

One Microsoft Way, Redmond, WA 98052

lovasz@microsoft.com

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