Colourful linear programming and simplicial depth

Antoine Deza (McMaster)

based on joint works with

Frédéric Meunier (ENPC)

Tamon Stephen (Simon Fraser)

Pauline Sarrabezolles (ENPC)

Feng Xie (Microsoft)

Colourful linear programming and simplicial depth

S, p general position

Carathéodory Theorem

Given a set S of n points in dimension d, then there exists

an open simplex generated by points in S containing p

Carathéodory Theorem

Given a set S of n points in dimension d, then there exists

an open simplex generated by points in S containing p

Given a set S of n points in dimension d, the simplicial

depth of p is the number of open simplices generated by

points in S containing p [Liu 1990]

Original Sorting Problem Simplicial Depth

depthS(p)=1

depthS(p)=2

depthS(p)=3

depthS(p)=4

depthS(p)=9

depthS (p)=5

Deepest point bounds in dimension 2 [Kárteszi 1955],

[Boros, Füredi 1984]

and [Bukh, Matoušek, Nivasch 2010]

3 32 2( ) max ( ) ( )

27 24pS

n nO n depth Op n

Deepest Point in Dimension 2

Deepest Point in Dimension d

1 1( ) max ( ) ( )

1( 1) 2 ( 1)!p

nO n depth n O

d dp n

Deepest point bounds in dimension d [Bárány 1982]

1 1( ) max ( ) ( )

1( 1) 2 ( 1)!p

nO n depth n O

d dp n

• tight upper bound

• lower bound uses Colourful Carathéodory theorem

… breakthrough [Gromov 2010] & further improvements

Colourful Carathéodory Theorem

Given colourful set S = S1 S2… Sd+1 in dimension d,

and p conv(S1) conv(S2) … conv(Sd+1), there

exists a colourful simplex containing p [Bárány 1982]

Colourful Simplicial Depth

depthS (p)=1

p conv(S1) conv(S2) … conv(Sd+1)

Given colourful set S = S1 S2… Sd+1 in dimension d, the

colourful simplicial depth of p is the number of open colourful

simplexes generated by points in S containing p

depthS (p)=2

depthS (p)=3

depthS (p)=16

( ) 1( ) max ( ) ( )

1( 1) 2 ( 1)!

ndO n depth n O n

[Bárány 1982]:

…breakthrough [Gromov 2010] & further improvements

,( ) min ( )

Sd tpe h pd

( ) 1d

max ( )1p

ndepth p c

[Bárány 1982]

[Wagner 2003]

[Gromov 2010]

simpler proofs: [Karazev 2012], [Matoušek, Wagner 2012]

d=2: [Boros, Füredi 1984], [Bukh 2006]

d=3: [Král, Mach, Sereni 2012] S, p general position

( 1)d d

( 1)!( 1)d

( 1)d d

Colourful Research Directions

S, p general position and |S1|,|S2|,…,|Sd+1| d+1

( )max ( ) ( )

1( 1)p

nddepth O n

Generalize the sufficient condition of Bárány for the

existence of a colourful simplex

Improve lower bound for

Computational approaches for for small d.

Obtain an efficient algorithm to find a colourful simplex :

Colourful Linear Programming Feasibility problem

,( ) min ( )

Sd tpe h pd

( )max ( ) ( )

1( 1)p

nddepth O n

,( ) min ( )

Sd tpe h pd

[Bárány 1982] Given colourful set S = S1 S2… Sd+1 in dimension d,

and p conv(S1) conv(S2) … conv(Sd+1), then there exists a

colourful simplex containing p

[Holmsen, Pach, Tverberg 2008] and [Arocha, Bárány, Bracho, Fabila,

Montejano 2009] Given colourful set S = S1 S2… Sd+1 in dimension

d, and p conv(Si Sj) for 1 ≤ i < j ≤ d +1, then there exists a colourful

simplex containing p

[Meunier, D. 2013] Given colourful set S = S1 S2… Sd+1 in

dimension d, if for 1 ≤ i < j ≤ d +1 there exists k≠i, k≠j, such that for all

xkSk the ray [xk p) intersects conv(Si Sj) in a point distinct from xk,

then there exists a colourful simplex containing p

Colourful Carathéodory Theorems

dimension d, if for 1 ≤ i < j ≤ d +1 there exists k≠i, k≠j, such that for all

xkSk the ray [xk p) intersects conv(Si Sj) in a point distinct from xk,

then there exists a colourful simplex containing p

dimension d, if for i ≠ j the open half-space containing p and defined by

an i-facet of a colourful simplex intersects Si Sj, then there exists a

dimension d, if for i ≠ j the open half-space containing p and defined by

an i-facet of a colourful simplex intersects Si Sj, then there exists a

further generalization in dimension 2

In a graph, if there is a vertex with an odd degree…

Given One, Get Another One

In a graph, if there is a vertex with an odd degree…

In a graph, if there is a vertex with an odd degree… then there is another one

Duoid / oik room partitioning Todd 1974, Edmonds 2009]

(Exchange algorithm: generalization of Lemke-Howson for finding a Nash

equilibrium for a 2 players game)

Polynomial Parity Argument PPA(D) [Papadimitriou 1994]

(Hamiltonian circuit in a cubic graph, Borsuk-Ulam, …)

[Meunier, D. 2013] Given colourful set S = S1 S2… Sd+1

in dimension d with |Si|=2, if there is a colourful simplex

containing p then there is another one

[Meunier, D. 2013] Any condition implying the existence

of a colourful simplex containing p actually implies that the

number of such simplices is at least d+1

( )max ( ) ( )

1( 1)p

nddepth O n

,( ) min ( )

Sd tpe h pd

[Bárány 1982]

,( ) min ( )

Sd tpe h pd

( )max ( ) ( )

1( 1)p

nddepth O n

Colourful Simplicial Depth Bounds

1 ( )d

[Bárány 1982] 1 ( )d d

,( ) min ( )

Sd tpe h pd

( )max ( ) ( )

1( 1)p

nddepth O n

[Bárány 1982]

[D.,Huang,Stephen,Terlaky 2006]

1 ( )d d 22 ( ) 1d d d

,( ) min ( )

Sd tpe h pd

( )d even for odd d

( )max ( ) ( )

1( 1)p

nddepth O n

[Bárány 1982]

1 ( )d d 22 ( ) 1d d d

,( ) min ( )

Sd tpe h pd

2 )max(3 , ) ( ) for 3

d dd d d

[Bárány, Matoušek 2007]

even for odd d

( )max ( ) ( )

1( 1)p

nddepth O n

1 ( )d d 22 ( ) 1d d d

,( ) min ( )

Sd tpe h pd

even for odd d

( )max ( ) ( )

1( 1)p

nddepth O n

[Bárány 1982]

3 ( ) for 3d d d

1 ( )d d 22 ( ) 1d d d

,( ) min ( )

Sd tpe h pd

2( 2)( ) for 8

[Stephen,Thomas 2008]

even for odd d

( )max ( ) ( )

1( 1)p

nddepth O n

[Bárány 1982]

3 ( ) for 3d d d

1 ( )d d 22 ( ) 1d d d

,( ) min ( )

Sd tpe h pd

max(3 , ) ( ) for 35

d dd d d

2( 2)( ) for 8

2( 1)( ) for 4

[D.,Stephen, Xie 2011]

[Bárány 1982]

[Stephen,Thomas 2008]

( )max ( ) ( )

1( 1)p

nddepth O n

( ) even for odd d d

,( ) min ( )

Sd tpe h pd

22( 1)

( ) 1 for 42

dd d d

(2) 5 (3) 10 (1) 2

2conjecture: ( ) 1d d

( )max ( ) ( )

1( 1)p

nddepth O n

,( ) min ( )

Sd tpe h pd

,( ) min ( )

Sd tpe h pd

,( ) min ( )

Sd tpe h pd

,( ) min ( )

Sd tpe h pd

,( ) min ( )

Sd tpe h pd

depthS(p)=1

,( ) min ( )

Sd tpe h pd

depthS(p)=2

,( ) min ( )

Sd tpe h pd

depthS(p)=3

,( ) min ( )

Sd tpe h pd

depthS(p)=4

,( ) min ( )

Sd tpe h pd

depthS(p)=5

23 ( ) 1d d d

,( ) min ( )

Sd tpe h pd

,( ) min ( )

Sd tpe h pd

9 (3) 10

,( ) min ( )

Sd tpe h pd

9 (3) 10

( ) even for odd d dp conv(S1) conv(S2) … conv(Sd+1)

,( ) min ( )

Sd tpe h pd

(3) 10

,( ) min ( )

Sd tpe h pd

(3) 10

,( ) min ( )

Sd tpe h pd

(3) 10

,( ) min ( )

Sd tpe h pd

(3) 10

,( ) min ( )

Sd tpe h pd

(3) 10

,( ) min ( )

Sd tpe h pd

depthS(p)=10

Transversal

color 1

color 2

color 3

colourful set of d points (one colour missing)

-transversal (o2,p2)

color 1

color 2

color 3

colourful set of d points (one colour missing)

2 -transversal (o2,p2) spans the antipode of g1

iff (o2,p2,g1) is a colourful simplex

antipode of g1

Transversal

Combinatorial (topological) Octahedra

color 1

color 2

color 3

pair of disjoint -transversals

octahedron [(o1,p1),(o2,p2)]

color 1

color 2

color 3

origin-containing octahedra

octahedron [(o1,p1),(o2,p2)]

colourful faces span the whole sphere if it contains

the origin (creating d+1 colourful simplexes)

Octahedron Lemma

color 1

color 2

color 3

octahedron not containing the origin

octahedron [(o1,p3),(o3,p1)] does not contain p

Octahedron Lemma

color 1

color 2

color 3

octahedron not containing the origin

octahedron [(o1,p3),(o3,p1)] spans any antipode an

even number of times

Octahedron Lemma

Given 2 disjoint transversals T1 and T2, and T1 spans x

(antipode of x),

either octahedron (T1,T2) contains p,

or there exists a transversal T≠T1 consisting of points

from T1 and T2 that spans x.

Octahedron Lemma

( )max ( ) ( )

1( 1)p

nddepth O n

,( ) min ( )

Sd tpe h pd

Computational Approach

edge: colourful simplex containing p

g1 o1 1S 2S 3S

combinatorial setting suggested by Imre Bárány

(d+1)-uniform (d+1)-partite hypergraph representation

of colourful point configurations

Computational Approach (d+1)-uniform (d+1)-partite hypergraph representation

1S 2S 3S

necessary conditions:

• every vertex belongs to at least 1 edge.

• even number of edges induced by subsets Xi of Si of size 2

reformulation of the Octahedron Lemma

Computational Approach (d+1)-uniform (d+1)-partite hypergraph representation

1S 2S 3S

if no hypergraph with t or less hyper-edges satisfies the 2

necessary conditions, then

computational proof that [D., Stephen, Xie 2013]

( )d t

(4) 14 isolated edge argument needed

Octahedral Systems n-uniform n-partite hypergraph (S1,...,Sn ,E) with |Si | ≥ 2 such

that the number of edges induced by subsets Xi of Si of size 2

for i=1,...n is even

Octahedral Systems

even number of edges if all |Si| are even for i = 1, ... , n

symmetric difference of 2 octahedral systems is octahedral

existence of non-realizable octahedral system

without isolated vertex

number of octahedral systems:

[D., Meunier, Sarrabezolles 2012]

| | (| | 1)

i iS S

Octahedral Systems

octahedral system without isolated vertex, |S1| = ... = |Sn| = m

has at least m(m+5)/2 – 11 edges, implying:

further analysis:

[D., Meunier, Sarrabezolles 2012]

( ) ( 1)( 6) / 2 11d d d

(4) 17

( )max ( ) ( )

1( 1)p

nddepth O n

[Bárány, Onn 1997] and [D., Huang, Stephen, Terlaky 2008]

,( ) min ( )

Sd tpe h pd

,( ) min ( )

Sd tpe h pd

2( 1)( 6) / 2 11 ( ) 1 for 5d d d d d

(2) 5 (3) 10 (1) 2

22 (5) 26

(4) 17

,( ) min ( )

Sd tpe h pd

2( 1)( 6) / 2 11 ( ) 1 for 5d d d d d

(2) 5 (3) 10 (1) 2

22 (5) 26

(4) 17

thank you

Tverberg Theorem

n points can be partitioned into colours, with

a point p in convex hull intersection. [Tverberg 1966]

combinations to choose d+1 colours.

max ( )1

O ndpdepthd

If each combination has at least colourful

simplices. [Bárány 82]

Colourful linear programming and simplicial depth

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