Splitting property via shadow systems

Kristóf BércziMTA-ELTE Egerváry Research

Group on Combinatorial Optimization

Erika R. KovácsDepartment of Operations

ResearchEötvös Loránd University

Péter CsikváriDepartment of Computer

ScienceEötvös Loránd University

László A. VéghDepartment of ManagementLondon School of Economics

VeszprémJune 2013

Outline of the talk

• Splitting property and multisets• Tuza’s conjecture• Turán numbers

Splitting property

(P,<) partially ordered set, H ⊆ PU(H) = {x∈P: ∃h∈H: x ≥ h} is the upper shadow of HL(H) = {x∈P: ∃h∈H: x ≤ h} is the lower shadow of H

Definition

Maximal antichain A has the splitting property, if ∃ partition A1 ⋃A2=A with U(A1) ⋃ L(A2) = P.(P,<) has the splitting property if every maximal antichain does.

Definition

H

Splitting property

Every dense maximal antichain in a finite poset has the splitting property.

Theorem (Ahlswede, Erdős and Graham ’95)

It is NP-hard to decide whether a given poset has the splitting property.

Theorem (Ahlswede, Erdős and Graham ’95)

A a

y

x

a’

Multisets Colours: = {1,…,k}

Multiset over : a multiset over : set of multisets over : sets in with r elementsa,c ∈ , a<c: a i ≤ ci i and a≠c (partial order on ) a is a lower shadow of cColour profile: vector encoding the multiset

Definition

1

1

2

22

233

3 33

2

∈ (2,3,1)(0,2,4)

has the splitting property.

Theorem (BCsKV ’12) 4

1

3

41

3 22

4

1

23

NOT DENSE!

Multisets 𝐌𝐤 ❑

𝐌𝐤 𝐤

𝐌𝐤 𝐤

+𝟏

𝐌𝐤 𝐤−𝟏

𝑨𝟏𝐤

𝑨𝟐𝐤

U()

L()

Proof

11

43

(2,0,1,1)

2

01

1

-1

2

0

1-1

20-1

1red red red

If result is → goes to

If result is → goes to

1

111

Tuza’s conjectureUndirected, simple graph G=(V,E).

Triangle packing: a set of pairwise edge-disjoint triangles.Triangle cover: a set of edges sharing an edge with all triangles.ν(G) = max cardinality of a triangle packingτ(G) = min cardinality of a triangle cover

Definition

ν(G) = 2τ(G) = 3

Tuza’s conjecture

τ(G) ≤ 2ν(G)

Conjecture (Tuza ‘81)

Best possible:

K4

K5

K5

K5 K

4K4

maximal 2-connected subgraphs

• Determining ν(G) and τ(G) are NP-complete

(Holyer ’81) (Yannakakis ’81)

Proved for various classes of graphs:• planar graphs• graphs with n nodes and ≥ n2 edges• chordal graphs without large complete subgraphs• 2-shadows of hypergraphs having girth at least 4• line-graphs of triangle-free graphs• τ(G) ≤ ν(G) for tripartite graphs • planar graphs not containing K4 ’s as subgraphs• each edge is contained in at most two triangles• odd-wheel-free graphs• triangle-3-colorable graphs

ν*(G)=

Tuza’s conjecture τ(G) ≤ (3-ε)ν(G) ( ε > )

Theorem (Haxell ‘99) τ(G) ≤ 2τ*(G)ν*(G) ≤ 2ν(G)

Theorem (Krivelevich ‘95)

τ(G)=3

τ* (G)=3

ν(G)=2

+ + + + 1=3

ν(H) ≤ ν*(H) = τ *(H) ≤τ (H)

r-packings and covers

H=(V,ε) (r-1)-uniform hypergraphr-block: complete sub-hypergraph on r nodesr-packing: set of disjoint r-blocksr-cover: set of hyperedges s.t. each r-block contains at least one

Idea 1

w:ε→R+

weighted r-packing: set of r-blocks s.t. each e is in at most w(e) νw(H) = max cardinality of a weighted r-packingτw(H) = min weight of an r-coverFractional versions ν*w(H) , τ*w(H) can be defined as usual.

Idea 2

(r-1)-uniform complete hypergraph on r+1 nodes and w ≡ 1

Extending the conjecture

τw(H) ≤ νw(H)

Conjecture Best possible:

τw(H) ≤ (r-1) τ *w(H)

Theorem (BCsKV ‘13)

Proof:

1. Colour V with (r-1) colours uniformly at random2. Choose hyperedges with colour profile in

νw(H) ≤ ν*w(H) = τ *w(H) ≤τ w(H)

Multisets 𝐌𝐤 ❑

𝐌𝐤 𝐤

+𝟏

𝐌𝐤 𝐤−𝟏

𝑨𝟏𝐤

𝑨𝟐𝐤

U()

L()

(r-1)-uniform complete hypergraph on r+1 nodes and w ≡ 1

Extending the conjecture

τw(H) ≤ νw(H)

Conjecture Best possible:

τw(H) ≤ (r-1) τ *w(H)

Theorem (BCsKV ‘13)

Proof:

1. Colour V with (r-1) colours uniformly at random2. Choose hyperedges with colour profile in

GIVES AN r-COVER

3. Show that the expected weight is not too large.

νw(H) ≤ ν*w(H) = τ *w(H) ≤τ w(H)

Turán number

Turán (n,t,r)-system: r-uniform hypergraph on n nodes s.t. every t-element subset of nodes spans an edge (r ≤ t ≤ n).T(n,t,r): minimum size of a Turán (n,t,r)-system (Pál Turán ’61).

Definition

Turán (6,5,3)-system

Turán number

tu(t,r) = Definitio

n

For any integers t>r,tu(t,r) ≤

Theorem (Sidorenko ’81)

• No exact value is known for t > r > 2 !!!

• Erdős ’81: $500 for a special, $1000 for the general case

For t=3, r=2 the optimal solution is

Theorem (Mantel 1907)

Weighted Turán number

w: → R+ , w*=w( ).Tw(n,t,r): minimum weight of a Turán (n,t,r)-system.

Definition

tw(t,r) = Definitio

n

For any t> r, tw(t,r) = tu(t,r).

Theorem (BCsKV’12)

Motivation In a weighted graph, there is an edge set with total weight ≤

covering each triangle.

Corollary

Proof:

1. Colour the nodes by two colours uniformly at random.2. Choose edges with endpoints having the same colour.

• e is chosen by probability expected cost of covering is □

Question: similar construction for the general case?

We need:

any t>r nodes spans an edge with choosen colour

profile

Proper choice• choose all edges with colour

profile in an ’extension’ of

Plan to get an (n,t,r)-system

Colour n nodes with r colours uniformly at

random.

Determine which colour profiles to

choose.

Determine the expected number of

choosen edges.

Colour n nodes with t-1 colours

uniformly at random.

2

01

1 0 0

0 0 t-r-

1

We need:

any t>r nodes spans an edge with choosen colour

profile

Choice 1• choose all edges with at

most r-1 colours too large…

Choice 2• choose all edges with colour

profile in still too large…BUT: can be improved!

Thank you for your attention!ご清聴ありがとうございました。