Splitting property via shadow systems

1. Splittingpropertyviashadowsystems Kristóf Bérczi MTA-ELTE EgerváryResearch Group onCombinatorialOptimization Erika R. Kovács Department of Operations Research Eötvös Loránd University Péter Csikvári Department of Computer Science Eötvös Loránd University László A. Végh Department of Management London School of Economics VeszprémJune 2013

2. Outline of thetalk • Splittingproperty and multisets • Tuza’s conjecture • Turán numbers

3. Splittingproperty Definition (P,<) partiallyorderedset, H ⊆ P U(H) = {x∈P: ∃h∈H: x ≥ h} is theuppershadow of H L(H) = {x∈P: ∃h∈H: x ≤h} is thelowershadow of H H Definition Maximalantichain A has thesplittingproperty, if∃partition A1 ⋃A2=A withU(A1)⋃ L(A2) = P. (P,<) has thesplittingpropertyifeverymaximalantichaindoes.

4. Splittingproperty Theorem (Ahlswede, Erdős and Graham ’95) Everydensemaximalantichainin a finiteposet has thesplittingproperty. y A a a’ x Theorem (Ahlswede, Erdős and Graham ’95) It is NP-hardtodecidewhether a givenposet has thesplittingproperty.

5. Multisets Definition Colours: = {1,…,k} Multiset over : amultiset over  : set of multisetsover : setsinwith r elements a,c ∈ , a<c: a i ≤ ciiand a≠c (partialorderon) a is a lowershadow of c Colourprofile: vectorencodingthemultiset (0,2,4) (2,3,1) ∈ 4 Theorem (BCsKV ’12) 1 has thesplittingproperty. 4 4 3 2 1 2 3 2 1 1 2 1 2 3 2 2 3 3 3 3 2 NOT DENSE! 3

6. Multisets U() L()

7. Proof 3 4 1 (2,0,1,1) 1 -1 -1 1 1 2 0 1 2 2 red red red 0 1 0 If result is → goes to Ifresult is → goesto -1 1 1 1 1

8. Tuza’s conjecture Undirected, simple graph G=(V,E). Definition Triangle packing: a set of pairwiseedge-disjointtriangles. Triangle cover: a set of edges sharing an edge with all triangles. ν(G) = maxcardinality of a trianglepacking τ(G) = min cardinality of a trianglecover ν(G) = 2 τ(G) = 3

9. Tuza’s conjecture maximal 2-connected subgraphs • Determiningν(G) andτ(G) areNP-complete • (Holyer ’81) (Yannakakis ’81) Best possible: K5 K4 Conjecture (Tuza ‘81) K5 K5 τ(G) ≤ 2ν(G) K4 K4 • Proved forvariousclasses of graphs: • planar graphs • graphs with n nodes and ≥n2 edges • chordalgraphs without large complete subgraphs • 2-shadows of hypergraphs having girth atleast 4 • line-graphs of triangle-free graphs • τ(G) ≤ ν(G) fortripartitegraphs • planar graphs not containing K4’s as subgraphs • each edge is contained in at most two triangles • odd-wheel-free graphs • triangle-3-colorablegraphs

10. Tuza’s conjecture Theorem (Haxell ‘99) Theorem (Krivelevich ‘95) τ(G) ≤ (3-ε)ν(G) ( ε> ) τ(G) ≤ 2τ*(G) ν*(G) ≤ 2ν(G) ν(G)=2 τ(G)=3 + + + ν*(G)= + 1=3 τ* (G)=3 ν(H) ≤ ν*(H) =τ *(H) ≤τ (H)

11. r-packings and covers Idea 1 H=(V,ε) (r-1)-uniformhypergraph r-block:completesub-hypergraphon r nodes r-packing:set of disjointr-blocks r-cover:set of hyperedgess.t. eachr-blockcontainsatleastone Idea 2 w:ε→R+ weightedr-packing:set of r-blockss.t. each e is inat most w(e) νw(H) = maxcardinality of a weightedr-packing τw(H) = min weight of an r-cover Fractionalversionsν*w(H) ,τ*w(H) can be definedasusual.

12. Extendingtheconjecture Conjecture Best possible: τw(H) ≤ νw(H) (r-1)-uniformcompletehypergraphonr+1nodes and w ≡ 1 Theorem (BCsKV ‘13) νw(H) ≤ ν*w(H) =τ *w(H) ≤τ w(H) τw(H) ≤ (r-1) τ *w(H) • Proof: • Colour V with(r-1) coloursuniformlyat random • Choosehyperedgeswithcolourprofilein

13. Multisets U() L()

14. Extendingtheconjecture Conjecture Best possible: τw(H) ≤ νw(H) (r-1)-uniformcompletehypergraphonr+1nodes and w ≡ 1 Theorem (BCsKV ‘13) νw(H) ≤ ν*w(H) =τ *w(H) ≤τ w(H) τw(H) ≤ (r-1) τ *w(H) • Proof: • Colour V with(r-1) coloursuniformlyat random • Choosehyperedgeswithcolourprofilein • GIVES AN r-COVER • Show thattheexpectedweight is nottoolarge.

15. Turán number Definition Turán (n,t,r)-system: r-uniformhypergraphon n nodess.t. everyt-elementsubset of nodesspans an edge (r ≤ t ≤ n). T(n,t,r): minimum size of a Turán (n,t,r)-system (Pál Turán ’61). Turán (6,5,3)-system

16. Turán number Theorem (Mantel1907) For t=3, r=2 theoptimalsolution is Definition tu(t,r) = • No exactvalue is knownfor t > r > 2 !!! • Erdős ’81: $500 for a special, $1000 forthegeneralcase Theorem (Sidorenko ’81) For anyintegerst>r, tu(t,r) ≤

17. Weighted Turán number Definition w: → R+ , w*=w( ). Tw(n,t,r): minimum weight of a Turán (n,t,r)-system. Definition tw(t,r) = Theorem (BCsKV’12) Forany t> r, tw(t,r) = tu(t,r).

18. Motivation Corollary In a weightedgraph, there is an edgesetwithtotal weight ≤ coveringeachtriangle. • Proof: • Colourthenodesbytwocoloursuniformlyat random. • Chooseedgeswithendpointshavingthesamecolour. • e is chosenbyprobability • expectedcost of covering is □ Question: similarconstructionforthegeneralcase?

19. Plantoget an (n,t,r)-system Colournnodeswithrcoloursuniformlyat random. Colournnodeswitht-1coloursuniformlyat random. • We need: • any t>r nodesspans an edgewithchoosencolourprofile • Choice 1 • choosealledgeswithat most r-1 colours • toolarge… • Choice 2 • choosealledgeswithcolourprofilein • stilltoolarge… • BUT: can be improved! • We need: • any t>r nodesspans an edgewithchoosencolourprofile • Properchoice • choosealledgeswithcolourprofilein an ’extension’ of Determinewhichcolourprofilestochoose. 1 2 0 0 t-r-1 Determinetheexpectednumber of choosenedges. 0 0 1 0

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