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David Pritchard Princeton Computer Science Department & Bla Bollobs, Thomas Rothvo, Alex Scott

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Cover-Decomposability -fold cover: covers every point times For some , can every -fold cover be decomposed into two covers?

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Cover-Decomposability -fold cover: covers every point times For some , can every -fold cover be decomposed into two covers?

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Cover-Decomposability Each instance is a combinatorial question: need to cover each region Combinatorial negative answers Normal setting: given coverage of fixed point set, get many covers of it

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Planar Cover-Decomposability Cover-decomposable (), if allowed shapes are Not cover-decomposable, if allowed shapes are Halfspaces (3)Lines Translates of any fixed convex polygon Translates of any non- convex quadrilateral Scaled translates of any fixed triangle (12) Axis-aligned rectangles Axis-aligned strips (3)Strips Unit discs (33??)Discs of mixed sizes Squares? Translates of any fixed convex set?

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The Basic Question

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Edge Cover Colouring Hypergraphs with edge size 2: graphs Can we guarantee many disjoint edge covers if is large enough? /2 by assignment problem: can orient edges s.t. each vertex is head of at least /2 edges

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Cover-Decomposition in Graphs =2 cd=1 =3 cd=2 =4 cd=3

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Proof of Guptas Theorem (by Alon-Berke-Buchin 2 -Csorba-Shannigrahi-Speckmann-Zumstein) Observation 1: bipartite case is easy Observation 2: every graph has a bipartite subgraph where each v retains degree at least /2 ceil(/2) from bipartitized edges floor(floor(/2)/2) from leftovers (assignment prob.)

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Main Results Hypergraphs with bounded edge size R cd /log R Tight asymptotically if = (log R) and always O(1)-factor from optimal Hypergraphs of paths in trees cd /13 Techniques: LLL, Chernoff, LPs

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The Dual Question Hypergraph duality: vertices edges A polychromatic colouring is a partition V = V 1 V 2 V k s.t. each edge contains all colours p(H) = cd(H*) p(H) 2 H has Property B

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Lovasz Local Lemma: There are any number of bad events, but each is independent of all but D others. LLL: If each bad event has individual probability at most 1/eD, then Pr[no bad events happen] > 0. Natural to try in our setting: randomly k-colour the edges /

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Edge size R v SS\{v}

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Improving the bound Known examples exhibit dichotomy: either cd is linear in , or the family is not at all cover-decomposable (/(log R + log )) is sub-linear Plvlgyi (2010): if family is closed under edge deletion & duplication, does decomposes into 2 covers for k imply decomposes into 3 covers for f(k) for some f?

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Splitting the Hypergraph (/log R) is already (/log R) if poly(R) Idea: partition edges to H 1,H 2,,H M with (H i ) poly(R), (H i ) ~ (H)/M =((H)/M/log R) covers ((H)/M/log R) covers M=3 ~/log R covers ((H i )/log R) covers

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Iterated Pairwise Splitting

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Beck-Fiala Theorem (81)

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Beck-Fiala Algorithm LP variables: S: 0 x S = 1 - y S 1 v: S:v S x S /2, S:v S y S /2 1. find extreme point LP solution 2. fix variables with values 0 or 1 3. discard all constraints involving R non-fixed variables Extreme point solution is defined by |H nonfixed | constraints, each var in R constraints; averaging terminates

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To the Trees For paths in trees, its analogous LP admits a similar counting lemma: extreme an integral variable or constraint with 6 nonfixed variables Also holds with edge-paths, or arc- paths in a bidirected tree

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Bad Trees Tree-hypergraphs with sibling edges in addition to path edges are not polychromatic (Pach, Tardos, Tth)

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Sparse Hypergraphs [Alon-Berke-Buchin 2 -Csorba-Shannigrahi-Speckmann-Zumstein] (, )-sparse hypergraph := incidences(U V, F H) |U|+|F| : -vertex-sparse incidences -edge-sparse incidences idea: shrink off blue ones, obtaining cd (-)/log vertices hyperedges bipartite incidence graph

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More Results Cover-decomposition with unit VC- dimension Cover-decomposition with their duals, which are vertex dicutsets in trees VC-dimension 2 is not cover- decomposable Big picture: no idea, but we have more positive/negative examples to work with

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Cover Scheduling