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