Polychromatic Numbers David Pritchard Princeton Computer Science Department amp Béla Bollobás Thomas Rothvoß Alex Scott CoverDecomposability δ fold cover covers every point ID: 242608
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Slide1
Cover-Decomposition and Polychromatic Numbers
David Pritchard
Princeton Computer Science Department
&
Béla
Bollobás
, Thomas
Rothvoß
,
Alex
ScottSlide2
Cover-Decomposabilityδ-fold cover: covers every point δ timesFor some δ, can every
δ
-fold cover be decomposed into two covers?Slide3
Cover-Decomposabilityδ-fold cover: covers every point δ timesFor some δ, can every
δ
-fold cover be decomposed into two covers?Slide4
Cover-DecomposabilityEach instance is a combinatorial question: need to cover each regionCombinatorial negative answersNormal setting: given coverage
of fixed point set, get many covers of itSlide5
Planar Cover-DecomposabilityCover-decomposable (δ), if allowed shapes are…
Not cover-decomposable, if
allowed shapes are…
Halfspaces
(3)Lines
Translates of any fixed convex polygonTranslates 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?Slide6
The Basic QuestionInput: A set system/hypergraph (V, H)each S ∈ H is a
hyperedge
S ⊆ V
A
cover-decomposition is a partitionH = H1 ⨄ H2
⨄ … ⨄ Hks.t. each Hi is a cover of V,
i.e. union(all hyperedges in Hi) = VGiven family, as coverage
δ grows, does cd := best possible k also grow?Slide7
Edge Cover ColouringHypergraphs with edge size 2: graphsCan we guarantee many disjoint edge covers if δ is large enough?
⌊
δ
/2⌋
by assignment problem:can orient edges s.t.each vertex is head of
at least ⌊δ/2⌋ edgesSlide8
Cover-Decomposition in GraphsR. P. Gupta, 1970s:In a graph, cd ≥ δ-1.
In a
multigraph
,
cd
≥ ⌊3δ
+1/4
⌋.Tight multigraph examples:
δ
=2 cd=1
δ
=3 cd=2
…
δ
=4
cd
=3Slide9
Proof of Gupta’s Theorem(by Alon-Berke-Buchin2-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 δ/2
ceil(
δ
/2) from
bipartitized
edges
floor(floor(
δ
/2)/2)
from
leftovers
(assignment prob.)Slide10
Main ResultsHypergraphs with bounded edge size Rcd ≳ δ/log R
Tight asymptotically if
δ
=
ω(log R) and always O(1)-factor from optimalHypergraphs of paths in trees
cd ≥ δ/13Techniques: LLL,
Chernoff, LPsSlide11
The Dual QuestionHypergraph duality: vertices ⇔ edgesA
polychromatic
colouring
is a partition
V = V1 ⨄ V2 ⨄ … ⨄ Vk
s.t. each edge contains all colours p(H) = cd(H*)
p(H) ≥ 2 ⇔ H has “Property B”Slide12
Lovasz Local Lemma:There are any number of “bad”events, but each is independentof all but D others.
LLL: If each bad event has individual probability at most
1/
eD
, thenPr[no “bad” events happen] > 0.Natural to try in our setting: randomly k-colour the edges
/Slide13
Edge size ≤ RPick some k, randomly k-colour edges.bad event: “vertex v misses colour c
.”
Dependence degree ≤ (max degree)×
R×k
set all degrees to δ by “shrinking”Analyze: Pr[v misses c] ≤ (1-
1/k)δ ≤ e-δ
/kNeed δRk × e
-δ/k < 1/e dependency degree
∴ cd ≥ Ω(δ
/(log R + log δ))
v
S
S
\{
v
}
→Slide14
Improving the boundKnown examples exhibit dichotomy: either cd is linear in δ, or the family is not at all cover-decomposable
Ω
(
δ/(log R + log δ
)) is sub-linearPálvölgyi (2010): if family is closed under edge deletion & duplication, does “decomposes into 2 covers for δ
≥ k” imply “decomposes into 3 covers for δ ≥ f(k)” for some f?Slide15
Splitting the HypergraphΩ(δ/log Rδ) is already Ω
(
δ
/log R)if
δ ≤ poly(R)Idea: partition edges to H1,H2,…,HM
with δ(Hi) ≤ poly(R), δ(Hi) ~
δ(H)/M
=
Ω
(
δ
(H)/M/log R) covers
Ω
(
δ
(H)/M/log R) covers
Ω
(
δ
(H)/M/log R) covers
M=3
~
δ
/log R covers
Ω
(
δ
(H
i
)/log R) coversSlide16
Iterated Pairwise SplittingSplit into pairs iteratively (M = 2K)
Each step: write H = H
+
⨄ H
- so that δ(V, H+), δ
(V, H-) ≥ δ(V, H)/2 - ε
with waste ε
minimizedEquivalent view: assign ±1 to edges, s.t. |total weight on each vertex| ≤
2ε
discrepancy!Slide17
Iterated Pairwise SplittingBest results obtained using discrepancy ≤ 2√Rln(RΔ)
(
Chernoff
bound, LLL)
yields main result cd ≳ δ
/log R.But we profit also from the Beck-
Fiala theorem, discrepancy < Rsince this approach extends to trees.Slide18
Beck-Fiala Theorem (‘81)Given hypergraph
with max edge size R, we can 2-partition the
hyperedge
set
s.t.
each vertex of initial degree d gets degree ≥ d/2 – R in each half.
LP formulation using indicator variables for each edge S: xS = 1 if S goes in first part,
yS = 1 otherwiseFractional degrees ΣS:v∈S
xS, ΣS:v∈S yS
Can we round the all-½ vector?Slide19
Beck-Fiala Algorithm
LP variables: ∀S: 0 ≤
x
S
= 1 - yS ≤ 1
∀v: ΣS:v∈S xS ≥ δ/2,
ΣS:v∈S yS ≥ δ/2
find extreme point LP solution“fix” variables with values 0 or 1discard all constraints involving ≤ R non-fixed variablesExtreme point solution is defined by
|Hnonfixed| constraints, each var in ≤ R
constraints; averaging ⇒ terminatesSlide20
To the TreesFor 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 treeSlide21
Bad
Trees
Tree-
hypergraphs
with
“sibling” edges in addition to path edges are
not polychromatic (Pach, Tardos, Tóth)Slide22
Sparse Hypergraphs[Alon-Berke-Buchin2-Csorba-Shannigrahi-Speckmann-Zumstein](
α
,
β)-sparse
hypergraph:= incidences(U ⊆ V, F ⊆ H) ≤ α|U|+β|F|
⇔: “α-vertex-sparse” incidences ⨄ “β-edge-sparse
” incidencesidea: shrink off blue ones, obtaining cd ≳ (δ
-α)/log β
vertices
hyperedges
bipartite
incidence
graph
≤
α
≤
β
Slide23
More ResultsCover-decomposition with unit VC-dimensionCover-decomposition with their duals, which are vertex dicutsets in treesVC-dimension 2 is not cover-decomposable
Big picture: no idea, but we have more positive/negative examples to work withSlide24
Cover SchedulingHyperedges are sensors monitoring V
with
nonuniform
b
attery
life
Each
hyperedge
S has battery life d
S
Goal: schedule
when each should
turn on, so V is covered from time 0 to T
How large can T be?
Result: Ω(min point coverage/R) schedule is possible
Open Q: improve R to log R!