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Cover-Decomposition and Cover-Decomposition and

Cover-Decomposition and - PowerPoint Presentation

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Cover-Decomposition and - PPT Presentation

Polychromatic Numbers David Pritchard Princeton Computer Science Department amp Béla Bollobás Thomas Rothvoß Alex Scott CoverDecomposability δ fold cover covers every point ID: 242608

log cover edge covers cover log covers edge edges vertex point fixed hypergraph trees decomposition degree partition set fold hypergraphs size translates

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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| ≤

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!