Randomized Algorithms for Cuts and - PowerPoint Presentation

Slide1

Randomized Algorithms for Cuts and Colouring

David Pritchard,NSERC Post-doctoral FellowSlide2

What Can Randomness Do?

Part 1: Check 3-edge-connectivityin a distributed network

Joint with Ramakrishna Thurimella

(Denver)Part 2: Find many disjoint setcovers as a function of min degree

Joint w/ Béla Bollobás

(Cambridge & Memphis), Thomas Rothvoß (MIT), Alex Scott (Oxford)Slide3

Distributed Computation

Vertices are computers that communicate using edges

initially, local/no knowledge

goal: compute global graph propertySlide4

Distributed Computation

Message passing happens in rounds

time complexity: # rounds elapsed

Diameter := maximum distance (# hops) between two nodes

e.g.,

Diam

= 5Slide5

Distributed Computation

Message passing happens in rounds

time complexity: # rounds elapsed

Diameter := maximum distance (# hops) between two nodesif message lengths are unrestricted,

we can compute anything in O(Diameter) rounds“CONGEST”

model: limit message lengths to O(log |V|) bitsSlide6

Known Time Complexities

Synchronizer w/polylog overhead, AP’90Breadth-first spanning tree:

O(

Diam) time“Universally optimal”Slide7

Known Time Complexities

Synchronizer w/polylog overhead, AP’90Breadth-first spanning tree:

O(

Diam) time“Universally optimal”

Depth-first spanning tree:O(|V|) time; no good lower boundMin-cost spanning tree (KP’98, PR’99):

O(√V log*V + Diam), Ω(√V/log V)Slide8

Main Result

Distributed

algorithm to check 3-edge-connectivity in

O(Diam) time

explicit: finds

2-edge-cutsapplication: reinforcementbeats prior O(Diam+V2); optimalMain tool: sample cycle space randomlySlide9

Main Tool: Cycle Space

connected network/graph (V, E)

(V, F)

Eulerian

if ∀v,

degF(v) is evenCycle space := the vector space {F : (V, F) is Eulerian}notation abuse: F is a subset of E and also its characteristic vector ∈ {0, 1}Ewhy is it a vector space?even deg. ⊕ even deg. ≡ even deg.

⊕

=Slide10

Random Sampling

Claim: if T is a spanning tree, any 0-1 vector on E\T extends uniquely to an

Eulerian subgraph

F

Corollary: sampling from the cycle space uniformly is as easy as sampling 2E\T

thick: T

green: in F

red: not in F

grey: undecidedSlide11

Cuts and Cycles

Claim. |

δ

(S)∩

F| is even for all Eulerian F

use Euler tour(s)!Claim. Unless E’ = δ(S) for some S,for a random F from the cycle space, |E’ ∩ F| is even exactly ½ of the time.Slide12

Finding 2-Edge Cuts? 2 edges {e

, e’} form a cut

Implies transitivity: if {e, f} and {f, g} are 2-edge cuts, so is {e, g}

Call equivalence classes “cut classes”

some edges not in any

2-edge-cuts

cut class

⇔ |{e, e’} ∩ F| even for all

Eulerian

F

⇔ e and e’ always both or neither in FSlide13

Algorithm for 2-Edge Cuts

Set k = O(log |V|).

Sample F

1

, F2,… Fkfrom cycle space.

Group equal rowsand output themas cut classes.{ei, ej} is a 2-edge-cut ⇔ ei, ej have equal rows, with error probability 2-kPr[any error] ≤ |E|22-k = 1/poly(V)F1 F2 F3 F4 F5 F6…

1 0 1 1

1 1…0 1 0 1 0 1…1 0 1 0 1 0…1 0 1 0 1 0…0 1 0 1 0 1… ⋮ ⋮ ⋮ ⋮ ⋮ ⋮e1e2e3e4e510110 ⋮Slide14

Questions

Leads to O(Diam + Δ/log V)-time algorithm for cut vertices

O(Diam + √V log*V) known before (

Thurimella ’97)Is O(Diam) possible or not?

Can this be derandomized

?in a distributed way?Slide15

Festival Scheduling

Set V of musicians, list of bands ⊆ V

A schedule maps each band to a day

Each musician must play

every

day

What is max # days in schedule?Slide16

The Basic QuestionInput: A

set system/hypergraph (V, H)each S ∈ H is a hyperedge S ⊆ V

A cover-decomposition is a partition

H = H1 ⨄ H2 ⨄ … ⨄ Hk

s.t. each Hi covers V, i.e. ⋃{S | S∈Hi

} = Vequivalently, a polychromatic coloringgoal: maximize #parts/#coloursSlide17

cd: Cover-Decomposition Number

cd(H) ≔ largest k for which there is a cover-decomposition into k partsEasy:

cd ≤ minimum degree ≕ δ

Does a converse hold? If δ ≥ 100 (H covers every point 100 times), can we get many disjoint covers?In general, no:

cd = 1 is still possibleBut for specific families…Slide18

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=1δ=3 cd=2…δ=4 cd=3Slide19

Ground set = finite X ⊆ Rd,

edges = subsets of X covered by shapesCover-Decomposition in GeometrySlide20

cd

=

Ω

(

δ

)

not cover-decomposable

Translates

of any

convex

polygon

Translates

of

any

non-convex quadrilateral

Axis-aligned strips

Axis-aligned rectangles

Halfspaces

in 2D

Unit strips in

2D

3D

orthants

4D orthants

Cover-Decomposition in Geometry

Do all

hypergraph families* satisfy this dichotomy? [Pálvölgyi

, Keszegh]*closed under edge deletion, duplication

Conjecture [

Pach, 1980]:for any fixed convex set S, there is δ

S, so that hypergraphs with a finite ground set in R2 and translates of S as edges, with δ ≥

δS, have cd(H) ≥ 2.“The family is cover-decomposable.” Slide21

Our ResultsHypergraphs

with bounded edge size Rcd ≳ δ/log R; this is tightTechniques: LLL, discrepancy, LPs

Hypergraphs of paths in trees

cd ≥ δ/5

Hypergraphs of VC-dimension ≤ Dcover-decomposable only for D = 1

Goal: find out how cover-decomposition number (cd) depends on minimum degree (δ) in as many natural hypergraph families as possible.Slide22

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, then

Pr[no “bad” events happen] > 0.Natural to try in our setting: randomly k-colour the edges

/Slide23

Edge size ≤ RPick some k, randomly k-

colour edges.bad event: “vertex v misses colour c.”

Dependence degree ≤ k×R×(max degree)

set all degrees to δ by “shrinking”Analyze: Pr[v misses c] ≤ (1-1

/k)δ ≤ e-δ/k

Need δRk × e-δ/k < 1/e dependency degree∴ cd ≥ Ω(δ/(log R + log δ))

v

SS\{v}→Slide24

Splitting the Hypergraph

Ω(δ/log Rδ) is already Ω(δ

/log R)if

δ ≤ poly(R)Idea: partition edges to H1,H2,…,H

M with δ(Hi) ≤ poly(R), δ(Hi

) ~ δ(H)/M

=

Ω(δ(H)/M/log R) coversΩ(δ(H)/M/log R) coversΩ(δ(H)/M/log R) covers

M=3

~δ/log R covers

Ω(δ(Hi)/log R) coversSlide25

Beck-

Fiala 1981: there is anassignment with discrepancy ≤ 2R

Iterated

Pairwise Splitting

Split H into H+, H-

so that∀v, ±: deg(v, H±) ≥ deg(v, H)/2 - εEquivalent view: assign ±1 to edges, s.t. |total weight on each vertex| ≤ 2εdiscrepancy!Slide26

Beck-

Fiala Algorithm

LP variables:

∀S: 0 ≤ xS = 1 - yS ≤ 1

∀v: ΣS:v∈S xS ≥

δ/2, ΣS:v∈S yS ≥ δ/2find extreme point LP solution“fix” variables with values 0 or 1discard all constraints involving ≤ R non-fixed variablesTermination lemmabasis of tight degree constraints has size ≤ |Hnonfixed|; each var is in ≤R constraintsSlide27

RemarksMaximum edge size R:

use better discrepancy bound to get right multiplicative constantConcentration/LLL instead of B-Fcd

≥ δ

/5 for paths in trees:B-F, different termination lemmalinear independence of basisSlide28

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 β-edge-sparse ones, obtaining cd ≳ (δ-α)/log β

vertices

hyperedgesbipartiteincidencegraph

≤

α

≤ β Slide29

Cover Scheduling

Hyperedges are sensors monitoring V

Each

hyperedge

S has battery life

dSGoal: schedule when each should turn on, so V is covered from time 0 to THow large can T be?(Cover-decomposition: d ≡ 1)

Result: Ω(min point coverage/R) schedule is possible

Open Q: improve R to log R!