Randomized Algorithms for Cuts and Colouring

David Pritchard,NSERC Post-doctoral Fellow

What Can Randomness Do?

Part 1: Check 3-edge-connectivityin a distributed networkJoint with Ramakrishna Thurimella

(Denver)

Part 2: Find many disjoint setcovers as a function of min degreeJoint w/ Béla Bollobás (Cambridge &

Memphis), Thomas Rothvoß (MIT), Alex Scott (Oxford)

Distributed ComputationVertices are computers that

communicate using edges♫ initially, local/no knowledge♫ goal: compute global graph property

Distributed Computation

Message passing happens in rounds♫ time complexity: # rounds elapsed

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

e.g., Diam = 5

Distributed Computation

Message passing happens in rounds♫ time complexity: # rounds elapsed

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

♫ if message lengths are unrestricted, we can compute anything in O(Diameter) rounds

♫ “CONGEST” model: limit message lengths to O(log |V|) bits

Known Time Complexities

Synchronizer w/polylog overhead, AP’90Breadth-first spanning tree:♫ O(Diam) time♫ “Universally optimal”

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 bound

Min-cost spanning tree (KP’98, PR’99):♫ O(√V log*V + Diam), Ω(√V/log V)

Main Result

Distributed algorithm to check 3-edge-connectivity in O(Diam) time

♫ explicit: finds 2-edge-cuts

♫ application: reinforcement♫ beats prior O(Diam+V2); optimal

Main tool: sample cycle space randomly

Main Tool: Cycle Space

connected network/graph (V, E)♫ (V, F) Eulerian if ∀v, degF(v) is even

Cycle space := the vector space {F : (V, F) is Eulerian}

♫ notation abuse: F is a subset of E and also its characteristic vector ∈ {0, 1}E

♫ why is it a vector space?♫ even deg. ⊕ even deg. ≡ even deg.⊕ =

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 Fred: not in Fgrey: undecided

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.

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 any2-edge-cuts

cut class

⇔ |{e, e’} ∩ F| even for all Eulerian F⇔ e and e’ always both or neither in F

Algorithm for 2-Edge Cuts

Set k = O(log |V|).Sample F1, F2,… Fk

from cycle space.Group equal rows

and output themas cut classes.

♫ {ei, ej} is a 2-edge-cut ⇔ ei, ej have equal rows, with error probability 2-k

♫ Pr[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… ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

e1

e2

e3

e4

e5

10110 ⋮

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?

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?

The Basic Question

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

A cover-decomposition is a partitionH = H1 H⨄ 2 … H⨄ ⨄ k

s.t. each Hi covers V, i.e. ⋃{S | S∈Hi} = V♫ equivalently, a polychromatic coloring

goal: maximize #parts/#colours

cd: Cover-Decomposition Number

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

♫ Easy: 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 possible♫ But for specific families…

Cover-Decomposition in Graphs

R. 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=3

Ground set = finite X ⊆ Rd,edges = subsets of X covered by shapes

Cover-Decomposition in Geometry

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 δS,

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.”

Our Results

Hypergraphs with bounded edge size R♫ cd ≳ δ/log R; this is tight

Techniques: LLL, discrepancy, LPsHypergraphs of paths in trees♫ cd ≥ δ/5

Hypergraphs of VC-dimension ≤ D♫ cover-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.

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

/

Edge size ≤ R

Pick 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))

vS S\{v}→

Splitting the Hypergraph

Ω(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) coversM=3

~δ/log R covers

Ω(δ(Hi)/log R) covers

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 - ε

♫ Equivalent view: assign ±1 to edges, s.t. |total weight on each vertex| ≤ 2ε

discrepancy!

Beck-Fiala Algorithm

LP variables: ∀S: 0 ≤ xS = 1 - yS ≤ 1

∀v: ΣS:v∈S xS ≥ δ/2, ΣS:v∈S yS ≥ δ/2

1. find extreme point LP solution2. “fix” variables with values 0 or 13. discard all constraints involving ≤ R

non-fixed variables♫ Termination lemma

♪ basis of tight degree constraints has size ≤ |Hnonfixed|; each var is in ≤R constraints

Remarks

Maximum edge size R:♫ use better discrepancy bound to get

right multiplicative constant♫ Concentration/LLL instead of B-F

cd ≥ δ /5 for paths in trees:♫ B-F, different termination lemma♪ linear independence of basis

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” incidences♫ idea: shrink off β-edge-sparse ones,

obtaining cd ≳ (δ-α)/log β vertices

hyperedges

bipartiteincidence

graph

≤ α

≤ β

Cover Scheduling

Hyperedges are sensors monitoring V♫ Each hyperedge S has battery life dS

♫ Goal: schedule when each should turn on, so V is covered from time 0 to T

♫ How large can T be?(Cover-decomposition: d ≡ 1)♫ Result: Ω(min point coverage/R)

schedule is possible♫ Open Q: improve R to log R!