Randomized Algorithms for Cuts and Colouring David Pritchard, NSERC Post-doctoral Fellow

What Can Randomness Do? Part 1: Check 3-edge-connectivity in a distributed network Joint with Ramakrishna Thurimella (Denver) Part 2: Find many disjoint set covers as a function of min degree Joint w/ Béla Bollobás (Cambridge & Memphis), Thomas Rothvoß (MIT), Alex Scott (Oxford)

Distributed Computation Vertices 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’90 Breadth-first spanning tree: ♫ O(Diam) time ♫ “Universally optimal”

Known Time Complexities Synchronizer w/polylog overhead, AP’90 Breadth-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+V 2 ); optimal Main tool: sample cycle space randomly

Main Tool: Cycle Space connected network/graph (V, E) ♫ (V, F) Eulerian if ∀ v, deg F (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 2 E\T thick: T green: in F red: not in F grey: 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 any 2-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 F 1, F 2,… F k from cycle space. Group equal rows and output them as cut classes. ♫ {e i, e j } is a 2-edge-cut ⇔ e i, e j have equal rows, with error probability 2 -k ♫ Pr[any error] ≤ |E| 2 2 -k = 1/poly(V) F 1 F 2 F 3 F 4 F 5 F 6 … … … … … ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ e1e2e3e4e5e1e2e3e4e ⋮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

cd: Cover-Decomposition Number

Cover-Decomposition in Graphs δ=2 cd=1 δ=3 cd=2 … δ=4 cd=3

Ground set = finite X ⊆ R d, 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 stripsAxis-aligned rectangles Halfspaces in 2DUnit strips in 2D 3D orthants4D 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 R 2 and translates of S as edges, with δ ≥ δ S, 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, LPs Hypergraphs 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 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 /

Edge size ≤ R v SS\{v} →

Splitting the Hypergraph Ω(δ/log Rδ) is already Ω(δ/log R) if δ ≤ poly(R) ♫ Idea: partition edges to H 1,H 2,…,H M with δ(H i ) ≤ poly(R), δ(H i ) ~ δ(H)/M =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) covers

Beck-Fiala 1981: there is an assignment with discrepancy ≤ 2R Iterated Pairwise Splitting

Beck-Fiala Algorithm LP variables: ∀ S: 0 ≤ x S = 1 - y S ≤ 1 ∀ v: Σ S:v ∈ S x S ≥ δ/2, Σ S:v ∈ S y S ≥ δ/2 1. find extreme point LP solution 2. “fix” variables with values 0 or 1 3. discard all constraints involving ≤ R non-fixed variables ♫ Termination lemma ♪ basis of tight degree constraints has size ≤ |H nonfixed |; 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-Buchin 2 -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 bipartite incidence graph ≤ α ≤ β

Cover Scheduling