David Pritchard Princeton Computer Science Department & Béla Bollobás, Thomas Rothvoß, Alex Scott

Cover-Decomposability δ-fold cover: covers every point δ times For some δ, can every δ-fold cover be decomposed into two covers?

Cover-Decomposability δ-fold cover: covers every point δ times For some δ, can every δ-fold cover be decomposed into two covers?

Cover-Decomposability Each instance is a combinatorial question: need to cover each region Combinatorial negative answers Normal setting: given coverage of fixed point set, get many covers of it

Planar Cover-Decomposability Cover-decomposable (δ), if allowed shapes are… Not cover-decomposable, if allowed shapes are… Halfspaces (3)Lines Translates of any fixed convex polygon Translates 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?

The Basic Question

Edge Cover Colouring Hypergraphs with edge size 2: graphs Can 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 ⌋ edges

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

Proof of Gupta’s Theorem (by Alon-Berke-Buchin 2 -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.)

Main Results Hypergraphs with bounded edge size R ♫ cd ≳ δ/log R Tight asymptotically if δ = ω(log R) and always O(1)-factor from optimal Hypergraphs of paths in trees ♫ cd ≥ δ/13 Techniques: LLL, Chernoff, LPs

The Dual Question Hypergraph duality: vertices ⇔ edges A polychromatic colouring is a partition V = V 1 V 2 … V k s.t. each edge contains all colours p(H) = cd(H*) p(H) ≥ 2 ⇔ H has “Property B”

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} →

Improving the bound Known 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-linear Pálvölgyi (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?

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

Iterated Pairwise Splitting

Beck-Fiala Theorem (‘81)

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 ♫ Extreme point solution is defined by |H nonfixed | constraints, each var in ≤ R constraints; averaging ⇒ terminates

To the Trees For 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 tree

Bad Trees Tree-hypergraphs with “sibling” edges in addition to path edges are not polychromatic (Pach, Tardos, Tóth)

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 blue ones, obtaining cd ≳ (δ-α)/log β vertices hyperedges bipartite incidence graph ≤ α ≤ β

More Results Cover-decomposition with unit VC- dimension Cover-decomposition with their duals, which are vertex dicutsets in trees VC-dimension 2 is not cover- decomposable Big picture: no idea, but we have more positive/negative examples to work with

Cover Scheduling