P a r t i al Col o rin g s of Uni modular Hypergraphs

1. Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

2. Partial Colorings of Unimodular Hypergraphs Overview • Introduction • Hypergraphs • Coloring hypergraphs (discrepancy) • Unimodular hypergraphs • Partial coloring • Partially coloring unimodular hypergraphs • Motivation • Result • Application Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

3. V j j j j ( ) V H E E V V E 2 4 5 µ = = = ; Introduction Hypergraphs • Hypergraph: • : finite set of vertices • : set of hyperedges vertices hyperedges Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

4. V ( ( ) f j g ) H V H E V V E E V E E 2 µ \ 2 ) = = V 0 0 ; ; 0 Introduction Hypergraphs • Hypergraph: • : finite set of vertices • : set of hyperedges • Induced subhypergraph: Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

5. ( ( ( ( ( ) ) ) P f ) ) j ( ( ( g ) j ) ) ( j ( ) ) j d d d d d d E V H H H H H E H E i i i i i i i 1 1 1 1 1 2 ¡ + +   s s s s : c c c c :   : : m s c m  n  a x v  s c   ! = = = = = = = = E E 2 E  ; ; 2 ; ; ; v ( ) ( ) ( ) E 1 1 1 + ¡ + ¡  = 1 ¡ = +1 +1 +1 -1 -1 Introduction Discrepancy of Hypergraphs • Color vertices s.t. all hyperedges are balanced: • “2-coloring” • “imbalance of hyperedge E” Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

6. j j j j j ( ( ( ( [ ) ) ) ] j f [ ] j g ) d H H E E H E E i i j i j 1 0 1 1 · · · · s c   n n ) ) = = = ; : : : … Introduction Unimodular Hypergraphs • Def: unimodular iff each induced subhypergraph has discrepancy at most one. • Remark: means • even “perfectly balanced” • odd “almost perfect”, “1” cannot be avoided The queen of low-discrepancy hypergraphs! Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

7. [ ] f [ f f g ] [ [ ] ] j g [ ] g f [ ] f g j [ ] g E V i i j j 1 £ £ £ [ 2 2 n : m n n n m m n = = = ; : : : ; Introduction Unimodular Hypergraphs: Examples • Intervals in . • Rows/Columns in a grid: • Bipartite graphs. Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

8. ( ( ) ( ( ) ) P f f ) g ( ) g d d E V V H H i i 0 0 1 1 0 0 1 ¡ +    v  s s : : v c c   v = = ! ! = = E ; 2 ; ; v -1 +1 ? -1 +1 0 Introduction Partial Coloring • Observe: is “caused” by the “odd” vertex in odd-cardinality hyperedges. • Plan: Don’t color all vertices! • “partial coloring” • vertices with are “uncolored” • , ... as before • Aim: , but doesn’t count! “Nice partial coloring” [Beck’s partial coloring method (1981)] Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

9. Partial Colorings of Unimodular Hypergraphs Existence of Nice Partial Colorings? • Clearly, not all hypergraphs have nice partial colorings: • Complete hypergraphs • Projective planes, hypergraphs constructed from Hadamard matrices (proof: the Eigenvalue argument works also for partial colorings) • Topic of this talk: Do at least unimodular hypergraphs have nice partial colorings? Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

10. ( ( ( [ [ [ ] ] ] f f f [ f [ ] g j j ] j [ [ ] g ] g ) ) g ) H H H i i i j i i j i j i 2 4 ¡ ¡ _ 2 2 n n n n n = = = = = ; ; ; : : Partial Colorings of Unimodular Hypergraphs Unimodular hgs with no nice partial coloring • “singletons” • “initial intervals” • “intervals of length 3 and 5” No hope for partial coloring?  Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

11. ( [ ] f f g j [ ] g ) H i i i i 1 2 2 + + ¡ 2 n n = ; ; ; +1 0 -1 +1 0 -1 +1 0 -1 Partial Colorings of Unimodular Hypergraphs Sometimes it works: • “length 3 intervals” • Rows and columns in the grid. • Uniform unimodular hypergraphs: All hyperedges contain the same number of vertices (needs proof). Question: When are there nice partial colorings? Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

12. ( ( ) ) P f = ( ) ( ) = g 6 k k k H E V 0 0 1 1 ¡ w w w : u w v = = ! E ; 2 ; : : : v Partial Colorings of Unimodular Hypergraphs Result • The following two properties are equivalent: • (i) has a perfectly balanced non-trivial (“nice”) partial coloring; • (ii) there are an integer k and non-trivial vertex weights • such that all hyperedges • have integral weight . 3/5 • Remark: The partial coloring in (i) colors at least half of the vertices with in (ii). • Application: “Randomly rounding rationals is as easy as rounding half-integers” [STACS 2007 ] 1/5 1/5 2/5 3/5 Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

13. f f f f f f f = g g g = = = ( g g = g = ) g ( = ) 6 k k k k B B A B A B A A B B ~ ~ ~ ~ n n n n n n ~ n ~ 1 0 0 0 0 0 0 0 0 1 1 1 1 2 1 1 1 0 2 2 2 1 2 2 ¡ ¡ + 2 2 2 2 2 2 2 ¼ ¼  x x y x y y x x x x x  x x x x  x : x y y y y x y = = = = = = i i ; ; ; ; ; ; ; ; ; ; : : : Application • IF: For all there is a RR such that • [low rounding errors w.r.t. matrix A] • [no rounding error w.r.t. totally unimodular matrix B] • THEN: For all rational there is a RR s.t. • “Proof”: • such that integral • Partial coloring: Exists such that • such that iff • RR of as above, • Repeat until . Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

14. ( ( ) ) P f = ( ) ( ) = g 6 k k k H E V 0 0 1 1 ¡ w w w : u w v = = ! E ; 2 ; : : : v Partial Colorings of Unimodular Hypergraphs Summary • The following two properties are equivalent: • (i) has a perfectly balanced non-trivial partial coloring; • (ii) there are an integer k and non-trivial vertex weights • such that all hyperedges • have integral weight . 3/5 • Remark: The partial coloring in (i) colors at least half of the vertices with in (ii). [Open Problem: How many?] • Author claims an application. 1/5 1/5 2/5 3/5 Thanks! Benjamin Doerr Partial Colorings of Unimodular Hypergraphs