1. Benjamin Doerr
Partial Colorings of Unimodular Hypergraphs

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

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

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

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

6. Unimodular Hypergraphs
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 H … H = ( [ n ] ; f i : j 1 g ) d i s c ( H ) 1 j E )  ( E = j E ) j  ( E = 1
The queen of low-discrepancy hypergraphs!
Benjamin Doerr
Partial Colorings of Unimodular Hypergraphs

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

8. Partial Coloring Introduction
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” d i s c ( H ) = 1 +1 -1 ?  : V ! f 1 ; + g +1 -1  ( v ) = v  ( E ) = P v 2 d i s c ( H ;  ) =  : V ! f g
[Beck’s partial coloring method (1981)]
Benjamin Doerr
Partial Colorings of Unimodular Hypergraphs

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

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

12. Result Partial Colorings of Unimodular Hypergraphs
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 H w : V ! f ; 1 = k ( ) g w ( E ) = P v 2
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 w ( u ) 6 =
1/5
2/5
3/5
Benjamin Doerr
Partial Colorings of Unimodular Hypergraphs

13. IF: For all there is a RR such that
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 x 2 f ; 1 = g n y 2 f ; 1 g n A x y B x = y y 2 f ; 1 g n x A x y B x = y x 2 f ; 1 = k : g n B x  2 f 1 ; g n B  = ~ x 2 f ; 1 = g n ~ x i = 1 2  i 6 = ~ y ~ x x : = ( 2 k ) ~ + y x 2 f ; 1 g n
Benjamin Doerr
Partial Colorings of Unimodular Hypergraphs

14. Thanks! Summary Partial Colorings of Unimodular Hypergraphs
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 H w : V ! f ; 1 = k ( ) g w ( E ) = P v 2
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 w ( u ) 6 =
1/5
2/5
Thanks!
3/5
Benjamin Doerr
Partial Colorings of Unimodular Hypergraphs