Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

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

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 µ 2 V j V = 5 vertices j E = 4 hyperedges Benjamin Doerr Partial Colorings of Unimodular Hypergraphs

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

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

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

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

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

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

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

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

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

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 ¼ y B x = y y 2 f ; 1 g n x A x ¼ 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

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