Partial Colorings of Unimodular Hypergraphs Benjamin Doerr (MPI Saarbrücken)

8 PostDoc Positions Where: MPI für Informatik (Saarbrücken, Germany) Group: Kurt Mehlhorn researchers in Discrete Maths and Algorithms Position: 1 or 2 years Reasonably paid, almost unlimited support No teaching duties, but teaching possible Deadline: January 31, 2007 Benjamin DoerrPartial Colorings of Unimodular Hypergraphs

Overview Introduction Hypergraphs Discrepancy Unimodular hypergraphs Partial coloring Partially coloring unimodular hypergraphs Motivation Result Application Benjamin DoerrPartial Colorings of Unimodular Hypergraphs Partial colorings of unimodular hypergraphs

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

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

Benjamin DoerrPartial Colorings of Unimodular Hypergraphs Color vertices s.t. all hyperedges are balanced: “2-coloring” “imbalance of hyperedge E” Discrepancy of Hypergraphs Â: V ! f ¡ 1 ; + 1 g  ( E ) : = P v 2 E  ( v ) +1  ( E ) = 1 + ¡ 1 ¡ 1 = ¡ 1 d i sc ( H ;  ) : = max E 2 E j  ( E ) j d i sc ( H ) : = m i n  d i sc ( H ;  ) d i sc ( H ;  ) = j  ( E ) j = = 2 d i sc ( H ) = d i sc ( H ;  ) = 1 Well studied problem, applications in maths and CS, famous papers by Roth, Beck, Lovász, Spencer, Matoušek,...

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

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

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! Introduction Benjamin DoerrPartial Colorings of Unimodular Hypergraphs Partial Coloring d i sc ( H ) = 1 +1 ? Â: V ! f ¡ 1 ; 0 ; + 1 g v  ( v ) = 0  ( E ) = P v 2 E  ( v ) d i sc ( H ;  ) = 0 Â: V ! f 0 g +1 0

Partial Colorings of Unimodular Hypergraphs Benjamin DoerrPartial Colorings of Unimodular Hypergraphs “singletons” “initial intervals” “intervals of length 3 and 5” Partial Coloring NOT always possible No hope for partial coloring?  H = ([ n ] ; ff i gj i 2 [ n ] g ) H = ([ n ] ; f [ i ] j i 2 [ n ] g ) H = ([ n ] ; f [ i :: j ] j j ¡ i = 2 _ j ¡ i = 4 g )

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

Partial Colorings Benjamin DoerrPartial 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. Result H w: V ! f 0 ; 1 = k ;::: ( k ¡ 1 )= k g w ( E ) = P v 2 E w ( v ) Remark: The partial coloring in (i) colors at least half of the vertices with in (ii). Application: “Rounding rationals is as easy as rounding half-integers” [STACS 2007?] w ( u ) 6 = 0 1/5 3/5 2/5

Application Benjamin DoerrPartial Colorings of Unimodular Hypergraphs IF: For all x Є {0,1/2}^n there is a y Є {0,1}^n such that Ax ≈ Ay [low rounding errors] Bx = By [sometimes no rounding error] some other nice features THEN: For all rational x there is a y Є {0,1}^n such that Ax ≈ Ay Bx = By some other nice features Heart of the proof: Partial coloring of unimodular hypergraphs

Partial Colorings of unimodular hypergraphs Benjamin DoerrPartial 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. Summary H w: V ! f 0 ; 1 = k ;::: ( k ¡ 1 )= k g w ( E ) = P v 2 E w ( v ) Remark: The partial coloring in (i) colors at least half of the vertices with in (ii). Application: “Rounding rationals is as easy as rounding half-integers” [STACS 2007?] w ( u ) 6 = 0 1/5 3/5 2/5 Thanks!