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Approximating the rectilinear crossing number Jacob Fox, J´ anos Pach, Andrew Suk September 17, 2016 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Crossing numbers Crossing number cr ( G ) = minimum number of crossing pairs of edges over all drawings of G . Rectilinear crossing number cr ( G ) = minimum number of crossing pairs of edges over all straight-line drawings of G � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

cr versus cr cr ( G ) ≤ cr ( G ) Theorem (F´ ary 1948) cr ( G ) = 0 if and only if cr ( G ) = 0 . Theorem (Bienstock and Dean 1993) There is a sequence of graphs G 1 , G 2 , . . . , G m , . . . such that cr ( G m ) = 4 cr ( G m ) ≥ m Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Rectilinear crossing number Computing cr ( G ) is NP-hard (Bienstock 1991) Open problem : Determine the asymptotic value of cr ( K n ). Theorem (´ Abrego et al. 2012 and Fabila-Monroy and L´ opez 2014) � n � � n � 0 . 379972 < cr ( K n ) < 0 . 380473 4 4 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Main result: approximating cr ( G ) Theorem (Fox, Pach, S. 2016) There is a deterministic n 2+ o (1) -time algorithm for constructing a straight-line drawing of any n-vertex graph G in the plane with cn 4 cr ( G ) + (log log n ) c ′ crossing pairs of edges. Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Crossing Lemma Lemma (Ajtai, Chv´ atal, Newborn, Szemer´ edi 1982 and Leighton 1983) Let G be a graph on n vertices and e edges. Then e 3 cr ( G ) ≥ 64 n 2 − 4 n Dense graph: | E ( G ) | ≥ ǫ n 2 , we have cr ( G ) = α n 4 + o ( n 4 ). Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

A (1 + o(1))-approximation for dense graphs Theorem (Fox, Pach, Suk 2016) There is a deterministic n 2+ o (1) -time algorithm for constructing a straight-line drawing of any n-vertex graph G with | E ( G ) | > ǫ n 2 , such that the drawing has at most cr ( G ) + o ( cr ( G )) crossing pairs of edges. Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Generalization to weighted graphs Edge weighted graphs. G = ( V , E ), w G : E → [0 , 1]. For a fixed drawing D , the weighted number of crossings is � w G ( e ) · w G ( e ′ ) ( e , e ′ ) ∈ X D � � � � � � � � 1 .5 .3 � � � � � � � � � � � � Example: 0.5 + 0.3 = 0.8 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Generalization to weighted graphs Edge weighted graphs. G = ( V , E ), w G : E → [0 , 1]. (each weight uses at most O (log n ) bits) For a fixed drawing D , the weighted number of crossings is � w G ( e ) · w G ( e ′ ) ( e , e ′ ) ∈ X D � � � � � � � � 1 .5 .3 � � � � � � � � � � � � Example: 0.5 + 0.3 = 0.8 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Rectilinear crossing number of edge-weighted graphs � w G ( e ) · w G ( e ′ ) cr ( G ) = min D ( e , e ′ ) ∈ X D If G is not weighted, w G ( e ) = 1 if e ∈ E ( G ) w G ( e ) = 0 if e �∈ E ( G ). Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Theorem (Frieze-Kannan 1999) For any ǫ > 0 , every graph G = ( V , E ) has a equitable vertex partition V = V 1 ∪ · · · ∪ V K , 1 /ǫ < K < 2 c ǫ − 2 , such that for all disjoint subsets S , T ⊂ V ( G ) � � � � e ( V i , V j ) | S ∩ V i || T ∩ V j | � < ǫ n 2 � � e ( S , T ) − � � ( n / K ) 2 � � 1 ≤ i , j ≤ K � � G = Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Theorem (Frieze-Kannan 1999) For any ǫ > 0 , every graph G = ( V , E ) has a equitable vertex partition V = V 1 ∪ · · · ∪ V K , 1 /ǫ < K < 2 c ǫ − 2 , such that for all disjoint subsets S , T ⊂ V ( G ) � � � � e ( V i , V j ) | S ∩ V i || T ∩ V j | � < ǫ n 2 � � e ( S , T ) − � � ( n / K ) 2 � � 1 ≤ i , j ≤ K � � G = Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Theorem (Frieze-Kannan 1999) For any ǫ > 0 , every graph G = ( V , E ) has a equitable vertex partition V = V 1 ∪ · · · ∪ V K , 1 /ǫ < K < 2 c ǫ − 2 , such that for all disjoint subsets S , T ⊂ V ( G ) � � � � e ( V i , V j ) | S ∩ V i || T ∩ V j | � < ǫ n 2 � � e ( S , T ) − � � ( n / K ) 2 � � 1 ≤ i , j ≤ K � � G = T S Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Theorem (Frieze-Kannan 1999) For any ǫ > 0 , every graph G = ( V , E ) has a equitable vertex partition V = V 1 ∪ · · · ∪ V K , 1 /ǫ < K < 2 c ǫ − 2 , such that for all disjoint subsets S , T ⊂ V ( G ) � � � � e ( V i , V j ) | S ∩ V i || T ∩ V j | � < ǫ n 2 � � e ( S , T ) − � � ( n / K ) 2 � � 1 ≤ i , j ≤ K � � G = T S Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Theorem (Frieze-Kannan 1999) For any ǫ > 0 , every graph G = ( V , E ) has a equitable vertex partition V = V 1 ∪ · · · ∪ V K , 1 /ǫ < K < 2 c ǫ − 2 , such that for all disjoint subsets S , T ⊂ V ( G ) � � � � e ( V i , V j ) | S ∩ V i || T ∩ V j | � < ǫ n 2 � � e ( S , T ) − � � ( n / K ) 2 � � 1 ≤ i , j ≤ K � � Theorem (Dellamonica et al. 2015) There is a determinist 2 2 ǫ − c n 2 -time algorithm for computing such a partition. Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Given a Frieze-Kannan regular partition P : V = V 1 ∪ · · · ∪ V K : V V V 3 V 1 4 2 G = ... V V V 6 K 5 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Given a Frieze-Kannan regular partition P : V = V 1 ∪ · · · ∪ V K : V V 3 V 1 V 4 2 G = P ... V V V 6 K 5 w G P ( uv ) = 0 if u , v ∈ V i , w G P ( uv ) = e G ( V i , V j ) if u ∈ V i , v ∈ V j ( n / K ) 2 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Given a Frieze-Kannan regular partition P : V = V 1 ∪ · · · ∪ V K : V V 1 V V 3 4 2 G = P ... V V V 6 K 5 w G P ( uv ) = 0 if u , v ∈ V i , w G P ( uv ) = e G ( V i , V j ) if u ∈ V i , v ∈ V j ( n / K ) 2 G ≈ G P : For S , T ⊂ V , | e G ( S , T ) − e G P ( S , T ) | < ǫ n 2 . Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

cr ( G ) versus cr ( G P ) Using the regularity lemma for same-type transversals (Fox-Pach-S.) Key Lemma Lemma (Fox, Pach, S. 2016) 1 4 C n 4 | cr ( G ) − cr ( G P ) | < ǫ Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Defining G / P Given a Frieze-Kannan regular partition P : V = V 1 ∪ · · · ∪ V K : V V V 3 V 1 4 2 G = P ... V V V 6 K 5 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Defining G / P Given a Frieze-Kannan regular partition P : V = V 1 ∪ · · · ∪ V K : �� �� � � � �� �� �� �� � � 2 3 4 1 G / P = �� �� � � �� �� ... �� �� � � �� �� K 5 6 V ( G / P ) = { 1 , 2 , . . . , K } w G / P ( ij ) = e G ( V i , V j ) if u ∈ V i , v ∈ V j ( n / K ) 2 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Defining G / P Given a Frieze-Kannan regular partition P : V = V 1 ∪ · · · ∪ V K : �� �� � � � �� �� �� �� � � 2 3 4 1 G / P = �� �� � � �� �� ... �� �� � � �� �� K 5 6 V ( G / P ) = { 1 , 2 , . . . , K } w G / P ( ij ) = e G ( V i , V j ) if u ∈ V i , v ∈ V j ( n / K ) 2 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

� � �� �� � � � � �� �� �� �� � � 2 3 4 1 G / P = �� �� ... �� �� � �� �� K 5 6 V V 3 V 1 V 4 2 G = P ... V V V 6 K 5 G P is an ( n / K )-blow up of G / P . Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Simple Lemma Lemma � n � 4 cr ( G / P ) ≤ cr ( G P ) K Proof. Consider a drawing of G P with cr ( G P ) weighted crossings. V V 3 V 1 V 4 2 G = P ... V V V 6 K 5 Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number

Simple Lemma Lemma � n � 4 cr ( G / P ) ≤ cr ( G P ) K Proof. Consider a drawing of G P with cr ( G P ) weighted crossings. �� �� �� �� � �� �� � � � � G = P � � � � � �� �� � G/P at least cr ( G / P ) weighted crossings. Jacob Fox, J´ anos Pach, Andrew Suk Approximating the rectilinear crossing number