Flag Algebras (hopefully simple basics) Bernard Lidick y - PowerPoint PPT Presentation

Flag algebras First try for Mantel More automatic approach Additional constraints Example - Mantel’s theorem version 1 Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue ≤ 1 . Assume = 0. (We want to conclude 2 .) � 2 � � � 0 ≤ 1 = 1 � � 1 − 2 v 1 − 4 v + 4 v + 4 n n v v v + 4 = 1 − 4 3 − 2 = 1 − 2 3 = 4 + 2 2 3 3 ≤ 1 − 2 11

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 Only and appear. 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 Only and appear. 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 Only and appear. 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 Only and appear. 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 Only and appear. 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 Only and appear. 12

Flag algebras First try for Mantel More automatic approach Additional constraints Example - stability for Mantel = 1 Assume = 0 and 2 . Goal is . − 2 0 ≤ 1 − 2 + o (1) 3 0 ≤ − 2 + o (1) 3 Only and appear. 12

Flag algebras First try for Mantel More automatic approach Additional constraints More automatic approach • How to use computer to guess the right equation for you. 13

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - example Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. 14

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - example Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 14

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - example Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 1 = + + + 14

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - example Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 1 = + + 14

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - example Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 1 = + + + 1 + 2 = 0 3 3 14

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - example Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 1 = + + + 1 + 2 = 0 3 3 ≤ 2 � � + + 3 14

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - example Theorem (Mantel 1907) 4 n 2 edges. A triangle-free graph contains at most 1 Assume edges are red and non-edges are blue. ≤ 1 Assume = 0. (We want to conclude 2 .) 1 = + + + 1 + 2 = 0 3 3 ≤ 2 � � + + 3 ≤ 2 3 14

Flag algebras First try for Mantel More automatic approach Additional constraints Example - Mantel’s theorem version 2 ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 15

Flag algebras First try for Mantel More automatic approach Additional constraints Example - Mantel’s theorem version 2 ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 Idea: find c 1 , c 2 , c 3 ∈ R such that for every graph G 0 ≤ c 1 + c 2 + c 3 . 15

Flag algebras First try for Mantel More automatic approach Additional constraints Example - Mantel’s theorem version 2 ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 Idea: find c 1 , c 2 , c 3 ∈ R such that for every graph G 0 ≤ c 1 + c 2 + c 3 . After summing together � 1 � � 2 � ≤ c 1 + 3 + c 2 + 3 + c 3 and � � (0 + c 1 ) , 1 3 + c 2 , 2 ≤ max 3 + c 3 . 15

Flag algebras First try for Mantel More automatic approach Additional constraints Example - Mantel’s theorem version 2 ≤ 1 Assume = 0. (We want to conclude 2 .) + 1 + 2 = 0 3 3 Idea: find c 1 , c 2 , c 3 ∈ R such that for every graph G 0 ≤ c 1 + c 2 + c 3 . After summing together � 1 � � 2 � ≤ c 1 + 3 + c 2 + 3 + c 3 and � � (0 + c 1 ) , 1 3 + c 2 , 2 ≤ max 3 + c 3 . c 3 < 0 15

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � � a � T � � � c 0 ≤ , , c b v v v v � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � � a � T � � � c 0 ≤ , , c b v v v v ? ? + 1 ? + 1 ? = a + b 2 c 2 c v v v v ? v × v = + o (1) v � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 Ordered � � a � T � � � c 0 ≤ , , c b v v v v ? ? + 1 ? + 1 ? = a + b 2 c 2 c v v v v ? v × v = + o (1) Unordered v ? v = 1 v × + o (1) � a 2 � c v � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 Ordered � � a � T � � � c 0 ≤ , , c b v v v v ? ? ? = a + b + c v v v ? v × v = + o (1) Unordered v ? v = 1 v × + o (1) � a 2 � c v � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � � a � T � � � c 0 ≤ , , c b v v v v ? ? ? = a + b + c v v v � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , c b n v v v v v = 1 ? ? ? � a + b + c n v v v v � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , c b n v v v v v = 1 ? ? ? � a + b + c n v v v v + a + 2 c + b + 2 c = a + b 3 3 � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , c b n v v v v v = 1 ? ? ? � a + b + c n v v v v + a + 2 c + b + 2 c = a 3 3 � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - candidates for c 1 , c 2 , c 3 � � a � T � � � 0 ≤ 1 c � , , c b n v v v v v = 1 ? ? ? � a + b + c n v v v v + a + 2 c + b + 2 c = a 3 3 c 1 = a , c 2 = a + 2 c , c 3 = b + 2 c 3 3 � a � c � 0 (matrix is positive semidefinite) c b 16

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a � c � 0 (matrix is positive semidefinite) c b 17

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a � c � 0 (matrix is positive semidefinite) c b 17

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max . 3 3 � a � c � 0 (matrix is positive semidefinite) c b 17

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max . 3 3 Try � a � � � c 1 / 2 − 1 / 2 = . c b − 1 / 2 1 / 2 17

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - using c 1 , c 2 , c 3 + 1 + 2 = 3 3 + a + 2 c + b + 2 c 0 ≤ a 3 3 � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max . 3 3 Try � a � � � c 1 / 2 − 1 / 2 = . c b − 1 / 2 1 / 2 It gives � 1 2 , 1 6 , 1 � = 1 ≤ max 2 . 2 17

Flag algebras First try for Mantel More automatic approach Additional constraints Flag algebras - optimizing a , b , c � a , 1 + a + 2 c , 2 + b + 2 c � ≤ max 3 3  Minimize d     subject to a ≤ d     1+ a +2 c  ≤ d  3 ( SDP ) 2+ b +2 c ≤ d  3   � �  a c   � 0    c b  ( SDP ) can be solved on computers using CSDP or SDPA. 18

Flag algebras First try for Mantel More automatic approach Additional constraints How to get stability? − 2 In first try we got 0 ≤ 1 − 2 which gives stability. 3 19

Flag algebras First try for Mantel More automatic approach Additional constraints How to get stability? We got � 1 � 2 , 1 6 , 1 = 1 ≤ max 2 . 2 which is ≤ 1 + 1 + 1 2 6 2 19

Flag algebras First try for Mantel More automatic approach Additional constraints How to get stability? ≤ 1 + 1 + 1 2 6 2 19

Flag algebras First try for Mantel More automatic approach Additional constraints How to get stability? ≤ 1 + 1 + 1 2 6 2 ≥ 1 Suppose G is an extremal graph ( 2 ). Then 1 ≤ 1 + 1 + 1 2 = 2 6 2 + 1 1 ≤ + 3 19

Flag algebras First try for Mantel More automatic approach Additional constraints How to get stability? ≤ 1 + 1 + 1 2 6 2 ≥ 1 Suppose G is an extremal graph ( 2 ). Then 1 ≤ 1 + 1 + 1 2 = 2 6 2 + 1 1 ≤ + 3 Combined with 1 = + + gives 0 ≤ − 2 3 19

Flag algebras First try for Mantel More automatic approach Additional constraints � 1 2 , 1 6 , 1 � = 1 ≤ max 2 2 � � = 1 Tells us that that if , then 2 • graphs with coefficients < 1 2 do not appear in the extremal example • subgraphs of extremal example should have 1 2 • gives possible subgraphs for extremal examples (if not known) • having 1 2 does not mean it appears in extremal example 20

Flag algebras First try for Mantel More automatic approach Additional constraints Additional constraints • Adding more constraints • Considering bigger (but still small) graphs may improve bounds 21

Flag algebras First try for Mantel More automatic approach Additional constraints Small experiment  Maximize    Mantel  subject to = 0   Solution is 1 2 . 22

Flag algebras First try for Mantel More automatic approach Additional constraints Small experiment  Maximize    Mantel  subject to = 0   Solution is 1 = p > 1 2 . But what if 2 ? 22

Flag algebras First try for Mantel More automatic approach Additional constraints Small experiment  Maximize    Mantel  subject to = 0   Solution is 1 = p > 1 2 . But what if 2 ?  Minimize    subject to ≥ p    22

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ p . 23

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ p . Theorem (Razborov ’08) � 2 � � � � � ( t − 1) t − 2 t ( t − p ( t + 1)) t + t ( t − p ( t + 1)) ≥ t 2 ( t + 1) 2 where t = ⌊ 1 / (1 − p ) ⌋ . Tight bound. 23

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ p . Theorem (Razborov ’08) � 2 � � � � � ( t − 1) t − 2 t ( t − p ( t + 1)) t + t ( t − p ( t + 1)) ≥ t 2 ( t + 1) 2 where t = ⌊ 1 / (1 − p ) ⌋ . Tight bound. Nontrivial application of FA. We will try simple approach for p = 0 . 6 23

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ p . Theorem (Razborov ’08) � 2 � � � � � ( t − 1) t − 2 t ( t − p ( t + 1)) t + t ( t − p ( t + 1)) ≥ t 2 ( t + 1) 2 where t = ⌊ 1 / (1 − p ) ⌋ . Tight bound. Nontrivial application of FA. We will try simple approach for p = 0 . 6 (but we will get worse bound). ≥ 0 . 1415009 . . . for p = 0 . 6 by Razborov 23

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ 0 . 6 . 24

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ 0 . 6 . = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } 24

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ 0 . 6 . = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } + 1 + 2 = 0 + 3 3 24

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ 0 . 6 . = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } + 1 + 2 0 . 6 ≤ = 0 + 3 3 24

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ 0 . 6 . = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } + 1 + 2 0 . 6 ≤ = 0 + 3 3 1 = + + + 24

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ 0 . 6 . = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } + 1 + 2 0 . 6 ≤ = 0 + 3 3 0 . 6 = 0 . 6 + 0 . 6 + 0 . 6 + 0 . 6 24

Flag algebras First try for Mantel More automatic approach Additional constraints Minimize subject to ≥ 0 . 6 . = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } + 1 + 2 0 . 6 ≤ = 0 + 3 3 0 . 6 = 0 . 6 + 0 . 6 + 0 . 6 + 0 . 6 � 1 � � 2 � 0 ≤ − 0 . 6 + 3 − 0 . 6 + 3 − 0 . 6 + 0 . 4 24

Flag algebras First try for Mantel More automatic approach Additional constraints = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } � 1 � � 2 � 0 ≤ − 0 . 6 + 3 − 0 . 6 + 3 − 0 . 6 + 0 . 4 25

Flag algebras First try for Mantel More automatic approach Additional constraints = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } � 1 � � 2 � 0 ≤ − 0 . 6 + 3 − 0 . 6 + 3 − 0 . 6 + 0 . 4 � � a � T � � � 0 ≤ 1 c � , , c b n v v v v v + a + 2 c + b + 2 c 0 ≤ a + b 3 3 25

Flag algebras First try for Mantel More automatic approach Additional constraints = 0 + 0 + 0 + ≥ min { 0 , 0 , 0 , 1 } − a + 2 c − b + 2 c 0 ≥ − a − b 3 3 � � � � � � 0 . 6 − 1 0 . 6 − 2 0 ≥ d 0 . 6 + + − 0 . 4 3 3  0  a c  � 0 (matrix is positive semidefinite) c b 0  0 0 d 26

Flag algebras First try for Mantel More automatic approach Additional constraints = 0 + 0 + 0 + − a + 2 c − b + 2 c 0 ≥ − a − b 3 3 � � 0 . 6 − 1 � � 0 . 6 − 2 � � 0 ≥ d 0 . 6 + + − 0 . 4 3 3 � � � 0 . 6 − 1 d − a + 2 c ≥ min 0 . 6 d − a , , 3 3 � � � 0 . 6 − 2 d − b + 2 c , 1 − 0 . 4 d − b 3 3 27

Flag algebras First try for Mantel More automatic approach Additional constraints � � � 0 . 6 − 1 d − a + 2 c ≥ min 0 . 6 d − a , , 3 3 � � � 0 . 6 − 2 d − b + 2 c , 1 − 0 . 4 d − b 3 3 Solution from CSDP: a = 6 × 0 . 1200006508849779385 a = 0 . 72 b = 6 × 0 . 05333290843810910981 b = 0 . 32 c = 6 × − 0 . 07999989818128358521 c = − 0 . 48 d = 1 . 400006454027185265 d = 1 . 4 � � ≥ min 0 . 12 , 0 . 453 , 0 . 12 , 0 . 12 = 0 . 12 28

Flag algebras First try for Mantel More automatic approach Additional constraints How to improve 0.12? Sample bigger graphs. Instead of 1 = + + + use 1 = + + + · · · + 29

Flag algebras First try for Mantel More automatic approach Additional constraints How to improve 0.12? Sample bigger graphs. Instead of 1 = + + + use 1 = + + + · · · + and include also M , P � 0 2 � T 2 , 2 , 2 , 2 , 2 , 2 , � � 2 � 1 1 1 1 1 1 1 1 0 ≤ M 2 � T 2 , 2 , 2 , 2 , 2 , 2 , � � 2 � 1 1 1 1 1 1 1 1 0 ≤ P 29

Flag algebras First try for Mantel More automatic approach Additional constraints How to improve 0.12? Sample bigger graphs. Instead of 1 = + + + use 1 = + + + · · · + and include also M , P � 0 2 � T 2 , 2 , 2 , 2 , 2 , 2 , � � 2 � 1 1 1 1 1 1 1 1 0 ≤ M 2 � T 2 , 2 , 2 , 2 , 2 , 2 , � � 2 � 1 1 1 1 1 1 1 1 0 ≤ P This gives ≥ 0 . 127815 . . . 29

Flag algebras First try for Mantel More automatic approach Additional constraints How to improve 0.12781. . . ? Sample even bigger graphs. Use K 5 instead of K 4 Include even more types and flags. 30

Flag algebras First try for Mantel More automatic approach Additional constraints How to improve 0.12781. . . ? Sample even bigger graphs. Use K 5 instead of K 4 Include even more types and flags. This gives ≥ 0 . 1333333 = 2 / 15 . 30