Lovsz theta function and its relationships with perfect graph theory - PowerPoint PPT Presentation

. . . . . . . . . . . . . . Lovász theta function and its relationships with perfect graph theory Arnaud Pêcher joint work with C. Bachoc and A. Thiery Univ. Bordeaux (LaBRI / INRIA RealOpt) Shanghai Jiao Tong University October 24th, 2016 A. Pêcher (Univ. Bordeaux) Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 27

. . . . . . . . . . . . Outline . 1 Lovasz’ theta function and perfectness 2 A closed formula. 3 Separating the values. 4 Proving the closed formula. A. Pêcher (Univ. Bordeaux) Motivation Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 27

. . . . . . . . . . . . . . . . Powers of chordless cycles C q A. Pêcher (Univ. Bordeaux) Motivation Shanghai, 2016 . . . . . . . . . . . . . . . . . . . 3 / 27 . . . . . p = q th power of the chordless cycle C p with p vertices C 2 C 3 C 9 9 9

. . . . . . . . . . . . . . . . Complements = circular-cliques p A. Pêcher (Univ. Bordeaux) Motivation Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . 4 / 27 . . . . Circular-Clique K p / q : vertices { 0 , 1 , · · · , p − 1 } and edges ij , s.t. q ≤ | i − j | ≤ p − q . Hence C q − 1 = K p / q . K 9/1 K 9/2 K 9/3 K 9/4 K 9/3 = C 2 K 9/4 = C 3 K 9/2 = C 9 9 9

. . . . . . . . . . . . . . . Lovász’s theta function is a real function such that, for every graph G : encoding) accuracy Explicitly known for a few families of graphs: perfect graphs (Sandwich Theorem), cycle graphs (Lovász 1978), Kneser graphs (Lovász 1979), square of cycle graphs (Brimkov et al 2000) A. Pêcher (Univ. Bordeaux) Motivation Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . 5 / 27 . . . . Some properties of Lovász’s theta function ϑ ϑ ( G ) is computable in polynomial time with given (polynomial space ω ( G ) ≤ ϑ ( G ) ≤ χ f ( G ) ≤ χ ( G ) (Sandwich Theorem) if G is homomorphic to H then ϑ ( G ) ≤ ϑ ( H ) Let ϑ p / q = ϑ ( K p / q )

. . . . . . . . . . . . From Perfect to Circular-perfect graphs . Circular chromatic number (Vince, 1988) Circular clique number (Zhu, 2000) Perfect Graph (Berge, 1960) Examples: bipartite graphs, chordal graphs, comparability graphs … (even weighted: Grötschel, Lovász, Schrijver 1981) A. Pêcher (Univ. Bordeaux) Motivation Shanghai, 2016 . . . . . . . . . . . . . . . 6 / 27 . . . . . . . . . . . . { } { } χ c ( G ) = inf k / d | G → K k / d ω c ( G ) = sup k / d | K k / d → G χ ( G ) = ⌈ χ c ( G ) ⌉ ω ( G ) = ⌊ ω c ( G ) ⌋ ω ( G ) ≤ ω c ( G ) ≤ ω f ( G ) = χ f ( G ) ≤ χ c ( G ) ≤ χ ( G ) A graph G is perfect if ∀ H ⊆ G , χ ( H ) = ω ( H ) . If G is perfect then ϑ ( G ) = ω ( G ) . Hence ω ( G ) = χ ( G ) is polytime.

. . . . . . . . . . . . . . From Perfect to Circular-perfect graphs Circular chromatic number (Vince, 1988) Circular clique number (Zhu, 2000) Circular-Perfect Graph (Zhu, 2000) Examples: perfect graphs, circular-cliques, outerplanar graphs … A. Pêcher (Univ. Bordeaux) Motivation Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . . . 6 / 27 . . . { } { } χ c ( G ) = inf k / d | G → K k / d ω c ( G ) = sup k / d | K k / d → G χ ( G ) = ⌈ χ c ( G ) ⌉ ω ( G ) = ⌊ ω c ( G ) ⌋ ω ( G ) ≤ ω c ( G ) ≤ ω f ( G ) = χ f ( G ) ≤ χ c ( G ) ≤ χ ( G ) A graph G is circular-perfect if ∀ H ⊆ G , χ c ( H ) = ω c ( H ) . ( ) If G is circular-perfect then ϑ ( G ) = ϑ where χ c ( G ) = p / q . K p / q Aim: use this equality to prove that χ c is polytime.

. . . . . . . . . . . . . . . . polynomial time: polyspace encoding. well-separated. A. Pêcher (Univ. Bordeaux) Motivation Shanghai, 2016 . . . . . . . . . . . . . 7 / 27 . . . . . . . . . . . Core of algorithm to compute χ c of circular-perfect graphs For circular perfect graphs, ϑ ( G ) ̸ = χ c ( G ) in general: √ ϑ ( C 5 ) = 5 < χ c ( C 5 ) = 2 . 5 . Strategy to compute χ c ( G ) for a circular-perfect graph G with n vertices in (1) compute ϑ ( G ) for some precision ϵ > 0 and denote by ϑ this value; (2) for every 1 ≤ p , q ≤ n , if | ϑ − ϑ ( K p / q ) | < ϵ , return p / q . Correct provided there is a unique pair ( p , q ) satisfying (2) and ϵ has Hence, roughly speaking, we need to prove that the values ϑ p / q are

. . . . . . . . . . . . Outline . 1 Lovasz’ theta function and perfectness 2 A closed formula. 3 Separating the values. 4 Proving the closed formula. A. Pêcher (Univ. Bordeaux) A closed formula Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 / 27

. . . . . . . . . . Previous solved cases Theorem - Lovász (1978) - q=2, p odd p cos p p . Proofs: Lovász: algebraic arguments Knuth (1994): linear program with two variables Theorem - Brimkov et al (2000) - q=3, p odd p cos cos p Proof: linear program with 3 variables + geometrical arguments A. Pêcher (Univ. Bordeaux) A closed formula Shanghai, 2016 . 9 / 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( ) π ( ) = ϑ ( C p ) = ϑ ( ) K p /2 π 1 + cos ( ) ( ))   ( 1 2 π 2 π 2 − cos p ⌊ p 3 ⌋ − cos ⌊ p 3 ⌋ + 1 ( ) = p  1 − ϑ  ( ( ) ) ( ( )) ) K p /3 ( 2 π 2 π p ⌊ p 3 ⌋ − 1 ⌊ p 3 ⌋ + 1 − 1

. . . . . . . . . . . . . . A closed formula q q p Theorem (Bachoc, P., Thiery 2010) q A. Pêcher (Univ. Bordeaux) A closed formula Shanghai, 2016 . . . . . . . . . . . . . . 10 / 27 . . . . . . . . . . . . ⌋ 2 π ( 2 k π ) (⌊ kp ) ∀ 0 ≤ k ≤ q − 1 , c k = cos , a k = cos p = 16 , q = 5 (∑ q − 1 ) A 0 ( c i ) with A 0 ( x ) = 2 q − 1 ∏ q − 1 ϑ ( K p / q ) = p i =1 ( x − a i ) i =0 A 0 (1)

. . . . . . . . . . . . Asymptotic behavior . q Corollary p q q q ) Corollary G A. Pêcher (Univ. Bordeaux) A closed formula Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . 11 / 27 . . ( ) 1 + ∑ q − 1 A 0 ( c i ) ϑ ( K p / q ) = p i =1 A 0 (1) If q ≥ 3 and p ≥ 4 q 3 / π then q − 4 e π 2 p ≤ ϑ ( K p / q ) ≤ p 3 (Upper bound is trivial as ϑ ( K p / q ) ≤ χ f ( K p / q ) = p For every ϵ > 0 , for every positive integer α , there is a positive integer ω such that for every circular-perfect graph G satisfying ω ( G ) ≥ ω and α ( G ) ≤ α , we ( ) have | ϑ − χ c ( G ) | ≤ ϵ .

. . . . . . . . . . . . Outline . 1 Lovasz’ theta function and perfectness 2 A closed formula. 3 Separating the values. 4 Proving the closed formula. A. Pêcher (Univ. Bordeaux) Injectivity Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 / 27

. . . . . . . . . . . . . . . . Separating result Theorem (Bachoc, P., Thiery (2013) The proof uses algebraic number theory and is in two steps: A. Pêcher (Univ. Bordeaux) Injectivity Shanghai, 2016 . . . . . . . . . . . . . 13 / 27 . . . . . . . . . . . Let p , p ′ , q , q ′ ≤ n such that p q ̸ = p ′ q ′ . � � Let ∆ = � ϑ ( K p ′ / q ′ ) − ϑ ( K p / q ) � . ∆ ≥ c − n 5 for some c > 0 Hence, computing χ c of circular-perfect graphs is polytime. (1) ∆ ̸ = 0 (2) if ∆ ̸ = 0 then ∆ ≥ c − n 5 for some c > 0

. . . . . . . . . . . . We have . p if and only if if and only if p p such that b is prime; A. Pêcher (Univ. Bordeaux) Injectivity Shanghai, 2016 . . . . . . . . . . . . . . . 14 / 27 . . . . . . . . . . . . ∆ = 0 : taking advantage of monotonicity q ≤ p ′ K p / q → K p ′ / q ′ (Bondy & Hell ’96) q ′ ϑ ( K p / q ) ≤ ϑ ( K p ′ / q ′ ) q < p ′ Assume ∆ = 0 : we have p q ′ and ϑ ( K p / q ) = ϑ ( K p ′ / q ′ ) = ϑ . [ ] q , p ′ b ∈ , ϑ ( K a / b ) = ϑ . Hence for every a q ′ [ ] q , p ′ b ∈ Take a q ′ b is coprime with a and a + 1 .

. . . . . . . . . . . . . . A flavour of algebraic number theory (1/2) Notations and definitions: x ; Some basic observations: k ); the set of algebraic integers is a ring. A. Pêcher (Univ. Bordeaux) Injectivity Shanghai, 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . 15 / 27 for every k , let ζ k = exp 2 i π / k ; let Φ be the Euler phi function: Φ( n ) ≤ n ; let Q ( ζ k ) denote the cylotomic field: the smallest complex field containing ζ k ; for every x ∈ Q ( ζ k ) , let polmin(x) ∈ Q [ X ] be the minimal polynomial of x is called an algebraic integer if polmin(x) ∈ Z [ X ] . Q ( ζ k ) is a vector space over Q whose dimension is Φ( k ) (hence at most