New Geometric Representations and Domination Problems on Tolerance - PowerPoint PPT Presentation

New Geometric Representations and Domination Problems on Tolerance and Multitolerance Graphs Archontia Giannopoulou George B. Mertzios School of Engineering and Computing Sciences, Durham University, UK Algorithmic Graph Theory on the Adriatic Coast June 16–19, 2015 Koper, Slovenia George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 1 / 23

Intersection graphs Definition An undirected graph G = ( V , E ) is called an intersection graph, if each vertex v ∈ V can be assigned to a set S v , such that two vertices of G are adjacent if and only if the corresponding sets have a nonempty intersection, i.e. E = { uv | S u ∩ S v � = ∅ } . George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 2 / 23

Intersection graphs Definition An undirected graph G = ( V , E ) is called an intersection graph, if each vertex v ∈ V can be assigned to a set S v , such that two vertices of G are adjacent if and only if the corresponding sets have a nonempty intersection, i.e. E = { uv | S u ∩ S v � = ∅ } . Definition A graph G is called an interval graph, if G is the intersection graph of a set of intervals on the real line. a a e b ⇔ b d c c d e George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 2 / 23

Tolerance graphs Definition (Golumbic, Monma, 1982) A graph G = ( V , E ) is called a tolerance graph, if there is a set I = { I v | v ∈ V } of intervals and a set t = { t v | v ∈ V } of positive numbers, such that uv ∈ E if and only if | I u ∩ I v | ≥ min { t u , t v } . George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 3 / 23

Tolerance graphs Definition (Golumbic, Monma, 1982) A graph G = ( V , E ) is called a tolerance graph, if there is a set I = { I v | v ∈ V } of intervals and a set t = { t v | v ∈ V } of positive numbers, such that uv ∈ E if and only if | I u ∩ I v | ≥ min { t u , t v } . I a I c I b a d I d ⇔ 0 1 2 3 4 5 6 7 8 9 10 c b t a = t c = 1 t b = 8 t d = 7 George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 3 / 23

Tolerance graphs Definition (Golumbic, Monma, 1982) A graph G = ( V , E ) is called a tolerance graph, if there is a set I = { I v | v ∈ V } of intervals and a set t = { t v | v ∈ V } of positive numbers, such that uv ∈ E if and only if | I u ∩ I v | ≥ min { t u , t v } . Definition A vertex v of a tolerance graph G = ( V , E ) with a tolerance representation � I , t � is called a bounded vertex, if t v ≤ | I v | . George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 3 / 23

Tolerance graphs Definition (Golumbic, Monma, 1982) A graph G = ( V , E ) is called a tolerance graph, if there is a set I = { I v | v ∈ V } of intervals and a set t = { t v | v ∈ V } of positive numbers, such that uv ∈ E if and only if | I u ∩ I v | ≥ min { t u , t v } . Definition A vertex v of a tolerance graph G = ( V , E ) with a tolerance representation � I , t � is called a bounded vertex, if t v ≤ | I v | . Otherwise, if t v > | I v | , v is called an unbounded vertex. George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 3 / 23

Multitolerance graphs Motivation and definition Tolerance graphs have important applications [Golumbic, Trenk, Tolerance graphs , 2004]: biology and bioinformatics (comparison of DNA sequences between organisms, e.g. in BLAST software) George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 4 / 23

Multitolerance graphs Motivation and definition Tolerance graphs have important applications [Golumbic, Trenk, Tolerance graphs , 2004]: biology and bioinformatics (comparison of DNA sequences between organisms, e.g. in BLAST software) interval − → DNA sub-sequence tolerance − → permissible number of errors George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 4 / 23

Multitolerance graphs Motivation and definition Tolerance graphs have important applications [Golumbic, Trenk, Tolerance graphs , 2004]: biology and bioinformatics (comparison of DNA sequences between organisms, e.g. in BLAST software) interval − → DNA sub-sequence tolerance − → permissible number of errors temporal reasoning, resource allocation, scheduling ... George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 4 / 23

Multitolerance graphs Motivation and definition Tolerance graphs have important applications [Golumbic, Trenk, Tolerance graphs , 2004]: biology and bioinformatics (comparison of DNA sequences between organisms, e.g. in BLAST software) interval − → DNA sub-sequence tolerance − → permissible number of errors temporal reasoning, resource allocation, scheduling ... In applications of BLAST, some genomic regions may be: biologically less significant, or George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 4 / 23

Multitolerance graphs Motivation and definition Tolerance graphs have important applications [Golumbic, Trenk, Tolerance graphs , 2004]: biology and bioinformatics (comparison of DNA sequences between organisms, e.g. in BLAST software) interval − → DNA sub-sequence tolerance − → permissible number of errors temporal reasoning, resource allocation, scheduling ... In applications of BLAST, some genomic regions may be: biologically less significant, or more error prone than others = ⇒ we want to treat several genomic parts non-uniformly. George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 4 / 23

Multitolerance graphs Motivation and definition Multitolerance graphs: t 1 t 2 I = [ ℓ, r ] : r t r ℓ ℓ t from left and right: different tolerances. George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 5 / 23

Multitolerance graphs Motivation and definition Multitolerance graphs: t 1 t 2 I = [ ℓ, r ] : r t r ℓ ℓ t from left and right: different tolerances. in the interior part: tolerate a convex combination of t 1 and t 2 . George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 5 / 23

Multitolerance graphs Motivation and definition Multitolerance graphs: t 1 t 2 I = [ ℓ, r ] : r t r ℓ ℓ t Formally: I ( I , ℓ t , r t ) = { λ · [ ℓ , ℓ t ] + ( 1 − λ ) · [ r t , r ] : λ ∈ [ 0, 1 ] } (convex hull of [ ℓ , ℓ t ] and [ r t , r ] ) George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 6 / 23

Multitolerance graphs Motivation and definition Multitolerance graphs: t 1 t 2 I = [ ℓ, r ] : r t r ℓ ℓ t Formally: I ( I , ℓ t , r t ) = { λ · [ ℓ , ℓ t ] + ( 1 − λ ) · [ r t , r ] : λ ∈ [ 0, 1 ] } (convex hull of [ ℓ , ℓ t ] and [ r t , r ] ) Set τ of tolerance intervals of I : either τ = I ( I , ℓ t , r t ) for two values ℓ t , r t ∈ I (bounded case), George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 6 / 23

Multitolerance graphs Motivation and definition Multitolerance graphs: t 1 t 2 I = [ ℓ, r ] : r t r ℓ ℓ t Formally: I ( I , ℓ t , r t ) = { λ · [ ℓ , ℓ t ] + ( 1 − λ ) · [ r t , r ] : λ ∈ [ 0, 1 ] } (convex hull of [ ℓ , ℓ t ] and [ r t , r ] ) Set τ of tolerance intervals of I : either τ = I ( I , ℓ t , r t ) for two values ℓ t , r t ∈ I (bounded case), or τ = R (unbounded case). George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 6 / 23

Multitolerance graphs Motivation and definition Multitolerance graphs: t 1 t 2 I = [ ℓ, r ] : r t r ℓ ℓ t Formally: I ( I , ℓ t , r t ) = { λ · [ ℓ , ℓ t ] + ( 1 − λ ) · [ r t , r ] : λ ∈ [ 0, 1 ] } (convex hull of [ ℓ , ℓ t ] and [ r t , r ] ) Set τ of tolerance intervals of I : either τ = I ( I , ℓ t , r t ) for two values ℓ t , r t ∈ I (bounded case), or τ = R (unbounded case). In a multitolerance graph G = ( V , E ) , uv ∈ E whenever: there exists a tolerance-interval Q u ∈ τ u such that Q u ⊆ I v , or there exists a tolerance-interval Q v ∈ τ v such that Q v ⊆ I u . George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 6 / 23

Complete classification in the hierarchy of perfect graphs perfect alternately weakly co-perfectly chordal orderable orientable cocomparability multitolerance bounded multitolerance tolerance trapezoid bounded tolerance parallelogram [Golumbic, Trenk, Tolerance Graphs , 2004] [Mertzios, SODA , 2011; Algorithmica , 2014] George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 7 / 23

Tolerance and multitolerance graphs Several NP-complete problems are known to be polynomially solvable on tolerance / multitolerance graphs George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 8 / 23

Tolerance and multitolerance graphs Several NP-complete problems are known to be polynomially solvable on tolerance / multitolerance graphs Some (few) algorithms used the (multi)tolerance representation: [Parra, Discr. Appl. Math. , 1998] [Golumbic, Siani, AISC , 2002] [Golumbic, Trenk, Tolerance Graphs , 2004] Most followed by the containment in weakly chordal / perfect graphs George Mertzios (Durham) Dominating Set on (multi)tolerance graphs AGTAC 2015 8 / 23