Constraint Satisfaction and Graph Theory Pavol Hell SIAM DM, June 2008 Pavol Hell Constraint Satisfaction and Graph Theory NP versus Colouring Problems in NP k−colouring problems NP−complete NP−complete . k > 2 P k=1,2 P Pavol Hell Constraint Satisfaction and Graph Theory Examples SAT 3-COL (x, y) ∈ {(1, 1), (2, 3)} and (x, z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)} Constraint Satisfaction Problems CSP Assign values to variables so that all constraints are satisfied Pavol Hell Constraint Satisfaction and Graph Theory 3-COL (x, y) ∈ {(1, 1), (2, 3)} and (x, z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)} Constraint Satisfaction Problems CSP Assign values to variables so that all constraints are satisfied Examples SAT Pavol Hell Constraint Satisfaction and Graph Theory (x, y) ∈ {(1, 1), (2, 3)} and (x, z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)} Constraint Satisfaction Problems CSP Assign values to variables so that all constraints are satisfied Examples SAT 3-COL Pavol Hell Constraint Satisfaction and Graph Theory Constraint Satisfaction Problems CSP Assign values to variables so that all constraints are satisfied Examples SAT 3-COL (x, y) ∈ {(1, 1), (2, 3)} and (x, z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)} Pavol Hell Constraint Satisfaction and Graph Theory NP versus CSP Problems in NP Problems in CSP NP−complete NP−complete . ? P P Pavol Hell Constraint Satisfaction and Graph Theory Generalizing Colouring 1 2 3 1 2 3 Pavol Hell Constraint Satisfaction and Graph Theory Which Problems / Complexities What is this problem? Generalizing Colouring 1 2 1 3 Pavol Hell Constraint Satisfaction and Graph Theory What is this problem? Generalizing Colouring 1 2 1 3 Which Problems / Complexities Pavol Hell Constraint Satisfaction and Graph Theory Generalizing Colouring 1 2 1 3 Which Problems / Complexities What is this problem? Pavol Hell Constraint Satisfaction and Graph Theory Homomorphism problem HOM(H) A colouring of a graph G without the above pattern is exactly a homomorphism to H Generalizing Colouring 1 2 1 = HOM(H) 2 3 1 3 H Pavol Hell Constraint Satisfaction and Graph Theory Generalizing Colouring 1 2 1 = HOM(H) 2 3 1 3 H Homomorphism problem HOM(H) A colouring of a graph G without the above pattern is exactly a homomorphism to H Pavol Hell Constraint Satisfaction and Graph Theory Generalizing Colouring Feder-Vardi Each constraint satisfaction problem is polynomially equivalent to HOM(H) for some digraph H Pavol Hell Constraint Satisfaction and Graph Theory NP versus CSP Problems in NP Problems in CSP NP−complete NP−complete . ? HOM(H) P P Pavol Hell Constraint Satisfaction and Graph Theory Which Problem Clique Cutset Problem Matrix partition problems (Feder-Hell-Motwani-Klein) Generalizing Colouring 1 2 1 3 Pavol Hell Constraint Satisfaction and Graph Theory Matrix partition problems (Feder-Hell-Motwani-Klein) Generalizing Colouring 1 2 1 3 Which Problem Clique Cutset Problem Pavol Hell Constraint Satisfaction and Graph Theory Generalizing Colouring 1 2 1 3 Which Problem Clique Cutset Problem Matrix partition problems (Feder-Hell-Motwani-Klein) Pavol Hell Constraint Satisfaction and Graph Theory NP versus MPP Problems in NP Matrix Partition Problems NP−complete NP−complete . ? P P Pavol Hell Constraint Satisfaction and Graph Theory Feder-Hell Given a 3-edge-coloured Kn, colour the vertices without a monochromatic edge Complexity ? (Feder-Hell-Kral-Sgall) Suspicious Problems Cameron-Eschen-Hoang-Sritharan Stubborn problem Pavol Hell Constraint Satisfaction and Graph Theory Complexity ? (Feder-Hell-Kral-Sgall) Suspicious Problems Cameron-Eschen-Hoang-Sritharan Stubborn problem Feder-Hell Given a 3-edge-coloured Kn, colour the vertices without a monochromatic edge Pavol Hell Constraint Satisfaction and Graph Theory Suspicious Problems Cameron-Eschen-Hoang-Sritharan Stubborn problem Feder-Hell Given a 3-edge-coloured Kn, colour the vertices without a monochromatic edge Complexity ? (Feder-Hell-Kral-Sgall) Pavol Hell Constraint Satisfaction and Graph Theory Which Problem Is H partitionable into a triangle-free graph and a cograph? Partition problems, generalized colouring problems, subcolouring problems, etc., Alekseev, Farrugia, Lozin, Broersma, Fomin, Nesetril, Woeginger, Ekim, de Werra, Stacho, MacGillivray, Yu, Hoang, Le, etc. Generalizing Colouring 1 2 2 1 1 2 2 Pavol Hell Constraint Satisfaction and Graph Theory Partition problems, generalized colouring problems, subcolouring problems, etc., Alekseev, Farrugia, Lozin, Broersma, Fomin, Nesetril, Woeginger, Ekim, de Werra, Stacho, MacGillivray, Yu, Hoang, Le, etc. Generalizing Colouring 1 2 2 1 1 2 2 Which Problem Is H partitionable into a triangle-free graph and a cograph? Pavol Hell Constraint Satisfaction and Graph Theory Generalizing Colouring 1 2 2 1 1 2 2 Which Problem Is H partitionable into a triangle-free graph and a cograph? Partition problems, generalized colouring problems, subcolouring problems, etc., Alekseev, Farrugia, Lozin, Broersma, Fomin, Nesetril, Woeginger, Ekim, de Werra, Stacho, MacGillivray, Yu, Hoang, Le, etc. Pavol Hell Constraint Satisfaction and Graph Theory What Problem is This? ? Generalizing Colouring 1 3 2 1 Pavol Hell Constraint Satisfaction and Graph Theory ? Generalizing Colouring 1 3 2 1 What Problem is This? Pavol Hell Constraint Satisfaction and Graph Theory Generalizing Colouring 1 3 2 1 What Problem is This? ? Pavol Hell Constraint Satisfaction and Graph Theory Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-Tardif Every problem in NP is polynomially equivalent to a pattern-forbidding colouring problem Every problem in CSP is polynomially equivalent to a pattern-forbidding colouring problem with patterns on single edges (or... trees) Some finite set of patterns corresponds to isomorphism complete problems. What does it look like? NP versus CSP Which Problems / Complexities How general are these "pattern-forbidding colouring problems"? Pavol Hell Constraint Satisfaction and Graph Theory Every problem in NP is polynomially equivalent to a pattern-forbidding colouring problem Every problem in CSP is polynomially equivalent to a pattern-forbidding colouring problem with patterns on single edges (or... trees) Some finite set of patterns corresponds to isomorphism complete problems. What does it look like? NP versus CSP Which Problems / Complexities How general are these "pattern-forbidding colouring problems"? Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-Tardif Pavol Hell Constraint Satisfaction and Graph Theory Every problem in CSP is polynomially equivalent to a pattern-forbidding colouring problem with patterns on single edges (or... trees) Some finite set of patterns corresponds to isomorphism complete problems. What does it look like? NP versus CSP Which Problems / Complexities How general are these "pattern-forbidding colouring problems"? Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-Tardif Every problem in NP is polynomially equivalent to a pattern-forbidding colouring problem Pavol Hell Constraint Satisfaction and Graph Theory (or... trees) Some finite set of patterns corresponds to isomorphism complete problems. What does it look like? NP versus CSP Which Problems / Complexities How general are these "pattern-forbidding colouring problems"? Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-Tardif Every problem in NP is polynomially equivalent to a pattern-forbidding colouring problem Every problem in CSP is polynomially equivalent to a pattern-forbidding colouring problem with patterns on single edges Pavol Hell Constraint Satisfaction and Graph Theory Some finite set of patterns corresponds to isomorphism complete problems. What does it look like? NP versus CSP Which Problems / Complexities How general are these "pattern-forbidding colouring problems"? Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-Tardif Every problem in NP is polynomially equivalent to a pattern-forbidding colouring problem Every problem in CSP is polynomially equivalent to a pattern-forbidding colouring problem with patterns on single edges (or... trees) Pavol Hell Constraint Satisfaction and Graph Theory NP versus CSP Which Problems / Complexities How general are these "pattern-forbidding colouring problems"? Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-Tardif Every problem in NP is polynomially equivalent to a pattern-forbidding colouring problem Every problem in CSP is polynomially equivalent to a pattern-forbidding colouring problem with patterns on single edges (or... trees) Some finite set of patterns corresponds to isomorphism complete problems. What does it look like? Pavol Hell Constraint Satisfaction and Graph Theory NP versus CSP Problems in NP Problems in CSP NP−complete NP−complete . ? P P Pavol Hell Constraint Satisfaction and Graph Theory NP versus CSP Pattern−forbidding Problems Pattern−forbidding Problems with Single−edge Patterns NP−complete NP−complete . ? P P Pavol Hell Constraint Satisfaction and Graph Theory NP versus MPP Pattern−forbidding Problems Pattern−forbidding Problems with Two−vertex Patterns NP−complete NP−complete . ? P P Pavol Hell Constraint Satisfaction and Graph Theory NP versus MPP Problems in NP Matrix Partition Problems NP−complete NP−complete . ? P P Pavol Hell Constraint Satisfaction and Graph Theory NP versus CSP Problems in NP Problems in CSP NP−complete NP−complete . ? P P Pavol Hell Constraint Satisfaction and Graph Theory Jeavons If POL(H) ⊆ POL(H0), then HOM(H0) reduces to HOM(H) The more polymorphisms H has, the more likely is HOM(H) to be polynomial Polymorphisms POL(H) for a digraph H f : V (H)k → V (H) such that ai bi ∈ E(H) ∀i =⇒