Min CSP on Four Elements: Moving Beyond Submodularity Peter Jonsson - PowerPoint PPT Presentation

Min CSP on Four Elements: Moving Beyond Submodularity Peter Jonsson 1 , Fredrik Kuivinen, Johan Thapper 2 1 Department of Computer and Information Science (IDA) Link¨ oping University, Sweden 2 Laboratoire d’Informatique de l’´ Ecole Polytechnique (LIX) ´ Ecole Polytechnique, France 17th International Conference on Principles and Practice of Constraint Programming (CP 2011) Perugia, September 2011 Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 1 / 19

Outline Problem Definition and Previous Work 1 Tractable Cases 2 Proof Sketch 3 Conclusion 4 Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 2 / 19

Problem Definition and Previous Work Outline Problem Definition and Previous Work 1 Tractable Cases 2 Proof Sketch 3 Conclusion 4 Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 3 / 19

Problem Definition and Previous Work Max CSP Definition ( CSP ( Γ ) ) Let Γ (the constraint language ) be a finite set of relations over a finite domain D . Instance: I = ( V , D , C ) , where V is a finite set of variables , and C is a finite set of constraints (( v 1 , . . . , v k ) , R ) , R ∈ Γ is a k -ary relation and v i ∈ V . Goal: Find an assignment σ : V → D such that ( σ ( v 1 ) , . . . , σ ( v k )) ∈ R , for every (( v 1 , . . . , v k ) , R ) ∈ C . Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 3 / 19

Problem Definition and Previous Work Max CSP Definition ( CSP ( Γ ) ) Let Γ (the constraint language ) be a finite set of relations over a finite domain D . Instance: I = ( V , D , C ) , where V is a finite set of variables , and C is a finite set of constraints (( v 1 , . . . , v k ) , R ) , R ∈ Γ is a k -ary relation and v i ∈ V . Goal: Find an assignment σ : V → D such that ( σ ( v 1 ) , . . . , σ ( v k )) ∈ R , for every (( v 1 , . . . , v k ) , R ) ∈ C . Definition ( Max CSP ( Γ ) ) Let Γ be a constraint language over a finite domain D . Instance: A CSP ( Γ ) -instance I = ( V , D , C ) . Goal: Find an assignment σ : V → D which maximises the number of satisfied constraints. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 3 / 19

Problem Definition and Previous Work CSP vs. Max CSP In general Max CSP ( Γ ) is harder than CSP ( Γ ) . Γ = {{ ( 0, 1 ) , ( 1, 0 ) }} CSP ( Γ ) is 2-colourability and Max CSP ( Γ ) is Max cut Γ = {{ ( a , b ) ∈ D 2 | a � = b }} ( k = | D | ) CSP ( Γ ) is k -colourability and Max CSP ( Γ ) is Max k -cut Γ = { x ∨ y , ¬ x ∨ y , ¬ x ∨ ¬ y } CSP ( Γ ) is 2- Sat and Max CSP ( Γ ) is Max 2-Sat Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 4 / 19

Problem Definition and Previous Work VCSP without mixed cost functions Definition Let Γ be a set of cost functions on D with values in Q ≥ 0 , and ∆ be a set of relations on D . (Weighted) VCSP ( Γ , ∆ ) is the following problem: Instance: A formal sum ∑ n i = 1 w i f i ( x i ) , and a finite set of constraints { ( y j , R j ) } , where w i ∈ Q ≥ 0 , f i ∈ Γ , R j ∈ ∆ , and x i , y j are matching lists of variables from V . Solution: A function σ : V → D such that σ ( y j ) ∈ R j for all j . Goal: Minimise ∑ n i = 1 w i f i ( σ ( x i )) . We write VCSP ( Γ ) when ∆ is empty. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 5 / 19

Problem Definition and Previous Work Min CSP using { 0, 1 } -valued functions Observation The problems Max CSP ( Γ ) and Min CSP ( {{ x �∈ R } | R ∈ Γ } ) are polynomial-time equivalent. (But the two problems may differ with respect to approximability.) In order to use the VCSP -framework, we have chosen to study Min CSP instead of Max CSP . Represent a k -ary relation R by its characteristic function f : D k → { 0, 1 } , with f ( x ) = 1 iff x ∈ R . Define Min CSP ( Γ ) ( Min CSP ( Γ , ∆ ) ) to be VCSP ( Γ ) ( VCSP ( Γ , ∆ ) ), when Γ consists of { 0, 1 } -valued functions only. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 6 / 19

Problem Definition and Previous Work Endomorphisms and cores for Min CSP Let Γ be a finite set of { 0, 1 } -valued cost functions. f : D → D is an endomorphism of Γ if h ( a ) = 0 = ⇒ h ( f ( a )) = 0 for every h ∈ Γ and tuple a ∈ D k . Γ is called a core if all of its endomorphisms are surjective. Lemma Let f be an endomorphism of Γ , D ′ = f ( D ) , and Γ ′ = { h | D ′ | h ∈ Γ } . Then Min CSP ( Γ ) and Min CSP ( Γ ′ ) are polynomial-time equivalent. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 7 / 19

Problem Definition and Previous Work Submodularity Definition A cost function h : D k → Q ≥ 0 is submodular w.r.t. a total order < on D if h ( a ) + h ( b ) ≥ h ( min ( a , b )) + h ( max ( a , b )) , for all a , b ∈ D k . min and max are taken w.r.t. to < and applied componentwise. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 8 / 19

Problem Definition and Previous Work Submodularity over arbitrary lattices Let ≤ be a partial order on D with a greatest lower bound ∧ ( meet ) and a least upper bound ∨ ( join ) for every pair of elements. We say that ( D ; ∧ , ∨ ) is a lattice. ( D ; min, max ) is called a chain on D and is a special case of a lattice . Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 9 / 19

Problem Definition and Previous Work Submodularity over arbitrary lattices Let ≤ be a partial order on D with a greatest lower bound ∧ ( meet ) and a least upper bound ∨ ( join ) for every pair of elements. We say that ( D ; ∧ , ∨ ) is a lattice. ( D ; min, max ) is called a chain on D and is a special case of a lattice . Definition A cost function h : D k → Q ≥ 0 is submodular w.r.t. a lattice ( D ; ∧ , ∨ ) if h ( a ) + h ( b ) ≥ h ( a ∧ b ) + h ( a ∨ b ) , for all a , b ∈ D k . ∧ and ∨ are applied componentwise. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 9 / 19

Problem Definition and Previous Work Submodularity with respect to distributive lattices A lattice ( D ; ∧ , ∨ ) is distributive if, for all x , y , z ∈ D , it satisfies x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) . Examples include chains (total orders) and products of chains, e.g., Theorem (Schrijver (2000); Iwata, Fleischer, and Fujishige (2001)) If Γ is submodular w.r.t. a distributive lattice, then VCSP ( Γ ) is tractable. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 10 / 19

Problem Definition and Previous Work Known classification results Full classifications of Min CSP ( Γ ) exist in the following cases: 2-element domains; Creignou (1995) 3-element domains; Jonsson, Klasson, and Krokhin (2006) Γ containing a single function; Jonsson and Krokhin (2007) Γ containing all unary functions; Deineko, Jonsson, Klasson, and Krokhin (2008) In each of these cases, provided that Γ is a core, Min CSP ( Γ ) is tractable if and only if Γ is submodular with respect to some chain on D . Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 11 / 19

Problem Definition and Previous Work Known classification results Full classifications of Min CSP ( Γ ) exist in the following cases: 2-element domains; Creignou (1995) 3-element domains; Jonsson, Klasson, and Krokhin (2006) Γ containing a single function; Jonsson and Krokhin (2007) Γ containing all unary functions; Deineko, Jonsson, Klasson, and Krokhin (2008) In each of these cases, provided that Γ is a core, Min CSP ( Γ ) is tractable if and only if Γ is submodular with respect to some chain on D . Min CSP ( Γ ) is also tractable when Γ is submodular with respect to any distributive lattice. Γ is submodular with respect to certain non-distributive lattices; Krokhin and Larose (2007), Kuivinen (2009) Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 11 / 19

Problem Definition and Previous Work Beyond submodularity Given these results, a tentative conjecture has been that, for a core Γ , Min CSP ( Γ ) is tractable when Γ is submodular with respect to some lattice, and this is the only source of tractability for Min CSP ( Γ ) . Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 12 / 19

Problem Definition and Previous Work Beyond submodularity Given these results, a tentative conjecture has been that, for a core Γ , Min CSP ( Γ ) is tractable when Γ is submodular with respect to some lattice, and this is the only source of tractability for Min CSP ( Γ ) . Our main result is a classification for the case when | D | = 4 which shows that the second part is false. In particular,             0 1 0 0 0 0 1 1     0 1 0 0 0 1 0 1             Γ =  ,  ,  ,  , .           1 1 1 1 1 0 1 0           0 1 0 0 1 1 0 0   Γ is a core which is not submodular with respect to any lattice on its domain, but Min CSP ( Γ ) is tractable. Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 12 / 19

Tractable Cases Outline Problem Definition and Previous Work 1 Tractable Cases 2 Proof Sketch 3 Conclusion 4 Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 13 / 19