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

Min CSP on Four Elements: Moving Beyond Submodularity

Peter Jonsson1, Fredrik Kuivinen, Johan Thapper2

1Department of Computer and Information Science (IDA)

Link¨

• ping University, Sweden

2Laboratoire 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

1

Problem Definition and Previous Work

2

Tractable Cases

3

Proof Sketch

4

Conclusion

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 2 / 19

Problem Definition and Previous Work

Outline

1

Problem Definition and Previous Work

2

Tractable Cases

3

Proof Sketch

4

Conclusion

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 ((v1, . . . , vk), R), R ∈ Γ is a k-ary relation and vi ∈ V . Goal: Find an assignment σ : V → D such that (σ(v1), . . . , σ(vk)) ∈ R, for every ((v1, . . . , vk), 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 ((v1, . . . , vk), R), R ∈ Γ is a k-ary relation and vi ∈ V . Goal: Find an assignment σ : V → D such that (σ(v1), . . . , σ(vk)) ∈ R, for every ((v1, . . . , vk), 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) ∈ D2 | 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

• f relations on D. (Weighted) VCSP(Γ, ∆) is the following problem:

Instance: A formal sum ∑n

i=1 wifi(xi), and a finite set of constraints

{(yj, Rj)}, where wi ∈ Q≥0, fi ∈ Γ, Rj ∈ ∆, and xi, yj are matching lists of variables from V . Solution: A function σ : V → D such that σ(yj) ∈ Rj for all j. Goal: Minimise ∑n

i=1 wifi(σ(xi)).

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 : Dk → {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 ∈ Dk. 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 : Dk → 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 ∈ Dk. 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 : Dk → 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 ∈ Dk. ∧ 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, Γ =            1 1 1 1 1 1 1     ,     1 1     ,     1 1     ,     1 1     ,     1 1            . 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

1

Problem Definition and Previous Work

2

Tractable Cases

3

Proof Sketch

4

Conclusion

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 13 / 19

Tractable Cases

Multimorphisms

Cohen, Cooper, Jeavons, and Krokhin (2006) introduced the first in a number of generalisations of polymorphisms for the purpose of investigating VCSP. Definition A pair of functions f , g : D2 → D is a (binary) multimorphism of a k-ary cost function h : Dk → Q≥0 if, for all tuples x, y ∈ Dk, h(x) + h(y) ≥ h(f (x, y)) + h(g(x, y)), where f and g are applied component-wise. It is a multimorphism of a valued constraint language if it is a multimorphism of every cost function of the language. If h is submodular w.r.t. some lattice (D; ∧, ∨), then the pair ∧, ∨ is a multimorphism of h.

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 13 / 19

Tractable Cases

1-defect chains

Definition Let (D; <) be a chain, and let b, c ∈ D be two distinct elements. Assume that f , g : D2 → D are two commutative operations such that: {x, y} = {b, c} = ⇒ f (x, y) = min<(x, y), g(x, y) = max<(x, y); {f (b, c), g(b, c)} ∩ {b, c} = ∅; and f (b, c) < g(b, c). We call (D; f , g) a 1-defect chain, and we say that (f , g) is a 1-defect chain multimorphism. f (b, c) g(b, c) b c f (b, c) b c g(b, c) g(b, c) f (b, c) b c a b c d

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 14 / 19

Tractable Cases

Tractability for 1-defect chain multimorphisms

Theorem If Γ has a 1-defect chain multimorphism, then VCSP(Γ) is tractable.

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 15 / 19

Tractable Cases

Tractability for 1-defect chain multimorphisms

Theorem If Γ has a 1-defect chain multimorphism, then VCSP(Γ) is tractable. D = {a, b, c, d} Γ =            1 1 1 1 1 1 1     ,     1 1     ,     1 1     ,     1 1     ,     1 1            . Γ is a core which is not submodular with respect to any lattice on D, but it has the following 1-defect chain multimorphisms. f (b, c) = a g(b, c) = d b c g(b, c) = d f (b, c) = a b c

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 15 / 19

Proof Sketch

Outline

1

Problem Definition and Previous Work

2

Tractable Cases

3

Proof Sketch

4

Conclusion

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 16 / 19

Proof Sketch

Expressive power, cores, and constants

A function h : Dk → Q≥0 is expressible over (Γ, ∆) if it can be obtained by (weighted) sums of costs functions from (Γ, ∆) and minimisation over variables. Let Γ, ∆fn denote the set of all such functions; the expressive power. Proposition Let Γ′ ⊆ Γ, ∆fn be a finite subset. Then VCSP(Γ′, ∆) is polynomial-time reducible to VCSP(Γ, ∆). Let CD = {{d} | d ∈ D}. Proposition Let Γ be a core over D. Then Min CSP(Γ, CD) is polynomial-time reducible to Min CSP(Γ).

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 16 / 19

Proof Sketch

Proof sketch

The proof proceeds as follows. WLOG, we can look at Min CSP(Γ, CD).

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 17 / 19

Proof Sketch

Proof sketch

The proof proceeds as follows. WLOG, we can look at Min CSP(Γ, CD). We define a graph G which encodes the binary multimorphisms of the binary functions in Γ, CDfn in certain independent sets. (This is inspired by a construction due to Kolmogorov and ˇ Zivn´ y (2010) for conservative finite-valued VCSP.)

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 17 / 19

Proof Sketch

Proof sketch

The proof proceeds as follows. WLOG, we can look at Min CSP(Γ, CD). We define a graph G which encodes the binary multimorphisms of the binary functions in Γ, CDfn in certain independent sets. (This is inspired by a construction due to Kolmogorov and ˇ Zivn´ y (2010) for conservative finite-valued VCSP.) From certain properties of this graph, it is easy to determine whether there is a reduction from Max cut.

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 17 / 19

Proof Sketch

Proof sketch

The proof proceeds as follows. WLOG, we can look at Min CSP(Γ, CD). We define a graph G which encodes the binary multimorphisms of the binary functions in Γ, CDfn in certain independent sets. (This is inspired by a construction due to Kolmogorov and ˇ Zivn´ y (2010) for conservative finite-valued VCSP.) From certain properties of this graph, it is easy to determine whether there is a reduction from Max cut. Otherwise we can argue, using the absence of these properties, that the binary functions in Γ, CDfn have one of a small number of binary mulitmorphisms which are known to imply tractability.

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 17 / 19

Proof Sketch

Proof sketch

The proof proceeds as follows. WLOG, we can look at Min CSP(Γ, CD). We define a graph G which encodes the binary multimorphisms of the binary functions in Γ, CDfn in certain independent sets. (This is inspired by a construction due to Kolmogorov and ˇ Zivn´ y (2010) for conservative finite-valued VCSP.) From certain properties of this graph, it is easy to determine whether there is a reduction from Max cut. Otherwise we can argue, using the absence of these properties, that the binary functions in Γ, CDfn have one of a small number of binary mulitmorphisms which are known to imply tractability. Finally, we need to show that these multimorphisms are in fact also multimorphisms of all higher-arity functions of Γ.

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 17 / 19

Proof Sketch

Main result

To summarise, we have the following classification. Theorem Let Γ be a core with domain D = {a, b, c, d}. If Γ is submodular w.r.t. a chain on D; if Γ has a 1-defect chain multimorphism; or if Γ is submodular w.r.t. a lattice isomorphic to , then Min CSP(Γ) is tractable. Otherwise Min CSP(Γ) is NP-hard.

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 18 / 19

Conclusion

Outline

1

Problem Definition and Previous Work

2

Tractable Cases

3

Proof Sketch

4

Conclusion

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 19 / 19

Conclusion

Conclusion

We have shown that for a core Γ over a 4-element domain, the problem Min CSP(Γ) is either tractable or NP-hard. An immediate consequence of our result is that submodularity is not the only source of tractability for Min CSP(Γ). This opens up several new directions for the complexity study of Min CSP. Our proof is not computer-assisted which has often been the case in previous classifications. The trade-off is that we cannot show APX-hardness (for Max CSP).

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 19 / 19

Conclusion

Conclusion

We have shown that for a core Γ over a 4-element domain, the problem Min CSP(Γ) is either tractable or NP-hard. An immediate consequence of our result is that submodularity is not the only source of tractability for Min CSP(Γ). This opens up several new directions for the complexity study of Min CSP. Our proof is not computer-assisted which has often been the case in previous classifications. The trade-off is that we cannot show APX-hardness (for Max CSP). Thank you for your attention!

Johan Thapper (LIX) Min CSP on Four Elements CP 2011 (Perugia) 19 / 19