Counting Quantifiers, Subset Surjective Functions, and Counting CSPs

Transcription

1 Counting Quantifiers, Subset Surjective Functions, and Counting CSPs Andrei A. Bulatov, Amir Hedayaty Simon Fraser University ISMVL 2012, Victoria, BC

2 2/19 Clones, Galois Correspondences, and CSPs Clones have been studied for ages Ivo s favorite! Recently found a nice application CSPs

3 3/19 Constraint Satisfaction Problem Decision: Given a conjunctive formula R decide if it is satisfiable 1( 2 x x, y) R ( z, x, ) K Counting: Given a conjunctive formula R 1( 2 x x, y) R ( z, x, ) K find the number of satisfying assignments If all relations are from set Γ, the problem is denoted CSP(Γ), #CSP(Γ)

4 4/19 CSPs: Example, Graph Coloring k-colorability, #k-colorability #k-colorability = #CSP( ), where is on a k-element set z x y t G v u #? K k (x y) (y z) (z x) (y t) (z u) (t v) (u v) (z t) (x v)

5 5/19 Approximate Counting Algorithm Alg is an ε-approximation algorithm for a counting CSP if it is polynomial time and for any instance P of the problem it outputs number Alg(P) such that # P Alg( P) # P < ε Alg is said to be an FPRAS if take P and ε as input and returns number Alg(P) satisfying the condition above in time polynomial in P and 1/ε

6 6/19 Boolean Counting and Approximation Theorem (Creignou, Herrmann, 1996) Let Γ be a constraint language over {0,1}. Then either #CSP(Γ) is solvable in polynomial time; or it is as hard as #SAT. Theorem (Dyer,Goldberg,Jerrum, 2007) Let Γ be a constraint language over {0,1}. Then either #CSP(Γ) is solvable in polynomial time; or it is as hard to approximate as #BIS; or it is as hard to approximate as #SAT.

7 7/19 More Easy Problems There are other easy problems H: #CSP(H) #Match (the # of matchings in a graph) Jerrum, Sinclair, 1996 #DNF (the # of sat. assignments of a DNF) Karp, Luby, Madras, 1989

8 8/19 PP-Definitions and CSP Reductions The co-clone Γ of a constraint language Γ consists of all relations pp-defined in Γ that is can be expressed using: Relations in Γ The equality relation over the domain of Γ Conjunction Existential quantification Jeavons, 1998: For a language Γ, and any finite, if Γ then CSP( ) is polynomial-time reducible to CSP(Γ)

9 9/19 More Definitions More Reductions Dalmau, B., 2003: If Γ then #CSP( ) is polynomial-time reducible to #CSP(Γ) Borner, et al., 2005: If is definable in Γ with universal quantifiers allowed then quantified CSP( ) is polynomialtime reducible to quantified CSP(Γ) B., Hedayaty, 2009: If is definable in Γ with quantifiers forbidden then quantified #CSP( ) is AP-reducible to quantified #CSP(Γ)

10 10/19 Invariance Definition A relation R is invariant under a k-ary operation f, if, for any tuples a 1,a 2,,a k R, the tuple obtained by applying f co-ordinatewise is a member of R. If R is invariant under f, then f is called a polymorphism of R. Pol(Γ) denotes the set of all polymorphisms of Γ Inv(C) denotes the set of all invariants of C PPol(Γ) denotes the set of all partial polymorphisms of Γ CPol(Γ) denotes the set of all surjective polymorphisms of Γ

11 11/19 Galois Correspondences BKKR, Geiger: For any constraint language Γ, over a finite domain, Γ = Inv(Pol(Γ)) Boerner et al.: Closure wrt pp-definitions + universal quantifiers gives Inv(CPol(Γ)) Fleischner, Rosenberg: Closure wrt pp-definitions existential quantifiers gives Inv(PPol(Γ))

12 12/19 AP-Reductions Algorithm Alg is an AP-reduction from problem A to problem B, if takes (P,ε) as input; uses A as an oracle; polynomial time; and satisfies the requirements of FPRAS whenever the oracle A satisfies the requirements of FPRAS.

13 13/19 Some AP-Reductions q q nq Every hom from the original instance extends to 3 if ( n )q mapped into the triangle, and to at most 3 1 otherwise n The total # of mappings is at most 6, so choosing q such n ( n 1) q nq that 6 3 << 3 we make the contribution of homs outside the triangle very small

14 14/19 Max-Quantifiers and AP-Reductions Let Φ( x1, K, xn, y1, K, ym ) be a conjunctive formula in language Γ, and for an assignment ϕ to x 1, K, x n let #ϕ denote the number of extensions of ϕ that satisfy Φ. Let M be the maximum of these values. We say that Φ is a max-implementation of R if ϕ ( x r ) R # ϕ = M Theorem If there is a max-implementation of R in Γ, then #CSP(Γ Γ {R}) is AP-reducible to #CSP(Γ)

15 15/19 Max-Co-Clones R( x, K, xn ) = max y1, K, ymφ( x1, K, xn, y1, K, y 1 m ) Problem: Find a Galois correspondence for max-co-clones

16 16/19 k-existential Quantifiers If Φ is a formula with free variables x 1, K, x n and y, a tuple, satisfies a 1 K,a n Ψ( x1, K, xn ) = k yφ( x1, K, xn, y) iff Φ( a 1, K, a, b) is true for at least k values b. n 1 = If D = k then k =

17 k-existential Co-Clones 17/19

18 18/19 K-Surjective Functions A (partial) function f : D n D is said to be k-surjective if for any A1, K, An D with A = = A n = k the set f ( A 1, K, A n contains at least k elements (or is empty) 1 K ) A function k-surjective for all k K is called K-surjective For a constraint language Γ and a set K N, the set of all K-surjective polymorphisms of Γ is denoted by m(k)-pol(γ)

19 19/19 Galois Correspondence Theorem For any constraint language Γ, for any K N, Inv( m( K) Pol( Γ)) = Γ K

20 Happy Birthday, Ivo!!