Counting Constraint Satisfaction Problems
This paper explores various counting problems, focusing on graph reliability under failure probabilities and the complexity of counting satisfiable structures in Boolean operations. Key themes include the characterization of #P-completeness, Turing reductions, and polynomial-time solvability related to constraint satisfaction problems (CSPs). Notable results are derived from bounded treewidth parameters and restrictions in various algebraic structures. The study also highlights the implications of polymorphisms on tractability and presents specific algorithms for linear equations and H-coloring.
Counting Constraint Satisfaction Problems
E N D
Presentation Transcript
Counting Constraint Satisfaction Problems Andrei A. Bulatov Simon Fraser University Dagstuhl 2009
Counting Problems: Graph Reliability s t Every connection fails with probability ½. What is the probability that the two distinguished nodes are connected? # subgraphs in which s and t are connected Prob = # all subgraphs
Counting Problems: Fourier Coefficients Let be a Boolean operation and Fourier coefficient is given by Observe that computing reduces to counting the zeroes of
Counting Problems Let L be a problem from NP, and M an NTM solving it. Then the corresponding counting problem is Instance: An instance I of L. Objective: How many accepting paths ofM(I)are there? M(I) This class of problems is called #P
Complexity Classes, Reductions Counting problems solvable in poly time: a subclass of FP Analog of NP: #P Problem A is Turing reducible to problem B if In other words, A can be solved in poly time using B as an oracle Problem A #P is #P-complete if every problem from #P is Turing reducible to A
Complexity Classes Another relevant complexity class: Reconsider the Graph Reliability problem: Every connection fails with probability 1/3. Let the graph have m edge Prob =
Left Side Restrictions #CSP(A,_) counting CSP with left side restricted #CSP(B) or #CSP(B) counting CSP with right side restricted Theorem (Dalmau, Jonsson, 2004) Under certain assumptions from parametrized complexity, for a class of structures A of bounded arity, the problem #CSP(A,_) is solvable in polynomial time if and only if A has bounded treewidth.
#3-SAT = #CSP( ) #k-coloring = #CSP( ) Right Side Restrictions NP-hard (if k 3) (#P-complete) #Linear Equations = #CSP( F ) polytime, FP H is complete bipartite or complete with loops polytime, FP #H-Coloring = #CSP(H) #P-complete (Dyer, Greenhill; 2001) otherwise
More Results #CSP(B)B is a 2-element structure, is solvable in polynomial time if and only if every relation from B can be represented as the set of solutions to a system of linear equations over GF(2). Otherwise it is #P-complete. (N.Creignou, M.Herrman, 1996) #H-Coloring H is an acyclic digraph, is solvable in polynomial time if and only if H satisfies certain conditions; it is #P-complete otherwise (Dyer,Goldberg,Paterson, 2007 )
More Results (cntd) #CSP(S)S corresponds to equations over a finite semigroup, is either solvable in polynomial time, or #P-complete (O.Klima, B.Larose, P.Tesson, )
Two Algorithms: 1 #Linear Equations (over a field F) Algorithm - Find the dimensionality k of the solution space (Gaussian elimination) - Output
i ii I II Two Algorithms: 2 #H-ColoringH is the following digraph
i ii U 2 E 1 4 2 W I II Two Algorithms: 2
PP-Definitions Theorem (B., Dalmau; 2003) If structures B and C are such that every relation of C is pp-definable in B, then #CSP(C) is Turing reducible to #CSP(B)
a b c x a z b c y PP-Definitions (cntd) Running example: H = (V; E) Relational product E E(x,y) = zE(x,y) E(z,y)
a b c x y z PP-Definitions (cntd) Another example
CSP(B), CSP(C), and CSP( ) are poly-time reducible to CSP(A) #CSP(B), #CSP(C), and #CSP( ) are Turing reducible to #CSP(A) Algebraic Theory: Polymorphisms If Pol(A) Pol(B), then CSP(B) is poly-time reducible to CSP(A) If Pol(A) Pol(B), then #CSP(B) is Turing reducible to #CSP(A)
Algebraic Theory: Varieties If CSP(A) is tractable then, for any finite B var(A), the problemCSP(B) is tractable If #CSP(A) is tractable then, for any finite B var(A), the problem#CSP(B) is tractable If, for a finite B var(A), CSP(B) is NP-complete, then CSP(A) is NP-complete If, for a finite B var(A), #CSP(B) is #P-complete, then #CSP(A) is #P-complete
The idempotent reduct of A is the algebra where F is the set of all idempotent operations of A If A is surjective, then CSP(A) is poly-time equivalent to CSP() #CSP(A) is Turing equivalent to #CSP() Algebraic Theory: Idempotency An algebra is called surjective if all its operations are surjective An operation f is called idempotent if f(x,…,x) = x An algebra is idempotent if all its operations are idempotent CSP(A) is poly-time equivalent to CSP(B) for some surjective B
If the disequality relation (on a set with at least 3 elements), or the relation is pp-definable in B, then CSP(B) is NP-complete Bad Relations and Good Polymorphisms If a reflexive but not symmetric binary relation is pp-definable in B then #CSP(B) is NP-complete If B has no Mal’tsev polymorphismthen #CSP(B) is #P-complete A ternary operation m(x,y,z) is said to be Mal’tsev if m(x,y,y) = m(y,y,x) = x If B has no non-unary polymorphism then CSP(B) is NP-complete
Good Polymorphisms Lemma A Boolean structure has a Mal’tsev polymorphism if and only if every relation from B can be represented as the set of solutions to a system of linear equations over GF(2). Lemma An undirected graph has a Mal’tsev polymorphism if and only if is complete bipartite or complete with all loops
Beyond Mal’tsev easy hard
Congruences A congruence of a structure B is an equivalence relation pp-definable in B (x,y) = zE(x,z) E(y,z) (x,y) = zE(z,x) E(z,y)
Congruences and Matrices 1 1 2 1
Theorem (B., Grohe, 2005) If , are congruences of a structure B and is greater than the number of classes of then #CSP(B) is #P-complete Matrices and Complexity
Corollary If there is an algebra B var(A) and congruences , of B such that is greater than the number of classes of , then #CSP(A) is #P-complete Matrices and Complexity (cntd) We call an algebra Asingular if the above conditions are not true for any B var(A) and any pair of congruences
a b c x y z u w v Congruences of Relations A congruence of a k-ary relation R pp-definable in structure B is an equivalence relation on R, which when considered 2k-ary relation is pp-definable in B
Matrices for Relations Let R be a relation pp-definable in structure B, , and congruences of R such that , . Then M(R;,;) denotes the matrix whose rows are indexed with -classes, columns are indexed with -classes, and the number in the intersection of ith row and jth column is the number of -classes in the intersection of ith class of and jth class of
Matrices of Relations (cntd) is the projectioncongruence: (x,y,z) and (u,v,w) are in if and only if x = u is the equality relation
Singularity Theorem If ,, are congruences of a relation R pp-definable in structure B such that , , and rank(M(R;,;)) is greater than the number of classes of then #CSP(B) is #P-complete We call a structure singular if the above conditions are not true for any pp-definable relation and any triple of congruences
Dichotomy Theorem #CSP(B) is solvable in polynomial time if and only if B is singular Otherwise it is #P-complete Theorem #CSP(A) is solvable in polynomial time if and only if A is singular Otherwise it is #P-complete
Find the number of homomorphisms recursively, modulo Algorithm More complicated case: H Con(H): Con(H):
Weighted #CSP Weights of a relational structure B with base set B: - vertex weight : B N - edge weight that maps every tuple from each relation of B to a natural (integer, rational, real, complex) number Given a structure A the weight of a homomorphism is computed as Then WCSP(B,,)
Weighted Dichotomy Theorem WCSP(B,,), where and are non-negative rational functions, is either solvable in polynomial time or #P-hard
B B 2 1 2 Getting Rid of Vertex Weights B Given a structure A, every homomorphism : A B gives rise to homomorphisms from A to We reduce WCSP(B,,) to WCSP (B ,,*)
a B b c Getting Rid of Edge Weights vertices: B ab ca 2 2 ba ac B B 2 bc cb relations: 1-1, 1-2, 2-1, 2-2 pairs x and y are in i-j iff i-th component of x equals j-th component of y weights: vertex weight of a vertex of B equals its edge weight in B
t w u v We reduce WCSP(B,,*) to WCSP( B ,*,) Edge Weights: Reduction 2-1 ut tv 2-1 A vw 2-2 1-1 2-1 uv
Open Problems • Can we decide, given a structure A, or an algebra A, if #CSP(A) • (resp., #CSP(A) ) is solvable in poly time? 2. Do something with negative, irrational, etc. weights
