A constraint programming approach to subgraph isomorphism

ions for sets. Filtering and propagators A simple algorithm exists to find a solution to a CSP. A straightforward idea is to enumerate all assignments of the variables with respect to their domains, and check if the constraints are satisfied. This strategy is correct, but the time to find a solution is exponential. Individual constraints can be used to remove values. Values in the domain of a CSP may violate some constraints of the CSP. Filtering removes values in the domain of D that have no support in a constraint c. A value d in D(x) has a support in c with respect to D if there exists an assignment A of the variables in D with x = d and A ⊆ c. The value d is also said to be consistent with the constraint c. A propagator is the implementation of a constraint that performs filtering. A propagator takes a domain D as input and returns a new domain where inconsistent values with respect to c are removed. Comparing the output of a propagator with its constraint calls for notations. A constraint c is a set of tuples indexed by variables, and can be viewed as a domain of a CSP. Set operations can be extended to domains of CSPs. Given a set operation ⋄, D1⋄D2 means D1(xi)⋄D2(xi) for all xi ∈ X. With these notations, properties of a propagator can be stated. Let D1 and D2 be two finite domains defined on the same set of vars. A propagator pc implementing a constraint c must respect the following conditions : • contractant: pc(D1) ⊆ D1 • monotone: D1 ⊆ D2 ⇒ pc(D1) ⊆ pc(D2) • correct: Sol(X,D, {c}) ⊆ Sol(X, pc(D), {c}) Successive applications of a propagator may narrow domains, that is p(p(D)) ⊆ p(D). A propagator p is said to be idempotent if p(p(D)) = p(D). This is important in the filtering process, as idempotent propagators need to be applied only once. Levels of consistency Propagators should remove inconsistent values with respect to a constraint. Doing so can however be NP-Complete. Moreover a polynomial optimal filtering algorithm is not always useful. This is the case, for example, when the density of solutions is high. 22 State of the Art of Exact Subgraph Isomorphism Removing values may be expensive, while enumeration leads quickly to a solution. This motivates the introduction of levels of consistency. The filtering performed by a propagator p for a constraint c may vary. We enumerate here a number of definitions of consistency, from the cheaper to the more expensive in time complexity, but with increasing filtering power. • checking: a propagator p is checking for a constraint c if (∀ x ∈ scope(c) : |D(x)| = 1) ⇒ ×xi∈XD(xi) ⊆ c. Intuitively, the propagator waits for the instantiation of all the variables in its scope, and check if the constraint c is verified for this assignment. • forward checking: a propagator is forward checking for a constraint c if all variables in scope(c) are instantiated but the variable xi and ∀ vi ∈ D(xi) : (v1, . . . , vi−1, vi, vi+1, . . . , vk) ∈ c. • (hyper) arc consistency: A propagator is hyper arc consistent with respect to a constraint c with scope(c) = x1, . . . , xk if ∀ 1 ≤ i ≤ k : ∀ vi ∈ Di, ∃ v1, . . . , vi−1, vi+1, . . . , vk ∈ D1 × · · · × Dk such that (v1, . . . , vk) ∈ c. For a binary constraint, it is called arc consistency, and for a unary constraint, node consistency. • (hyper) bound consistency: The idea behind bound consistency is to consider the bounds of the domains. In total ordered finite domains, there exits a minD(x) and a maxD(x) elements, and we may prune only on those bounds. This idea leads to several definitions of bound consistency. – bound consistency: A propagator is bound consistent with respect to a constraint c with scope(c) = x1, . . . , xk if ∀ 1 ≤ i ≤ k : ∃ v1, . . . , vi−1, vi+1, . . . , vk ∈ [minD1, maxD1] × · · · × [minDk, maxDk] such that (v1, . . . , vi−1, minDi, vi+1, . . . , vk) ∈ c and (v1, . . . , vi−1, maxDi, vi+1, . . . , vk) ∈ c. – range bound consistency: A propagator is range bound consistent with respect to a constraint c with scope(c) = x1, . . . , xk if ∀ 1 ≤ i ≤ k ∀ vi ∈ [minDi, maxDi] ∃ v1, . . . , vi−1, vi+1, . . . , vk ∈ [minD1, maxD1] × · · · × [minDk, maxDk] such that (v1, . . . , vi−1, vi, vi+1, . . . , vk) ∈ c. – domain bound consistent: A propagator is domain bound consistent with respect to a constraint c with scope(c) =