The Polytope of Tree-Structured Binary Constraint Satisfaction Problems

We correct a result that we recently published in this conference series on the polytope of Binary Constraint Problems (BCPs). We had claimed that the so-called ”support formulation” would characterize the convex hull of all feasible solutions to tree-structured BCPs. We show that this claim is not accurate by providing a small counter example. We then show that the respective polytope defines a facet of the stable-set polytope of a perfect graph which allows us to perform LP inference in polynomial time. 1 Binary Constraint Satisfaction Definition 1 (Binary Constraint Satisfaction Problem). – A binary constraint problem (BCP) is a triplet 〈V,D,C〉, where V = {X1, . . . , Xn} denotes the finite set of variables, D = {D1, . . . , Dn} denotes a set of n finite sets of possible values for these variables (Di is called the domain of variables Xi), and C = {C1, . . . , Cm} is the set of constraints, where Cj : Dj1 × Dj2 → Bool specifies which simultaneous assignments of values to the variables Xj1 and Xj2 are allowed. The set {Xj1 , Xj2} is called the scope of constraint Cj . – An assignment for a BCP P = 〈V,D,C〉 is a function σ : V → ⋃ i≤n Di. A solution to a BCP P = 〈V,D,C〉 is an assignment σ such that σ(Xi) ∈ Di for all 1 ≤ i ≤ n and such that Cj(σ(Xj1), σ(Xj2)) = true for all 1 ≤ j ≤ m. The set of all solutions to a BCP P is denoted by Sol(P). Note how, in contrast to the custom in integer programming, in CP the term “binary” is used to express that all constraints affect just two variables, while the size of the domain of each variable is not limited! The fact that the arity of the constraints is limited to two allows us to state constraints simply as sets of allowed pairs Rj1,j2 = {(k, l) | Xj1 = k, Xj2 = l ok}, or, alternatively, as sets of forbidden pairs Rj1,j2 = {(k, l) | Xj1 = k, Xj2 = l forbidden}. It is easy to see that the general BCP is NP-hard. One simple way is to reduce from graph coloring where each node is modeled as a variable that must be assigned a color such that adjacent nodes are not colored identically (i.e., the corresponding constraint on each edge {i, j} is a not-equal constraint Ri,j = {(k, k) | ∀ k}). Conversely, every binary constraint problem can be visualized ⋆ This work was supported by the National Science Foundation through the Career: Cornflower Project (award number 0644113). as a constraint network where each node corresponds to a variable and an edge connects two nodes iff there exists a constraint over the corresponding variables. Of course, the exact semantic of the constraints is lost in that visualization. However, it is a well-known fact that any BCP whose corresponding constraint network is a tree can be solved in polynomial time [4, 5]. 2 The Support Formulation In [1], we devise an IP model for BCPs by using linear constraints to specify that, when a variable Xj1 takes value k, variable Xj2 must take a value that is consistent with Xj1 = k. In that way, we enforce that each variable assignment is supported by a correct assignment to adjacent variables (by which we mean variables that share a constraint). The IP then reads:1 SIP = max X