#2SAT is a classical #P-complete problem. We present here, a novel algorithm for given a 2-CF §, to build a minimum spanning tree for its constraint graph G§ assuming dynamic weights on the edges of the input graph.
Counting combinatorial objects over graphs has been an interesting and important area of research in Mathematics, Physics, and Computer Sciences. Counting problems, being mathematically interesting by themselves, are closely related to important practical problems. For instance, reliability issues are often equivalent to counting problems [2, 3, 5].
The techniques for building minimum spanning trees have been developed assuming static weights on the edges of the graph. But for the #2SAT problem [4, 6], instead of static weights we consider dynamic weights determined by the signs of the edges, as well as the number of partial models associated with the nodes. We address the construction of the minimum spanning tree of a constraint graph considering such dynamic weights. Let § be a 2-CF, the constraint graph of § is the undirected graph G§ = (V (§); E(§)), with V (§) = À(§) and E(§) = ffÀ(x); À(y)g : fx; yg 2 §g, i.e. the vertices of G§ are the variables of §, and for each clause fx; yg in § there is an edge fÀ(x); À(y)g 2 E(§). Each edge has associated an ordered pair (s1; s2) of signs assigned as labels. For example, the signs s1 and s2 for the clause fx _ yg are related to the signs of the literals x and y respectively, then s1 = ¡ and s2 = + and the edge is denoted as: x ¡ + y which is equivalent to y + ¡ x.
A connected component of G is a maximal induced subgraph of G. We say that the set of connected components of § are the subformulas corresponding to the connected components of G§ . From now on, when we mention a 2-CF §, we assume that § is a connected component graph. 3.
Linear procedures for #2SAT We present here, some procedures for computing the number of models of a formula for basic topologies of a graph.
Procedure A: If § is a path:
The ¯rst pair (®0; ¯0) is (
(®i; ¯i) =
8(¯i¡1 ;¹i¡1 ) if (²i; ±i) = (0; 0)
>
><(¹i¡1 ;¯i¡1 ) if (²i; ±i) = (0; 1)
>(®i¡1;¹i¡1 ) if (²i; ±i) = (1; 0)
>:(¹i¡1 ;®i¡1) if (²i; ±i) = (1; 1)
(
Let us consider the class P of Boolean functions where their constraint graphs are paths. For example, ¯gures (1a) and (1b) represent Boolean path functions. We denote as Pn+ a monotone path with n nodes and where each variable appears with the same sign (like in (1a)), while Pn¡+ is the n path where the occurrences of the variables are with complementary signs (like in (1b)), we call to this last Boolean formula a changing sign path. Let #+ : IN ! IN be the function which for a n 2 IN , it considers a monotone path Pn+ 2 P holding that #+(n) = #SAT (Pn+). We can estimate the rate of growth of #+ since it corresponds with the growth of #SAT (Pn+) when the number of variables n growth. As #SAT (Pn+) 2 O(Án) with Á = ³ 1+2p5 ´ ¼ 1:618 [1], then, #+(n) 2 O ³³ 1+2p5 ´n´. Let #¡+ : IN ! IN be a function which for a n 2 IN it considers a changing sign path Pn¡+ 2 P, and #¡+ holds that #¡+(n) = #SAT (Pn¡+). Thus, the rate of growth of the function #¡+ is the same that the growth of #SAT (Pn¡+) when the number of variables n growth, and as #SAT (Pn¡+) 2 O(n) [1], then #¡+ 2 O(n). Comparing the rates of growth for P + and P ¡+, we obtain O(#SAT (Pn+)) > O(#SAT (Pn¡+)).
Procedure B: If § is a tree: Let § be a 2-CF where its associated constraint graph G§ is a tree. We denote with (®v; ¯v) the pair associated with the node v (v 2 G§ ). We compute #SAT (§) considering the methodology used in [1].
Example 2 Let B7+ a monotone balanced tree, like in (2a). And B+¡ the binary
7
balanced tree where each internal node has complementary signs on its incident
child-edges. Applying Count M odels f or trees to both trees, it starts
assigning the pair (
(
+
(
- (
(
+
(
(8,8) = 16 b)
(
(
(
(
For monotone Boolean formulas whose constraint graph is a balanced binary tree, like in example 2, we can express how the number of models changes according with the number of nodes of the tree. E.g. for a monotone binary tree with 3 nodes (®3; ¯3) = (22; 12) = (4; 1). The following balanced binary tree has 7 nodes and (®7; ¯7) = (52; 42) = (25; 16). The following tree has 15 nodes and (®15; ¯15) = (412; 252) = (1681; 625). In general, we obtain the following recurrence relation for the balanced monotone binary trees Bn+ with n nodes: #SAT(Bn+) = Fn2=2 + Fn4=4 being Fi the i-th Fibonacci number. This recurrence relation has an order of growth of O(2(3=4)n).
On the other hand, the upper bound for the class of binary formulas where there are complementary signs on the child-edges on each internal node, tree denoted as Bn+¡ is given by #SAT (Bn+¡) 2 O(2 n2 ). The previous upper bounds establish a hierarchy for #SAT according to the topology of the constraint graphs, given as: O(#SAT (Pn¡+)) · O(#SAT (Bn+¡)) · O(#SAT (Pn+)) · O(#SAT (Bn+)). 4.
Building the Minimum Spanning Tree Let A1; : : : ; An be the sequence of initial charges obtained by the procedures (A) or (B) for computing #SAT(§). Now, we build a new sequence of pairs which represent the ¯nal charges (or just the charges) Bn; : : : ; B1, being Bi the charge of the variable xi 2 À(§), computed as:
Bn = An
Bn¡i = balance(An¡i; Bn¡i+1); i = 1; :::; n ¡ 1
(
In the case that the coe±cients P0 or P1 are not integer numbers, the following formulas are applied for computing the charge of x.
ax = (by ¡ ay); bx = (¹y ¡ ax) if(s1; s2) = (¡; ¡) ax = (ay ¡ by); bx = (¹y ¡ ax) if(s1; s2) = (¡; +) bx = (by ¡ ay) ; ax = (¹y ¡ bx) if(s1; s2) = (+; ¡) bx = (ay ¡ by) ; ax = (¹y ¡ bx) if(s1; s2) = (+; +)
Note that the essence of the rules in balance consists in applying the inverse
operation utilized via recurrence (
A Procedure for Building Spanning Trees for #2SAT Given a 2-CF § and its constraint graph G§, a minimum spanning tree, denoted by T§, is a tree containing all vertices of G§ and such that #SAT (A§) ¸ #SAT (§) and #SAT (A§) is minimum into the set of all spanning trees of G§.
When G§ contains cycles, our proposal works like the well known Kruskal's
algorithm. An initial spanning tree A§ = (V (G§); P Edges) is formed by all
vertices of G§ because all vertices are connected components by themselves, and
all pendant edge of G are edges of the spanning tree (if there are not pendant
edges then an empty set is initially assigned to P Edges).
(
X1
X2 (
The ¯rst step of our procedure consists in detecting the potential edges which could generate change of signs when a node will be visited,(see ¯g. 3b). In each step, each edge with one of its end-points in a node of A§ and the other endpoint in a node not included in A§ . While the procedure detects an edge e 2 E which generate change of signs on the incident edges of a same node, such edge e is added to A§ . When there are more than one edge E generating change of signs or there are not such edges the procedure reviews how increase the number of models when an edge e 2 E would be added to A§ , for this in each node an inverse counting is calculated ,(see ¯g. 3c). The resulting spanning tree (see ¯g. 3d) with a total number of models of 7=6+1.
The edge e 2 E which infers a minimal increment on #SAT(A§ ) with respect to any other edge in E, is selected to be added to A§ . Notice that the increment on the number of models depends mainly of the signs of e as well as the charge of the two-endpoints of e. There are a set of strategies used for detecting the edges in (E(G§ ) ¡ E(A§ )) which infer minimal increment on #SAT(A§ ). Such strategies are: In general if two edges e1 and e2 generate the same increment on the number of models of A§ e1 is preferred over e2 if e1 could bring about a change of signs, in the following steps, on its incident node, if the edge e connects A§ with one extra-node and generates complementary signs on one of its endpoints, then e is an optimal selection, and it is preferable to obtain a path than a tree when the signs on the edges are the same. Let P Edges = fe 2 E(G§) : e is a pendant edge g; All Edges := E(G) ¡ P Edges; Cs := ;; fSet of potential edgesg A§ := (V (G); P Edges); fall pendant edges are part of the Treeg CandEdge = SelectC andidateEdges(All Edges) fSet of candidate semi-edges g F irstEdge = U nionC andidateEdges(CandEdge) flooking for a complete edgeg if F irstEdge <> ; then
E(A§) := E(A§) [ fF irstEdgeg; Initialize(V ect M odels; 1; 1) ;
Count M odels(A§; V ect M odels); InverseOrderCount(A§; V ect M odels); end if while (All Edges <> ;) do
Count M odels(All Edges; V ect M odels); fcount models by each potential edgeg Sel Edge = minfV ect M odelsg; fselect the edge which minimum incrementalg if jSel Edgej > 1 then
Cs = F ind(T est; A§); flooking for edges which could generate change of signsg T est = complete(Cs); fchoose edges generating change of sign on the nodesg Sel Edge = F irst(T est); fSelect an edge keeping potential change of signg end if All Edges := All Edges ¡ fSel Edgeg; E(A§) := E(A§) [ fSel Edgeg; All Edges := All Edges ¡ Edges Cycles(A§; All Edges); fdelete cyclesg end while 5.
We consider a new line of researching of building spanning trees with dynamic weights determined by the signs of the edges and the partial number of models associated with the endpoints of the edges. This consideration allows us, given a 2-CF §, to build its minimum spanning tree A§ such that #SAT(A§ ) is a minimum upper bound for #SAT(§). To build e±ciently the minimum spanning tree of a 2-CF is very helpful in the research for determining frontiers between e±cient and exponential counting procedures for the #2SAT problem.