On oblivious branching programs with bounded repetition that cannot efficiently compute CNFs of bounded treewidth
In this paper we study complexity of an extension of ordered binary decision diagrams (obdds) called -obdds on cnfs of bounded (primal graph) treewidth. In particular, we show that for each there is a class of cnfs of treewidth for which the equivalent -obdds are of size . Moreover, this lower bound holds if -obdd is non-deterministic and semantic. Our second result uses the above lower bound to separate the above model from sentential decision diagrams (sdds). In order to obtain the lower bound, we use a structural graph parameter called matching width. Our third result shows that matching width and pathwidth are linearly related.
Ordered Binary Decision Diagrams obdds is a famous representation of Boolean functions being actively investigated from both applied and theoretical perspective. The theoretical research, among other things, has resulted in many upper and lower bounds of obdd size realizing various classes of functions .
One such an upper bound, established in  states that a cnf of treewidth of its primal graph can be represented by an obdd of size . In terms of parameterized complexity, this is an xp upper bound, that is the degree of the polynomial depends on . A natural open question is whether this upper bound can be improved to an fpt upper bound, i.e. one of the form , where is a universal constant.
This question is of a particular interest in the area of knowledge compilation because of the recent introduction of Sentential Decision Diagrams (sdds)  for which an fpt upper bound does hold. sdds share with obdds a number of nice properties and have a good potential to replace obdds in applications. Yet obdd-related machinery is much more developed (one reason for that is that obdds have been investigated for a much longer time) and hence it is interesting to say if this gap between upper bounds can be significantly tightened by finding a better upper bound for obdds.
In , we answered this question negatively by demonstrating that for each there is a class of cnfs of primal graph treewidth at most for which the size of equivalent obdds is . In this paper, which can be considered as a follow-up version of , our motivation is to see how far the obdd can be extended so that the above lower bound would hold for that extended model in a way that the lower bound in  would follow as a special case. As a result, we extend obdds as follows. First, for an arbitrary (but fixed) constant we use -obdds instead obdd. That is, we allow each variable to occur at most times along each computational path, however the occurrences are ordered as concatenated copies of the same fixed permutation (in this setting the obdd is simply -obdd). Second, we allow the model to be non-deterministic. Roughly speaking, this means that instead of applying this restriction on a branching program, the restriction is applied on a switching and rectifier network. Third, we allow this restriction to be semantic, i.e to hold only for consistent paths that do not contain opposite occurrences of the same variables. The in-consistent paths are not constrained at all. We call the resulting model Nondeterministic Semantic -obdd and abbreviate it -nsobdd. In particular, we show that for each fixed there is a class of cnfs (in fact, the same class as we used in ) for which the smallest -nsobdd is of size . Clearly, the lower bound for obdds follows if we substitute .
The above lower bound shows that -nsobdds are inherently different from sdd with respect to representation of cnf of bounded treewidth. Our second result shows that this difference can, in fact, be turned into a (non-parameterized) separation. In particular by, essentially, setting to , we obtain a class of cnf that can be represented by polynomial size sdds but require -nsobdd of quasipolynomial size.
Our third result is related to the way the main lower bound is obtained. In particular, the cnfs we consider for the sake of obtaining lower bounds, correspond to undirected graphs. We introduce a graph parameter called matching width and show that the size of -nsobdd equivalent to the considered cnf is exponential in the matching width of the corresponding graph. Then we show that there are graphs for which the matching width is times larger than their treewidth. The lower bound readily follows from the combination of these results. The relationship between matching width and treewidth suggests that the former is similar to pathwdith. Our third result shows that this is indeed true, that is pathwidth and matching width are linearly related.
The last result might seem a little bit out of scope. The reason why we provide it in this paper is that matching width has already been used several time to obtain lower bounds [12, 11, 5]. So, it is interesting to see how it is connected to well known graph parameters. To the best of our knowledge  is the first paper where matching width for used for lower bounds, so a follow-up version of , seems the natural place for showing how matching width is connected to pathwidth.
Let us overview the related work. The -obdd have been considered in  with exponential lower bound provided for several functions. The -obdd model is known to be more powerful than the ordinary obdd. In particular, Theorem 7.2.2. of  provides a class of functions polynomial for -obdd and exponential even for Free Binary Decision Diagrams (fbdd) (that is, read-once branching programs). Moreover, it is known that increse of adds computational power. In particular, it has been demonstrated in  that for each there is a class of functions computable by poly-size -obdds and requiring exponential size -obdds. Interesting refinements of this hierarchy involving width of branching programs have been proposed in [1, 9].
It is also known that non-determinism adds power to obdd. In particular, Theorem 10.2.3. of  demonstrates a class of functions that can be computed by poly-size non-deterministic obdds, yet require exponential size fbdds. We are not aware of the any existing research specifically on non-deterministic -obdds. They are obviously a special case of non-deterministic read -times branching programs and hence exponential lower bounds (e.g. ) apply to them. It is well known that semantic rather than syntactic restriction adds a lot of power if the obliviousness requirement is dropped. In particular,  demonstrates a class of functions that can be computed by poly-size semantic non-deterministic read-once branching programs but require exponential size if ‘semantic’ is replaced by ‘syntactic’. In fact, no super-polynomial lower bound is known for the former. We are not aware, however, if the semantic restriction adds any power to non-deterministic obdds. The lower bound of  has been generalized in  to a different direction than the one considered in this paper: namely the obliviousness was dropped. In particular, it has been shown that the non FPT lower bound holds for non-deterministic read-once branching programs.
Matching width can be seen as a special case of maximum matching width introduced in  when the underlying tree is a caterpillar. It has been shown in  that maximum matching width is linearly related to the treewidth. The linear relationship between matching width and pathwdith, established in this paper, looks natural in this context.
The rest of the paper is structured as follows. Section 2 introduces the necessary background. Section 3 states the lower bound on -nsobdds along with the separation from sdd. The lower is proved in Section 4. The proof of linear relationship between matching width and pathwidth is provided in Section 5.
In this paper by a set of literals we mean one that does not contain an occurrence of a variable and its negation. For a set of literals we denote by the set of variables whose literals occur in . If is a Boolean function or its representation by a cnf or obdd, we denote by the set of variables of . A truth assignment to on which is true is called a satisfying assignment of . A set of literals represents the truth assignment to where variables occurring positively in (i.e. whose literals in are positive) are assigned with and the variables occurring negatively are assigned with .
A non-deterministic branching program is a directed acyclic graph dag with one root and one leaf . Some of the edges of are labelled with literals of variables. A path of is consistent if it does not have two edges labelled with opposite occurrences of the same variable. This gives us possibility to define , the set of literals labelling the edges of a consistent path . A consistent root-leaf path of is also called a computational path. The function computed by is defined as follows. Let be an assignment to the variables of . Then is a satisfying assignment of if and only if there is a computational path of such that .
Special classes of non-deterministic branching programs can be defined by putting restrictions on properties of their root-leaf paths. A restriction is semantic if it is applied to computational paths only and syntactic if it is applied to all the root-leaf paths. In order to define the restriction we use in this paper, we need an additional notation.
Let be a permutation of variables and let be a sequence of literals of (some) variables occurring in . We say that is ordered according to if for any two variables and occurring in , the occurrence of is ordered before the occurrence of in if and only if is ordered before in . For instance if then is ordered according to .
Let be paths of a directed graph . Then if is obtained by appending to the end of . For example, suppose that a path is represented by a sequence of its vertices and let . Then and also and also . This definition is naturally extended to a a decomposition of into an arbitrary number of paths.
We consider a class of non-deterministic branching programs for which there is a permutation of its variables and a constant such that each computational (that is, the restriction is semantic) path of can be represented as so that on each each variable occurs at most once and the sequence of literals labelling the edges along is ordered according to .
We call this class of branching programs Nondeterministic Semantic -obdd and abbreviate it -nsobdd. Notice that the ordering imposed on computational paths is more restrictive than the one imposed on read--times oblivious branching programs. Indeed, in the latter case, variables can occur along a path in an arbitrary (though the same for all paths) order. In our case, however, the order of occurrences is determined by concatenation of copies of the same permutation of variables.
Given a cnf , its primal graph has the set of vertices corresponding to the variables of . Two vertices are adjacent if and only if there is a clause of where the corresponding variables both occur.
Given a graph , its tree decomposition is a pair where is a tree and is a set of bags corresponding to the vertices of . Each is a subset of and the bags obey the rules of union (that is, ), containment (that is, for each there is such that ), and connectedness (that is for each , the set of all such that induces a subtree of ). The width of is the size of the largest bag minus one. The treewidth of is the smallest width of a tree decomposition of . If is a path then is a path decomposition The pathwidth of a graph is the smallest width of its path decomposition.
Figure 1 shows a graph and its tree decomposition. The width of this tree decomposition is since the size of the largest bag is .
3 Lower bound parameterized by treewidth
In this section, given two integers and , we define a class of cnfs, roughly speaking, based on complete binary trees of height where each node is associated with a clique of size . Then we prove that the treewidth of the primal graphs of cnfs of this class is linearly bounded by . Further on, we state the main technical theorem (proven in the next section) that claims that the smallest -nsobdd size for cnfs of this class exponentially depends on . Finally, we re-interpret this lower bound in terms of the number of variables and the treewidth to get the lower bound announced in the Introduction.
Let be a graph. A graph based cnf denoted by is defined as follows. The set of variables consists of variables for each and variables for each . The set of clauses consists of clauses for each . In other words, the variables of correspond to the vertices and edges of . The clauses correspond to the edges of .
Denote by a complete binary tree of height . Let be the graph obtained from by associating each vertex with a clique of size and, for each edge of , making all the vertices of the cliques associated with and mutually adjacent. Denote by .
Figure 2 shows and . To avoid shading the picture of with many edges, the cliques corresponding to the vertices of are marked by circles and the bold edges between the circles mean that that there are edges between all pairs of vertices of the corresponding cliques.
Let . Then the treewidth of the primal graph of is at most .
Proof. The primal graph of can be obtained from by adding one vertex for each edge of and making this vertex adjacent to the ends of . Let be a tree decomposition of of size at most . For each vertex , add a new vertex to adjacent to the vertex whose bag contains the ends of . Associate with a bag containing and the ends of . It is not hard to see that as a result we obtain a tree decomposition of the primal graph of . Also, as the size of each new bag is and , the width of the tree decomposition remains at most . So, it remains to show that the treewidth of is at most .
Consider the following tree decomposition of . is just . We look upon as a rooted tree, the centre of being the root. The bag of each node contains the clique of corresponding to . In addition, if is not the root vertex then also contains the clique corresponding to the parent of . Observe that satisfies the connectivity property. Indeed, each vertex appears in the bag corresponding to its ‘own’ clique and the cliques of its children. Clearly, the set of nodes corresponding to the bags induce a connected subgraph. The rest of the tree decomposition properties can be verified straightforwardly. We conclude that is indeed a tree decomposition of . As the size of each bag is at most , the width of is at most .
The following is the main technical result whose proof is given in the next section.
The size of obdd computing is at least .
The following lemma reformulates the statement of Theorem 1 in terms of the number of variables of and .
Let be the number of variables of . Then the size of -nsobdd computing is at least .
Proof. Recall that has nodes. For each node of , has variables corresponding to the vertices of the clique of plus variables corresponding to the edges of this clique. In addition, if is a non-root node then it is associated with variables connecting the clique of with the clique of its parent. Thus each node of is associated with at most variables and hence the total number of variables . Thus .
It follows from Theorem 1 that the size of a -nsobdd computing is at least as required.
Two lower bound parameterized by the treewdith now easily follows.
For each there is an infinite sequence of cnfs of treewidth at most of their primal graphs such that for each the size of -obdd computing it is at least , where is the number of variables of . In particular, for and every fixed , we get the earlier obtained obdd lower bound of as a special case.
Proof. For an odd , consider the cnfs for all and for an even , consider the cnfs for all . By Lemma 1, the treewidth of the primal graph of is at most and of at most . Thus the treewidth requirement is satisfied regarding these classes.
Taking into account Lemma 2 and performing simple algebraic calculation, we observe the -nsobdd size is lower-bounded by .
There is an infinite family of functions that can be computed by SDDs of size and for which the smallest -NSOBDD are of size (for each fixed )
Proof Consider functions . Let us compute the number of variables of . Following the calculation as in Lemma 2, we observe that
Denote by and by . Then .
In particular, and hence for a sufficiently large . Substituting instead and in the lower bound provided by Theorem 1 gives us lower bound which is for every fixed .
4 Lower bound parameterized by the matching width
The central concept we use for the proof of Theorem 1 is that of matching width. A matching of a graph is a set of edges of such that no two edges are incident to the same vertex. Let be a permutation of the set of vertices of a graph . Let be a prefix of (i.e. all vertices of are ordered after ). Let us call the matching width of , the size of the largest matching consisting of the edges between and (we take the liberty to use sequences as sets, the correct use will be always clear from the context). Further on, the matching width of is the largest matching width of a prefix of . Finally the matching width of , is the smallest matching width of a permutation of .
Consider a path of vertices so that is adjacent to for . The matching width of permutation is since between any suffix and prefix there is only one edge. However, the matching width of the permutation is as witnessed by the partition and . Since the matching width of a graph is determined by the permutation having the smallest matching width, and, since the graph has some edges, there cannot be a permutation of matching width , we conclude that the matching width of this graph is .
The main ‘engine’ for establishing the lower bound for Theorem 1 is the following theorem, stating a lower bound on the size of a -nsobdd.
Let be a graph of matching width at least and let be a -nsobdd computing . Then .
(Lemma 2 of ) For any , the matching width of is at least .
Now, we are ready to prove Theorem 1
Proof of Theorem 1 According to Theorem 4, the size of implementing is at least where is the matching width of Replace by the lower bound on the matching width of provided by Theorem 5. The required lower bound immediately follows.
4.1 Proof of Theorem 4
Let be a permutation of the vertices of where precedes if and only if precedes in te underlying permutation of . We refer to as the permutation of corresponding to . Let be a prefix of such that there is a matching of such that all of belong to and all of do not. Such a prefix exists by definition of matching width and our assumption that matching width of is at least .
Let be the set of all assignments to the variables of satisfying the following conditions.
Each for .
For each , the occurrences of and have distinct signs (if the former occurs positively the latter occurs negatively and if the former occurs negatively the latter occurs positively).
All the variables besides are assigned positively.
Then the following statements are easy to observe.
Each is a satisfying assignment of .
Proof. For the first statement, note that all the clauses besides are clearly satisfied by because is assigned positively by construction. The clauses are also satisfied by because one of , is assigned positively. This proves the first statement.
There are ways to assign variables . By definition of each such assignment can be extended to an element of and these elements are clearly all distinct. This proves the second statement.
In light of the first statement of Observation 1, for each we can identify a computational path of such that . For each we are going to identify a sequence of its vertices of length at most and to show that for distinct , the sequences associated with and are distinct. In light of the second statement of Observation 1, it will follow that contains at least sequences of nodes of length . As the number of such sequences is at most , it will immediately follow that .
In order to define a sequence of vertices associated with each , we need some preparation. Let be an arbitrary computational path of and let be subpaths of such that and the following holds for each .
Each variable occurs at most once as a label of .
The labels on are ordered according to .
Note that the required exists according to definition of -nsobdd.
Further on, let be subpaths of such that for each , the following holds.
For each , can occur only in (not in ).
For each , can occur only in (not in ).
Note that exists. Indeed, by definition of , all the variables occur in before all the variables . Therefore, if both and occur on , we can identify the last edge of labelled by a variable of and let to be the prefix of ending at the head of . If only variables of occur on then let and be the last vertex of . If only variables of occur on then let be the first vertex of and . Finally, if no variables occur on , the partition can be arbitrary.
Let be the respective ends of . We call the separation vector of and the decomposition of w.r.t. .
Remark. Note that there may be more than one possible satisfying the above conditions and hence can have several separation vectors. We just pick an arbitrary one and call it the separation vector.
The separation vectors of paths are these very sequences mentioned in the proof plan above. Now we are going to prove that distinct paths have different separation vectors.
Let be two distinct elements of . Then and have different separation vectors.
Proof. Assume that and have the same separation vector . Let be a variable having opposite assignments in and . (Such a variable necessarily exists because the assignments of are determined by assignments of . So, if the assignments if each has the same occurrence in both and , the same is true regarding each , and hence , a contradiction). Assume w.l.o.g. that occurs negatively in and positively in .
Let and be the respective decompositions of and w.r.t. to . Let be the path obtained from by replacement of each with even by .
is a computational path.
Proof. We need only to verify that there is no variable occurring both positively and negatively on . By definition of , each variable has the same occurrence in both and . As and , cannot have distinct occurrences in and . By definition of the decomposition w.r.t. the separation vector, a variable with cannot occur in with even . It follows that in , can only occur on which is a subgrpah of . As is a computational path, it does not contain opposite literals of and hence neither does . Due to the same reason, a variable with cannot occur on with even . It follows that in can only occur on which is a subgrpah of . As is a computational path, it does not contain opposite literals of and hence neither does .
is disjoint with .
Proof. By definition of occurs negatively in both and and hence it cannot occur positively in nor in and hence, in turn it cannot occur positively in composed of subpaths of and . Since , a literal of cannot occur on (by definition of the decomposition w.r.t. the separation vector). If a literal of occurs on then it is an element of and hence negative by assumption. Similarly, a literal of cannot occur on (since ). If a literal of occurs on then it is an element of and hence negative (by assumption, and hence by definition of .
By Claim 1 and definition of , an arbitrary extension of is a satisfying assignment of . By Claim 2, there is an extension of containing all of . However, falsifies clause existing since is an edge of . This contradiction shows that our initial assumption that and have the same separation vector is incorrect and hence the lemma holds.
5 Matching width vs. pathwidth
In this section we will show that the matching width, , of a graph is linearly related to its pathwidth, . It particular, we will show that .
Let us extend our notation. The maximum matching size of a graph is denoted by . Let be an ordering of vertices of . For , we denote by and by . The superscript can be omitted if the ordering is clear from the context. We denote by or by , if the ordering is clear from the context, the graph with the set of vertices and the set of edges . In other words the edges of are exactly those edges of that have one end in and one end in . With this notation in mind, the matching width of can be stated as follows.
If we denote by the set of all permutations of vertices of then
Recall that a vertex cover (VC) of graph is a set of vertices incident to all of its edges. The smallest size of vertex cover of is denoted by .
Observe that each is a bipartite graph because and , partitioning its set of vertices are indepdent sets of . It is well known that for a bipartite graph the size of the smallest vertex cover equals the size of maximum matching, that is . Hence can be restated as follows
Now we are bready to prove an upper bound on .
For any graph , .
Proof. Let be a path decomposition of of width . Let be the vertices of chronologically listed as they occur along . Recall that are the bags of the decomposition and the size of each bag is at most .
Now we are going to define a permutation of for which we will show that , which will imply the theorem because, by definition .
For , let be the smallest number such that . Let is an arbitrary permutation of such that whenever . It is not hard to see that such an order indeed exists. For instance, can be created as follows. Arbitrary order the vertices of . For each , suppose that the vertices have been already ordered and let be the corresponding permutation. Then create a permuation of by arbitrary ordering the vertices of and appending them to the end of .
We are going to show that for each , is a vertex cover of that is for each either or . Observe that this will imply the desired statement that . Indeed, by definition, there is such that . Combining with the claim we are going to prove, we will have
the first and the second inequalities follow from the definitions of and pathwidth, respectively.
Pick and let . Assume w.l.o.g. that and . Then, by definition of , and . If the equality occurs regarding any of them, say then, by definition of function , . Thus it remains to consider the case where and .
By the containment property of the path decomposition, there is such that . By definition of , and hence . To preserve the connectedness property, must occur in all bags for . In particular, since and , , as required.
Next we are going to show that . For this we need the following definition.
Let be a permutation of . For each , , let be a smallest VC of . The sets are called settled w.r.t. if for each ,
The following lemma is proved in Section 5.1.
For each permutation of there are that are settled w.r.t. .
Now we are ready for the theorem.
For any graph , .
Proof. Let be a permutation of such that . Let be the smallest VCs of , respectively, that are settled w.r.t. .
Our candidate for path decomposition of width is a pair where is a path and is a set of bags defined as follows.
For , .
In the rest of the proof we demonstrate that is indeed a path decomposition of having width at most . This amounts to proving the following statements.
satisfies the union property. Indeed, by construction, for , .
satisfies the containment property. Indeed, let and assume w.l.o.g. that . This means that is an edge of each of and hence each of has a non-empty intersection with .
Assume that . Then, by construction, , satisfying the containment property. Assume next that . Then, by construction, , satisfying the containment property. If none of the above assumptions hold then and . It follows that there is such that and . Then by construction, , satisfying the containment property.
satisfies the connectedness property. Assume by contradiction that the connectedness property is violated. That is, there is a vertex and such that , , and . We assume that is smallest possible subject to this property, that is, .
Since , . That is .
It follows that . Indeed, if , this follows by definition of . Otherwise, notice that since are settled, . As we know that , we conclude that
As , . By Definition 1, . We claim that . This claim will imply that and hence by the minimality of (as all the neighbours of are already there). This is a contradiction to our assumption, confirming correctness of the connectedness property. It thus remains to prove the claim.
Let . As and , and hence . Consequently, . As , . Thus is an edge of with one end in , the other in . Hence is an edge of , that is confirming the claim and the connectedness property as specified in the previous paragraph.
The width of is at most . That is, we have to show that for each , . By definition, for . According to ((3)) . Thus, in our case, . It follows that for , . Clearly, the same upper bound applies to and .
5.1 Proof of Lemma 4
Let be a bipartite graph with set of vertices and the set of edges , all having one end in the other end in . In order to prove Lemma 4, we need the following three auxiliary statements.
Let be a smallest VC of a . Let . Let be a smallest VC of . Then is a smallest VC of .
Proof. is a VC of . Indeed, none of the edges covered are covered by and hence they are covered by .
If we assume that is not a smallest VC of then . That is, , from where we conclude that . That is, is a VC of smaller than in contradiction to the definition of .
Let be a bipartite graph and let be such that there is