An Efficient Algorithm for Solving the 2-MAXSAT Problem

To develop an efficient algorithm to find a truth assignment that maximizes the number of satisfied conjunctions in formula $D$ = $D_{1}$ $\lor$ …, $\lor$ $D_{n}$, where each $D_{i}$ ($i$ = 1, …, $n$) is a conjunction of variables $c$ ($\in$ $V$), we need to represent each $D_{i}$ as a sequence of variables (referred to as a variable sequence). For this purpose, we introduce a new notation:

($c_{j}$, *) = $c_{j}$ $\lor$ $\neg$$c_{j}$ = true,

In this way, the above $D$ can be rewritten as a new formula in DNF as follows:

Doing this enables us to represent each $D_{i}$ as a variable sequence, but with all the negative literals being removed. It is because if the variable in a negative literal is set to true, the corresponding conjunction $D_{i}$ must be false, and our goal is to establish a graph in which each node represents a variable and each path $p$ corresponds to a truth assignment satisfying $D_{i}$ (by which any variable on $p$ is set true while all those varibles not on $p$ are set false). Obviousely, in such a graph, any variable appearing in a negative literal should not be involved since any path through such a variable corresponds a truth assignment not satisfying $D_{i}$.

See Table I for illustration.

First, we pay attention to the variable sequence for $D_{2}$ (the second sequence in the second column of Table  I), in which the negative literal $\neg$$c_{4}$ (in $D_{2}$) is elimilated. In the same way, you can check all the other variable sequences.

Now it is easy for us to compute the appearance frequencies of different variables in the variable sequences, by which each ($c$, *) is counted as a single appearance of $c$ while any negative literals are not considered, as illustrated in Table II, in which we show the appearance frequencies of all the variables in the above $D$.

According to the variable appearance frequencies, we will impose a global ordering over all variables in $D$ such that the most frequent variables appear first, but with ties broken arbitrarily. For instance, for the $D$ shown above, we can specify a global ordering like this: $c_{2}$ $\to$ $c_{3}$ $\to$ $c_{1}$ $\to$ $c_{4}$ $\to$ $c_{5}$ $\to$ $c_{6}$. Here, $c_{2}$ is most frequent and then appears first. The other variables have the same frequency. So, we simply impose a fixed order on them: $c_{3}$ $\to$ $c_{1}$ $\to$ $c_{4}$ $\to$ $c_{5}$ $\to$ $c_{6}$.

Following this general ordering, each conjunction $D_{i}$ in $D$ can be represented as a sorted variable sequence as illustrated in the third column of Table I, where the variables in a sequence are ordered in terms of their appearance frequencies such that more frequent variables appear before less frequent ones. In addition, a start symbol $\#$ and an end symbol $\$$ are used as sentinals for technical convenience. In fact, any global ordering of variables works well (i.e., you can specify any global ordering of variables), based on which a graph representation of assignments can be established. However, ordering variables according to their appearance frequencies can greatly improve the efficiency when searching a graph constructed over all the variable sequences for conjunctions in $D$ to find solusions since more variables from different conjunctions can be merged together.

Later on, by a variable sequence, we always mean a sorted variable sequence. Also, we will use $D_{i}$ and the variable sequence for $D_{i}$ interchangeably without causing any confusion.

In addition, for our algorithm, we need to introduce a graph structure to represent all the truth assignments for each $D_{i}$ ($i$ = 1, …, $n$) (called a $p$*-graph), under which $D_{i}$ evaluates to true. In the following, however, we first define a simple concept of $p$-graphs for ease of explanation.

In Fig. 1(a), we show such a $p$-graph for $D_{1}$ = $d_{0}$$d_{1}$$d_{2}$$d_{3}$$d_{4}$$d_{5}$$d_{6}$$d_{7}$ = $\#.$($c_{2}$, *).($c_{3}$, *).$c_{1}$.$c_{4}$.($c_{5}$, *).($c_{6}$, *).$\$$. Beside a main path going through all the variables in $D_{1}$, there are four off-path edges (edges not on the main path), referred to as spans attached to the main path, corresponding to ($c_{2}$, *), ($c_{3}$, *), ($c_{5}$, *), and ($c_{6}$, *), respectively. Each span is represented by the subpath covered by it. For example, we will use the subpath $<$$v_{0}$, $v_{1}$, $v_{2}$$>$ (subpath going three nodes: $v_{0}$, $v_{1}$, $v_{2}$) to stand for the span connecting $v_{0}$ and $v_{2}$; $<$$v_{1}$, $v_{2}$, $v_{3}$$>$ for the span connecting $v_{1}$ and $v_{3}$; $<$$v_{4}$, $v_{5}$, $v_{6}$$>$ for the span connecting $v_{4}$ and $v_{6}$, and $<$$v_{5}$, $v_{6}$, $v_{7}$$>$ for the span connecting $v_{6}$ and $v_{7}$. By using spans, the meaning of ‘*’s (it is either 0 or 1) is appropriately represented since along a span we can bypass the corresponding variable (then its value is set to 0) while along an edge on the main path we go through the corresponding variable (then its value is set to 1).

In fact, what we want is to represent, in an efficient way, all those truth assignments for each $D_{i}$ ($i$ = 1, …, $n$), under which $D_{i}$ evaluates to true. However, $p$-graphs fail to do so since when we go through from a node $v$ to another node $u$ through a span, $u$ must be selected. If $u$ represents a ($c$, *) for some variable name $c$, the meaning of this ‘*’ is not properly rendered. It is because ($c$, *) indicates that $c$ is optional, but going through a span from $v$ to ($c$, *) makes $c$ always selected. So, the notation ($c$, *), which is used to indicate that $c$ is optional, is not correctly implemented.

For this reason, we introduce another concept, $p$*-graphs, described as below.

Let $s_{1}$ = $<$$v_{1}$, …, $v_{k}$$>$ and $s_{2}$ = $<$$u_{1}$, …, $u_{l}$$>$ be two spans attached onto a same path. We say, $s_{1}$ and $s_{2}$ are overlapped, if $u_{1}$ = $v_{j}$ for some $j$ $\in$ $\{$1, …, $k$ - 1$\}$, or if $v_{1}$ = $u_{j^{\prime}}$ for some $j^{\prime}$ $\in$ $\{$1, …, $l$ - 1$\}$. For example, in Fig. 1(a), $<$$v_{0}$, $v_{1}$, $v_{2}$$>$ and $<$$v_{1}$, $v_{2}$, $v_{3}$$>$ are overlapped. $<$$v_{4}$ $v_{5}$, $v_{6}$$>$ and $<$$v_{5}$, $v_{6}$, $v_{7}$$>$ are also overlapped.

Here, we notice that if we had one more span, $<$$v_{3}$, $v_{4}$, $v_{5}$$>$, for example, it would be connected to $<$$v_{1}$, $v_{2}$, $v_{3}$$>$, but not overlapped with $<$$v_{1}$, $v_{2}$, $v_{3}$$>$. Being aware of this difference is important since the overlapped spans imply the consecutive ‘*’s, just like $<$$v_{1}$, $v_{1}$, $v_{2}$$>$ and $<$$v_{1}$, $v_{2}$, $v_{3}$$>$, which correspond to two consecutive ‘*’s: ($c_{2}$, *) and ($c_{3}$, *). Therefore, the overlapped spans exhibit some kind of transitivity. That is, if $s_{1}$ and $s_{2}$ are two overlapped spans, the $s_{1}$ $\cup$ $s_{2}$ must be a new, but bigger span. Applying this operation to all the spans over a $p$-path, we will get a ’transitive closure’ of overlapped spans.

Let $S$ be the set of all spans over the main path $p$ for a certain conjunction. The transive closure of $S$, denoted as $S$*, is another set of spans $S$* = $\{$$s_{1}$, $s_{2}$, …, $s_{l}$$\}$ for sime $l$, which contains the whole $S$ and is with each $s_{i}$ satisfying one of the following two conditions:

1. $s_{i}$ $\in$ $S$, or

2. There exist $j$, $k$ ($\neq$ $i$) such that $s_{j}$ and $s_{k}$ are overlapped and $s_{i}$ = $s_{j}$ $\cup$ $s_{k}$.

Based on the above discussion, we give the following definition.

In Fig. 1(b), we show the $p$*-graph with respect to the $p$-graph shown in Fig. 1(a).

As another example, consider $D_{2}$ = $\#.$$c_{2}$.$c_{3}$.($c_{1}$, *).($c_{5}$, *).($c_{6}$, *).$\$$. Its $p$-graph is shown in Fig. 1(c) and its $p$*-graph in Fig. 1(d), in which we notice that we have span $<$$u_{2}$, $u_{3}$, $u_{4}$, $u_{5}$$>$ (representing two consecutive ‘*’s) due to two overlapped spans: $<$$u_{2}$, $u_{3}$, $u_{4}$$>$ and $<$$u_{3}$, $u_{4}$, $u_{5}$$>$. Further, we have span $<$$u_{2}$, $u_{3}$, $u_{4}$, $u_{5}$, $u_{6}$$>$ (representing three consecutive ‘*’s) due to $<$$u_{2}$, $u_{3}$, $u_{4}$, $u_{5}$$>$ and $<$$u_{4}$, $u_{5}$, $u_{6}$$>$. In the same way, we can check all the other spans in Fig. 1(d).

The purpose of the $p$*-graph for a certain conjunction $D_{i}$ is to represent all the truth assignments, under each of which $D_{i}$ evaluates to true. Specifically, in $P$* each root-to-leaf path $p$ corresponds to a truth assignment, by which each variable on $p$ is set to true while any other variables are set false.

Concerning $p$*-graphs, we have the following lemma.

For example, in Fig. 1(b), the path: $v_{0}$ $\to$ $v_{3}$ $\to$ $v_{4}$ $\to$ $v_{5}$ $\to$ $v_{7}$ represents a truth assignment: $\{$$c_{1}$ = 1, $c_{2}$ = 0, $c_{3}$ = 0, $c_{4}$ = 1, $c_{5}$ = 1, $c_{6}$ = 0$\}$, under which $D_{1}$ evaluates to true. In Fig.  1(d), the path: $u_{0}$ $\to$ $u_{1}$ $\to$ $u_{2}$ $\to$ $u_{6}$ represents another truth assignment: $\{$$c_{1}$ = 0, $c_{2}$ = 1, $c_{3}$ = 1, $c_{4}$ = 0, $c_{5}$ = 0, $c_{6}$ = 0$\}$, under which $D_{2}$ evaluates to true. We can examine all the paths in these two graphs and find that Lemma 1 always holds for them.

To find a truth assignment to maximize the number of satisfied $D_{j}^{\prime}$s in $D$, we will first construct a trie-like structure $G$ over $D$, and then search $G$ bottom-up to find answers.

Let $P_{1}$*, $P_{2}$*, …, $P_{n}$* be all the $p$*-graphs constructed for all $D_{j}$’s in $D$, respectively. Let $p_{j}$ and $S_{j}$* ($j$ = 1, …, $n$) be the main path of $P_{j}$* and the transitive closure over its spans, respectively. We will construct $G$ in two steps.

In the first step, we will establish a trie [3, 15], denoted as $T$ = $trie$($R$) over $R$ = $\{$$p_{1}$, …, $p_{n}$$\}$ as follows.

If $|R|$ = 0, $trie$($R$) is, of course, empty. For $|R|$ = 1, $trie$($R$) is a single node. If $|R|$ $>$ 1, $R$ is split into $r$ (possibly empty) subsets $R_{1}$, $R_{2}$, …, $R_{r}$ so that each $R_{i}$ ($i$ = 1, …, $r$) contains all those sequences with the same first variable name. The tries: $trie(R_{1}$), $trie(R_{2}$), …, $trie(R_{r}$) are constructed in the same way except that at the $k$th step, the splitting of sets is based on the $k$th variable name (along the global ordering of variables). They are then connected from their respective roots to a single node to create $trie$($R$).

In Fig. 2, we show the trie constructed for the variable sequences given in the third column of Table I. In such a trie, special attention should be paid to all the leaf nodes each labeled with $\$$, representing a conjunction (or a subset of conjunctions), which can be satisfied under the truth assignment represented by the corresponding main path. For example, the subset $\{$$D_{1}$, $D_{3}$, $D_{5}$$\}$ associated with $v_{7}$ is satisfiable under the truth assignment represented by the path from $v_{0}$ to $v_{7}$. Such a path is also called a tree path.

The main advantage of tries is to cluster common parts of variable sequences together to avoid possible repeated checking. Then, if variable sequences are sorted according to their appearance frequencies, more variables will be clustered. More importantly, this idea can also be applied to the variable subsequences (as will be seen later), over which some dynamical tries can be recursively constructed, leading to a polynomial-time algorithm for solving the problem.

Each node $v$ in the trie stands for a variable $c$, referred to as the label of $v$ and denodeted as $l$($v$) = $c$; and each edge $e$ is referred to as a tree edge, labeled with a set of integers representing all the variable sequences going through $e$, denoted as $s(e)$. For example, $s(v_{0},v_{1})$ = $\{$1, 2, 3, 4, 5, 6$\}$. It is because all the variable sequences given in Table I need to pass through this edge to reach their respective leaf nodes. In the same way, you can check all the other labels associated with tree edges.

In regard to the tree paths, we have the following lemma.

Finally, we will associate each node $v$ in the trie $T$ with a pair of numbers (pre, post) to speed up recognizing ancestor/descendant relationships of nodes in $T$, where pre is the order number of $v$ when searching $T$ in preorder and post is the order number of $v$ when searching $T$ in postorder.

These two numbers can be used to characterize the ancestor/descendant relationships in $T$ as follows.

• -

  Let $v$ and $v\textquoteright$ be two nodes in $T$. Then, $v\textquoteright$ is a descendant of $v$ iff pre($v\textquoteright$) > pre($v$) and post($v\textquoteright$) < post($v$).

For the proof of this property of any tree, see Exercise 2.3.2-20 in [14].

For instance, by checking the label associated with $v_{2}$ against the label for $v_{9}$ in Fig. 2, we see that $v_{2}$ is an ancestor of $v_{9}$ in terms of this property. Specifically, $v_{2}$’s label is (3, 12) and $v_{9}$’s label is (10, 6), and we have 3 < 10 and 12 > 6. We also see that since the pairs associated with $v_{14}$ and $v_{6}$ do not satisfy the property, $v_{14}$ must not be an ancestor of $v_{6}$ and vice versa.

In the second step, we will add all $S_{i}$* ($i$ = 1, …, $n$) to the trie $T$ to construct a trie-like graph $G$, as illustrated in Fig. 3. This trie-like graph is constructed for all the variable sequences given in Table I, in which each span is associated with a set of numbers used to indicate what variable sequences the span belongs to. For example, the span $<$$v_{0}$, $v_{1}$, $v_{2}$$>$ (in Fig. 3) is associated with three numbers: 1, 5, 6, indicating that the span belongs to 3 conjunctions: $D_{1}$, $D_{5}$, and $D_{6}$. In Fig.  3, however, the labels for all tree edges are not shown for a clear illustration.

In addition, each $p$*-graph itself is considered to be a simple trie-like graph.

Concerning the paths in a trie-like graph, we have a lemma similar to Lemma 2.

From Fig. 3, we can see that although the number of truth assignments for $D$ is exponential, they can be represented by a graph with polynomial numbers of nodes and edges. In fact, in a single $p$*-graph, the number of edges is bounded by O($m^{2}$). Thus, a trie-like graph over $n$ $p$*-graphs has at most O($nm^{2}$) edges.

In a next step, we will search $G$ bottom-up level by level to seek all the possible largest subsets of conjunctions which can be satisfied by a certain truth assignment.

First of all, we call each node in $T$ with more than one child a branching node. For instance, node $v_{3}$ with two children $v_{4}$ and $v_{8}$ in $G$ shown in Fig. 3 is a branching node. For the same reason, $v_{2}$ and $v_{1}$ are another two branching nodes. Note that $v_{0}$ is not a branching node since it has only one child in $T$ (although it has more than one child in $G$.)

Around the branching node, we have two very important concepts defined below.

For $\textit{RS}^{v,u}_{s}$[$c$], node $u$ is also called its anchor node while any node in $\textit{RS}^{v,u}_{s}$[$c$] is called a reachable node of $u$.

For instance, for node $v_{2}$ in Fig.  3, which is on the tree path from root to $v_{3}$ (a branching node), we have two RSs with respect to $v_{3}$:

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  $\textit{RS}^{v_{3},v_{2}}_{\{2,5\}}$[$c_{5}$] = $\{$$v_{5}$, $v_{8}$$\}$,

• -

  $\textit{RS}^{v_{3},v_{2}}_{\{2,5\}}$[$c_{6}$] = $\{$$v_{6}$, $v_{9}$$\}$.

We have $\textit{RS}^{v_{3},v_{2}}_{\{2,5\}}$[$c_{5}$] due to two spans $v_{2}$ $\xrightarrow{5}$ $v_{5}$ and $v_{2}$ $\xrightarrow{2}$ $v_{8}$ going out of $v_{2}$, respectively reaching $v_{5}$ and $v_{8}$ on two different $p$*-graphs in $G$[$v_{3}$] with $l$($v_{5}$) = $l$($v_{8}$) = ‘$c_{5}$’. We have $\textit{RS}^{v_{3},v_{2}}_{\{2,5\}}$[$c_{6}$] due to another two spans going out of $v_{2}$: $v_{2}$ $\xrightarrow{5}$ $v_{6}$ and $v_{2}$ $\xrightarrow{2}$ $v_{9}$ with $l$($v_{6}$) = $l$($v_{9}$) = ‘$c_{6}$’.

Hence, $v_{2}$ is not only the anchor node of $\{$$v_{5}$, $v_{8}$$\}$, but also the anchor node of $\{$$v_{6}$, $v_{9}$$\}$.

In general, we are interested only in those RSs with |RS| $\geq$ 2 since any RS with $|\textit{RS}|$ = 1 only leads us to a leaf node in $T$, and no larger subsets of conjunctions can be found. In fact, going through a span with the corresponding $|\textit{RS}|$ = 1, we cannot get any new answers. So, in the subsequent discussion, by an RS, we mean an RS with |RS| $\geq$ 2.

The definition of this concept for a branching node $v$ itself is a little bit different from any other node on the tree path (from root to $v$). Specifically, each of its RSs is defined to be a subset of nodes reachable from a span or from a tree edge. So, for $v_{3}$ we have:

• -

  $\textit{RS}^{v_{3},v_{3}}_{\{2,5\}}$[$c_{5}$] = $\{$$v_{5}$, $v_{8}$$\}$,

• -

  $\textit{RS}^{v_{3},v_{3}}_{\{2,5\}}$[$c_{6}$] = $\{$$v_{6}$, $v_{9}$$\}$,

respectively due to span $v_{3}$ $\xrightarrow{5}$ $v_{5}$ and tree edge $v_{3}$ $\rightarrow$ $v_{8}$ going out of $v_{3}$ with $l$($v_{6}$) = $l$($v_{8}$) = ‘$c_{5}$’; and two spans $v_{3}$ $\xrightarrow{5}$ $v_{6}$ and $v_{3}$ $\xrightarrow{2}$ $v_{9}$ going out of $v_{3}$ with $l$($v_{6}$) = $l$($v_{8}$) = ‘$c_{6}$’. Here, we notice that the label for the tree edge $v_{3}$ $\rightarrow$ $v_{8}$ is 2 since this tree edge belongs to $D_{2}$ (see Fig. 2).

Concerning RSs, we have the following lemma, which is important for the construction of trie-like subgraphs.

If $RS_{s}^{v,u}[c]$ $\subset$ $RS_{s}^{v,u^{\prime}}[c]$, we say, $RS_{s}^{v,u^{\prime}}[c]$ is larger than $RS_{s}^{v,u}[c]$.

Based on the concept of reachable subsets through spans, we are able to define another more important concept, upper boundaries, given below.

1. 1.

   Each $u_{g}$ (1 $\leq$ $g$ $\leq$ $f$) appears in some $\textit{RS}^{v,v_{i}}_{s}$[$c$] (1 $\leq$ $i$ $\leq$ $k$), where $c$ is a label and $|\textit{RS}^{v,v_{i}}_{s}$[$c$]$|$ > 1.

2. 2.

   For any two nodes $u_{g}$, $u_{g^{\prime}}$ ($g$ $\neq$ $g^{\prime}$), they are not related by the ancestor/descendant relationship.

Fig.  4 gives an intuitive illustration of this concept.

As a concrete example, consider $v_{5}$ and $v_{8}$ in Fig.  3. They make up an upBound with respect to $v_{3}$ (a branching node), based on which we will construct a trie-like graph over two subgraphs, rooted at $v_{5}$ and $v_{8}$, respectively. This can be done in a way similar to the construction of $G$ over all the initial $p$*-graphs (which then hints a recursive process to do the task). Here, we remark that $v_{4}$ is not included since it is not invlved in any RS with respect to $v_{3}$ with $|\textit{RS}|$ $\geq$ 2. In fact, the truth assignment with $v_{4}$ being set to true satisfies only the conjunctions associated with leaf node $v_{10}$. This has already been determined when the initial trie is built up in the first step.

Mainly, the following operations will be carried out when encountering a branching node $v$.

• •

  Calculate all RSs with respect $v$.

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  Calculate the upBound in terms of RSs.

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  Make a recursive call of the algorithm on a subgraph which is constructed over all the $p$*-subgraphs each rooted at a node on the corresponding upBound.

See the following example for illustration.