url - Universität zu Lübeck

7.3. NP-COMPLETENESS FOR PATH EXPRESSIONS WITH NOT 135
Now we build two path expressions p and p ′ corresponding to F A and F B as follows:
Both p and p ′ have the same prefix, for instance /a. The formulas are attached
as a qualifier to the a node. The logical operators ∧ and ∨ are translated to AND
and OR operators from XPath. Each positive variable of F A respectively F B is interpreted
as the existence of a matching child element. Each negative variable is
covered by a NOT operator.
Because F A and F B are always satisfiable individually Mod(p) and Mod(p ′ ) are
not empty.
If p(t) ∩ p ′ (t) ≠ ∅ it means that there is an a node that fulfills the two qualifiers of p
and p ′ . But this means that F A and F B are satisfied simultaneously. Hereby, the
reduction 3SAT ≤ pol XIP is finished; the conversion from a Boolean formula to a
path expression can be done in polynomial (even linear) time.
Finally, XIP NOT is in NP because a nondeterministic Turing-machine may guess a
satisfying assignment if it exists. Because the XIP NOT is NP-hard and belongs to
NP it is NP-complete.
7.3.1 Algorithm and Complexity of XIP NOT
Our algorithm solving XIP NOT for two path expressions p, p ′ ∈ XP {[],∗,//,NOT } is
based on the introduced algorithm checking the intersection of linearized path
expressions (see figure 7.3). If the linearized path expressions of p and p ′ have
an intersection we now have to check if their additional qualifiers and keys are
mutually exclusive. This implies that we have to check the satisfiability of p ∧ p ′ .
Because of the NP-completeness of XIP NOT this step has an exponential worst
case complexity: every permutation of true/false that is assigned to the variables
in the qualifiers has to be generated and checked if it satisfies p ∧ p ′ . If we find
no satisfying configuration the intersection remains empty.
The algorithm uses a simple optimization step that reduces the number of variables
that have to be checked: A variable may only be responsible for mutual
exclusiveness if it appears negated and non-negated in the set of qualifiers of p
and p ′ . Therefore, we can remove all variables from the expressions that are only
negated or non-negated. This simple optimization has a similar aim as the Davis
Putnam algorithm [23] for deciding 3SAT. The extended algorithm is presented in
figure 7.4:
In the first step the algorithm checks whether the linearized path expressions
have an intersection. If this is not the case the algorithm stops and returns false.
In lines 2 to 4 we reduce the numbers of relevant variables by removing the trivial
ones. The function getNegVars returns all variables from a path expression
that appear negated. The function getPosVars is defined vice versa. In step 5 the
algorithm iterates over all possible true/false values for the remaining variables