Download Computing q-Horn Strong Backdoor Sets: a preliminary

Computing q-Horn Strong Backdoor Sets: a
preliminary experimentation
Richard Ostrowski and Lionel Paris!
LSIS CNRS UMR 7296
Université d’Aix Marseille
{richard.ostrowski, lionel.paris}@lsis.org
Abstract. Identifying and exploiting hidden problem structures is recognized as a fundamental way to deal with the intractability of combinatorial problems. A particular structure called (Strong) Backdoor has
been identified in the context of the satisfiability problem. Connections
has been established between Backdoors and problem hardness leading
to a better approximation of the worst case time complexity. Strong
Backdoor sets can be computed for any tractable class. In [1], a method
for the approximation of Strong Backdoor sets for the Horn-Sat fragment was proposed. This approximation is realized in two steps. First,
the best Horn renaming of the original CNF formula, in term of number
of clauses, is computed. Then a Horn Strong Backdoor set is extracted
from the non Horn part of the renamed formula. In this article, we extend
this method to the class of q-Horn [2] which is an extension of the Horn
class and 2-SAT. Experimental results show that the size of the Strong
Backdoor set with respect to the q-Horn fragment is slightly better than
for the Horn fragment.
1
Introduction
Propositional satisfiability (SAT) is the problem of deciding whether a Boolean
formula in conjunctive normal form (CNF) is satisfiable or not. SAT is one of
the most studied NP-Complete problems because of its theoretical and practical
importance. Encouraged by the impressive progress in practical solving of SAT,
various applications (planning, formal verification, equivalency and logical circuits simplifications, etc.) are encoded and solved using SAT. Most of the best
complete solvers are based on the Conflict Driven Clause Learning scheme introduced by Grasp [3], efficiently implemented in Chaff [4] and popularized by
Minisat [?]. Such a basic algorithm is enhanced with many important pruning
techniques such as learning, extended use of Boolean constraints propagation,
preprocessing, symmetries detection and exploitation, etc.
The other important class of satisfiability algorithms concerns local search
based methods. These techniques can not prove unsatisfiability, since search
space is explored in a non systematic way (e.g. [5, 6]). However, impressive performance are obtained on hard and large satisfiable instances (including random
instances).
During last years, several authors have focused on detecting possible hidden
structures inside SAT instances (e.g backbones [7], Backdoors [8, 9], equivalences
!
This work was supported by the French National Research Agency under grant
TUPLES (ANR-2010-BLAN-0210).
[10] and functional dependencies [11]), cardinality constraints [12] allowing to
explain and improve the efficiency of SAT solvers on large real-world instances.
Other important theoretical results like heavy tailed phenomena [8] and backbones [13] were also obtained leading to a better understanding of problem hardness. The efficiency of satisfiability solvers can be in part explained using these
results.
In this paper, we focuses our attention on the Strong Backdoor sets computation problem. Let us recall that a set of variables forms a Backdoor for a
given formula if there exists an assignment to these variables such that the simplified formula can be solved in polynomial time. Such a set of variables is called
a Strong Backdoor, if any assignment to these variables leads to a tractable
sub-formula.
As all optimization problems, computing the smallest Strong Backdoor is an
NP-hard problem. Approximating (in polynomial time) the ”smallest” Strong
Backdoor set is an interesting and important research issue.
In this article, we are interested in computing a Strong Backdoor set for the
tractable q-Horn class. The q-Horn class, generalizing the important classes of
2-SAT, Horn, and disguised Horn formulas, has been introduced in [2]. In [14],
the authors proposed a linear algorithm for recognizing a q-Horn formula. We
implemented this algorithm to determine whether a formula is q-Horn and if it
is not, we compute a Strong Backdoor. This Strong Backdoor set corresponds
to a deletion Strong Backdoor set introduced in [15]. In this article, the authors
propose a algorithm with a time complexity exponential in the size of the Strong
Backdoor set. This huge complexity prevents it from being tractable. That is why
we suggest approximating it with a competitive algorithm.
The paper is organized as follows. First we give preliminary definitions linked
to the paper context. Then, we recall the principle of the algorithm to determine whether a formula is q-Horn. If the formula is not a q-Horn formula, we
calculate a Strong Backdoor set by translating the associated feedback vertex
set problem into a MAX-SAT problem. Finally, we present some experimental
results comparing the size of the Strong Backdoor computed when considering
the tractable class of Horn (or disguised Horn) and q-Horn. To conclude, the
scope of these results is discussed and some paths for future works are given.
2
Preliminary Definitions and Notations
Let B be a Boolean (i.e. propositional) language of formulas built in the standard
way, using usual connectives (∨, ∧, ¬, ⇒, ⇔) and a set of propositional variables.
A CNF formula Σ is a set (interpreted as a conjunction) of clauses, where a
clause is a set (interpreted as a disjunction) of literals.
A literal is a positive (l) or negated (¬l or l) propositional variable. Let us
recall that any Boolean formula can be translated
to CNF using linear Tseitin
!
encoding [16]. The size of CNF Σ is defined by c∈Σ |c| where |c| is the number of
literals in c. A unit (resp. binary) clause is a clause of size 1 (resp. 2). A unit literal
is the unique literal of a unit clause. A unit clause C = {l} is said to be positive
if its unit literal l is positive (l ∈ V). We note nbV ar(Σ) (resp. nbCla(Σ))
the number of variables (resp. clauses) of Σ. V(Σ) (resp. L(Σ)) is the set of
variables (resp. literals) occurring in Σ. A set of literals S ⊂ L(Σ) is consistent
iff ∀l ∈ S, ¬l ∈
/ S. Given a set of literals S, we define ∼ S = {¬l|l ∈ S}. A literal
l ∈ L(Σ) is called monotone if ¬l ∈
/ L(Σ). We note V + (Σ) (resp. V − (Σ)) as the
set of variables occurring in Σ at least one time with positive (resp. negative)
polarity. For each literal l, OccΣ (l) represents the set {C ∈ Σ|l ∈ C} i.e. the
set of clauses of Σ in witch literal l appears. We will simply write Occ(l) if no
confusion can be made.
A truth assignment of a Boolean formula Σ is an assignment of truth values
{true, f alse} to its variables. A variable x is satisfied (resp. falsified) under I
if I[x] = true (resp. I[x] = f alse). A truth assignment I can be represented
as a set of literals. A literal l ∈ I (resp. ¬l ∈ I) if I[l] = true (resp. I[l] =
f alse). When considering partial assignments, in witch some variables are not yet
instantiated, those uninstantiated variables does not occur in the interpretation,
neither in positive polarity, nor in negative one. A model of a formula Σ is a truth
assignment I such that I(Σ) = {}. Accordingly, SAT consists in determining if
the formula admits a model, or not.
3
Algorithm for q-Horn formulas detection
A CNF formula Σ is a q-Horn formula if it exists a certifying!
function β: L(Σ)
→ {0, 1/2, 1} with β(l)=1-β(l) for every l ∈ V(Σ) such that l∈C β(l) ≤ 1 for
every clause C of Σ. In [14], the authors proposed a linear time algorithm to
determine if a CNF formula is a q-Horn formula. The principle is to convert
the set of clauses into a directed graph G and then to calculate the Strongly
connected components. Depending on the component, a value among 0,1/2 or 1
is assigned to each literal of the formula. The set of variables corresponding to the
literals assigned to 0 or 1 is called H, whereas the set of variables corresponding
to the literals assigned to 1/2 is called Q. Each clause is finally analyzed to verify
that the sum of the value of the literals is less or equal to 1 (meaning that each
clause must contain at most 2 literals of L(Q)).
4
Computing a Strong Backdoor set
When a formula is not a q-Horn formula, we use the values returned by the previous algorithm for each literal to calculate a Strong Backdoor set. The principle
is to focus on the clauses for which the sum of the value of the literals (provided
by the certifying function) is greater than 1. This means that there is at least
3 literals per clause having the value 1/2. The Strong Backdoor corresponds to
the smallest set of variables of Q such that, when withdrawing them from the
formula, every clauses contain at most 2 variables of L(Q). This problem can be
reduced easily to the optimization version of the Feedback Vertex Set problem,
known to be N P − Hard [17].
Different approaches are used to solve the FVS problem efficiently [18]. Recently, [19, 20] proposed new local search methods. All these approaches are
based on a graph representation. As the implementation of their algorithms are
not public, we reduce the FVS problem into a weighted MAX-SAT problem
where efficient solvers are available [21, 22].
The reduction consists in:
– Extracting the set of variables corresponding to the literals of Q in all the
clauses containing more than 2 of them.
– Considering these sets as positive clauses.
– For all these clauses, encoding a set of clauses expressing the cardinality
constraint ”at most 2 among all the concerned literals”. All of the added
clauses have an infinite weight
– Adding the unit clauses containing the positive literals with a weight of 1.
– Using MaxWalkSAT [22] to find the smallest Strong Backdoor (variables set
to 1).
5
Experimental results
In order to validate our approach, we computed Strong Backdoor sets with
respect to q-Horn using the proposed algorithm, and compared the results in
terms of size, to the results obtained with respect to Horn, using the approach
proposed in [1]. For the experimentation, we selected more that 5000 instances
issued from the last SAT competitions and satLib, and we ran both methods on
them. In the amount of instances, we excluded all the random ones, and those
with a huge size. The results that we obtained are shown in figure 1. Each point
represents an instance, and both coordinate are equals to the size of the smallest
q-Horn Strong Backdoor set and the size of the smallest Horn Strong Backdoor
set. All the points located above the med-line represent the instances for which
the Horn Strong Backdoor sets are bigger that the q-Horn ones.
As we can see, the results are not as propitious as we would like. Indeed, as
q-Horn is a strict generalization of Horn, we would have expected that in the
worst case, the q-Horn Strong Backdoor set would have the same size as the
Horn one. But this is not the case!
We investigated the situation to find an explanation, and discovered this
particular case:
Lets consider the set of clauses {l1 ∨ l2 ∨ ¬l3 ∨ ¬l4 , ¬l1 ∨ ¬l2 ∨ l3 , ¬l1 ∨ ¬l2 , ∨l4 }.
We can see that only the first clause is not Horn and contains 2 positive literals.
Thus, a Horn Strong Backdoor is obtained by removing one of these literals and
then, the size of the set is one. On the other hand, according to the recognition
algorithm of q-Horn formulas, we obtain that all the variables are assigned to
1/2. Thus, we need to remove at least 2 variables in the first clauses and 1 in
the second one. Hence, the minimal size of the q-Horn Strong Backdoor set is
two. By increasing the size of this simple example, we can easily explain the fact
that sometimes (quite often unfortunately), the Horn Strong Backdoor sets are
smaller.
6
Conclusions and perspectives
In this work we presented an original way to compute Strong Backdoor sets with
respect to q-Horn clauses, which is one of the largest tractable class known. As
a reminder, we have:
Horn ! Renameable − Horn ! q − Horn " 2 − SAT
As far as we know, this is the first attempt to exploit ”in practice” the qHorn class. We saw that most of the time, the size of the Strong Backdoor sets
computed with our method provides smaller sets than regrading Horn formulas.
However, if in the case of Horn clauses, the exploitation of the Strong Backdoors
were trivial, due to the fact that solving a Horn Formula is done by using unit
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Fig. 1. Size of the Strong Backdoor for horn and q-Horn sub-formulas on some instances
propagation (and all the solvers do unit propagation at each node), in the case of
q-Horn clauses, we have to implement a dedicated method in the chosen solver.
This represents our main perspective: find a way to exploit the Strong Backdoors
sets obtained by our new method.
Another perspective would be to tackle the phenomenon discovered during
the experimentations. Indeed, q-Horn including strictly Horn, there must be a
way to guarantee that the Strong Backdoor sets obtained with the q-Horn class
will strictly be smaller than the one obtained with Horn formulas. The fact that,
in our method we only deal with variables in Q to compute our Strong Backdoor
set, may be the cause of the problem. Maybe we should find a way to integrate
the variables of H in the computation of the Strong Backdoor set to circumvent
the problem.
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