16 research outputs found
Treewidth with a Quantifier Alternation Revisited
In this paper we take a closer look at the parameterized complexity of existsforall SAT, the prototypical complete problem of the class Sigma_2^p, the second level of the polynomial hierarchy. We provide a number of tight fine-grained bounds on the complexity of this problem and its variants with respect to the most important structural graph parameters. Specifically, we show the following lower bounds (assuming the ETH):
- It is impossible to decide existsforall SAT in time less than double-exponential in the input formula\u27s treewidth. More strongly, we establish the same bound with respect to the formula\u27s primal vertex cover, a much more restrictive measure. This lower bound, which matches the performance of known algorithms, shows that the degeneration of the performance of treewidth-based algorithms to a tower of exponentials already begins in problems with one quantifier alternation.
- For the more general existsforall CSP problem over a non-boolean domain of size B, there is no algorithm running in time 2^{B^{o(vc)}}, where vc is the input\u27s primal vertex cover.
- existsforall SAT is already NP-hard even when the input formula has constant modular treewidth (or clique-width), indicating that dense graph parameters are less useful for problems in Sigma_2^p.
- For the two weighted versions of existsforall SAT recently introduced by de Haan and Szeider, called exists_kforall SAT and existsforall_k SAT, we give tight upper and lower bounds parameterized by treewidth (or primal vertex cover) and the weight k. Interestingly, the complexity of these two problems turns out to be quite different: one is double-exponential in treewidth, while the other is double-exponential in k.
We complement the above negative results by showing a double-exponential FPT algorithm for QBF parameterized by vertex cover, showing that for this parameter the complexity never goes beyond double-exponential, for any number of quantifier alternations
Graph Homomorphism, Monotone Classes and Bounded Pathwidth
A recent paper describes a framework for studying the computational
complexity of graph problems on monotone classes, that is those omitting a set
of graphs as a subgraph. If the problems lie in the framework, and many do,
then the computational complexity can be described for all monotone classes
defined by a finite set of omitted subgraphs. It is known that certain
homomorphism problems, e.g. -Colouring, do not sit in the framework. By
contrast, we show that the more general problem of Graph Homomorphism does sit
in the framework.
The original framework had examples where hard versus easy were NP-complete
versus P, or at least quadratic versus almost linear. We give the first example
of a problem in the framework such that hardness is in the polynomial hierarchy
above NP. Considering a variant of the colouring game as studied by Bodlaender,
we show that with the restriction of bounded alternation, the list version of
this problem is contained in the framework. The hard cases are
-complete and the easy cases are in P.
The cases in P comprise those classes for which the pathwidth is bounded.
Bodlaender explains that Sequential -Colouring Construction Game is in P on
classes with bounded vertex separation number, which coincides with bounded
pathwidth on unordered graphs. However, these graphs are ordered with a playing
order for the two players, which corresponds to a prefix pattern in a
quantified formula. We prove that Sequential -Colouring Construction Game is
Pspace-complete on some class of bounded pathwidth, using a celebrated result
of Atserias and Oliva.
We consider several locally constrained variants of the homomorphism problem.
Like -Colouring, none of these is in the framework. However, when we
consider the bounded-degree restrictions, we prove that each of these problems
is in our framework
Does Treewidth Help in Modal Satisfiability?
Many tractable algorithms for solving the Constraint Satisfaction Problem
(CSP) have been developed using the notion of the treewidth of some graph
derived from the input CSP instance. In particular, the incidence graph of the
CSP instance is one such graph. We introduce the notion of an incidence graph
for modal logic formulae in a certain normal form. We investigate the
parameterized complexity of modal satisfiability with the modal depth of the
formula and the treewidth of the incidence graph as parameters. For various
combinations of Euclidean, reflexive, symmetric and transitive models, we show
either that modal satisfiability is FPT, or that it is W[1]-hard. In
particular, modal satisfiability in general models is FPT, while it is
W[1]-hard in transitive models. As might be expected, modal satisfiability in
transitive and Euclidean models is FPT.Comment: Full version of the paper appearing in MFCS 2010. Change from v1:
improved section 5 to avoid exponential blow-up in formula siz
Generalizing Consistency and other Constraint Properties to Quantified Constraints
Quantified constraints and Quantified Boolean Formulae are typically much
more difficult to reason with than classical constraints, because quantifier
alternation makes the usual notion of solution inappropriate. As a consequence,
basic properties of Constraint Satisfaction Problems (CSP), such as consistency
or substitutability, are not completely understood in the quantified case.
These properties are important because they are the basis of most of the
reasoning methods used to solve classical (existentially quantified)
constraints, and one would like to benefit from similar reasoning methods in
the resolution of quantified constraints. In this paper, we show that most of
the properties that are used by solvers for CSP can be generalized to
quantified CSP. This requires a re-thinking of a number of basic concepts; in
particular, we propose a notion of outcome that generalizes the classical
notion of solution and on which all definitions are based. We propose a
systematic study of the relations which hold between these properties, as well
as complexity results regarding the decision of these properties. Finally, and
since these problems are typically intractable, we generalize the approach used
in CSP and propose weaker, easier to check notions based on locality, which
allow to detect these properties incompletely but in polynomial time
Backdoors to Acyclic SAT
Backdoor sets, a notion introduced by Williams et al. in 2003, are certain
sets of key variables of a CNF formula F that make it easy to solve the
formula; by assigning truth values to the variables in a backdoor set, the
formula gets reduced to one or several polynomial-time solvable formulas. More
specifically, a weak backdoor set of F is a set X of variables such that there
exits a truth assignment t to X that reduces F to a satisfiable formula F[t]
that belongs to a polynomial-time decidable base class C. A strong backdoor set
is a set X of variables such that for all assignments t to X, the reduced
formula F[t] belongs to C.
We study the problem of finding backdoor sets of size at most k with respect
to the base class of CNF formulas with acyclic incidence graphs, taking k as
the parameter. We show that
1. the detection of weak backdoor sets is W[2]-hard in general but
fixed-parameter tractable for r-CNF formulas, for any fixed r>=3, and
2. the detection of strong backdoor sets is fixed-parameter approximable.
Result 1 is the the first positive one for a base class that does not have a
characterization with obstructions of bounded size. Result 2 is the first
positive one for a base class for which strong backdoor sets are more powerful
than deletion backdoor sets.
Not only SAT, but also #SAT can be solved in polynomial time for CNF formulas
with acyclic incidence graphs. Hence Result 2 establishes a new structural
parameter that makes #SAT fixed-parameter tractable and that is incomparable
with known parameters such as treewidth and clique-width.
We obtain the algorithms by a combination of an algorithmic version of the
Erd\"os-P\'osa Theorem, Courcelle's model checking for monadic second order
logic, and new combinatorial results on how disjoint cycles can interact with
the backdoor set
Decomposing quantified conjunctive (or disjunctive) formulas
Model checking---deciding if a logical sentence holds on a structure---is a basic computational task that is well known to be intractable in general. For first-order logic on finite structures, it is PSPACE-complete, and the natural evaluation algorithm exhibits exponential dependence on the formula. We study model checking on the quantified conjunctive fragment of first-order logic, namely, prenex sentences having a purely conjunctive quantifier-free part. Following a number of works, we associate a graph to the quantifier-free part; each sentence then induces a prefixed graph, a quantifier prefix paired with a graph on its variables. We give a comprehensive classification of the sets of prefixed graphs on which model checking is tractable based on a novel generalization of treewidth that generalizes and places into a unified framework a number of existing results
Lower Bounds for QBFs of Bounded Treewidth
The problem of deciding the validity (QSAT) of quantified Boolean formulas
(QBF) is a vivid research area in both theory and practice. In the field of
parameterized algorithmics, the well-studied graph measure treewidth turned out
to be a successful parameter. A well-known result by Chen in parameterized
complexity is that QSAT when parameterized by the treewidth of the primal graph
of the input formula together with the quantifier depth of the formula is
fixed-parameter tractable. More precisely, the runtime of such an algorithm is
polynomial in the formula size and exponential in the treewidth, where the
exponential function in the treewidth is a tower, whose height is the
quantifier depth. A natural question is whether one can significantly improve
these results and decrease the tower while assuming the Exponential Time
Hypothesis (ETH). In the last years, there has been a growing interest in the
quest of establishing lower bounds under ETH, showing mostly problem-specific
lower bounds up to the third level of the polynomial hierarchy. Still, an
important question is to settle this as general as possible and to cover the
whole polynomial hierarchy. In this work, we show lower bounds based on the ETH
for arbitrary QBFs parameterized by treewidth (and quantifier depth). More
formally, we establish lower bounds for QSAT and treewidth, namely, that under
ETH there cannot be an algorithm that solves QSAT of quantifier depth i in
runtime significantly better than i-fold exponential in the treewidth and
polynomial in the input size. In doing so, we provide a versatile reduction
technique to compress treewidth that encodes the essence of dynamic programming
on arbitrary tree decompositions. Further, we describe a general methodology
for a more fine-grained analysis of problems parameterized by treewidth that
are at higher levels of the polynomial hierarchy