13 research outputs found
Existence versus Exploitation: The Opacity of Backbones and Backdoors Under a Weak Assumption
Backdoors and backbones of Boolean formulas are hidden structural properties.
A natural goal, already in part realized, is that solver algorithms seek to
obtain substantially better performance by exploiting these structures.
However, the present paper is not intended to improve the performance of SAT
solvers, but rather is a cautionary paper. In particular, the theme of this
paper is that there is a potential chasm between the existence of such
structures in the Boolean formula and being able to effectively exploit them.
This does not mean that these structures are not useful to solvers. It does
mean that one must be very careful not to assume that it is computationally
easy to go from the existence of a structure to being able to get one's hands
on it and/or being able to exploit the structure.
For example, in this paper we show that, under the assumption that P
NP, there are easily recognizable families of Boolean formulas with strong
backdoors that are easy to find, yet for which it is hard (in fact,
NP-complete) to determine whether the formulas are satisfiable. We also show
that, also under the assumption P NP, there are easily recognizable sets
of Boolean formulas for which it is hard (in fact, NP-complete) to determine
whether they have a large backbone
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
Understanding and Enhancing CDCL-based SAT Solvers
Modern conflict-driven clause-learning (CDCL) Boolean satisfiability (SAT) solvers routinely
solve formulas from industrial domains with millions of variables and clauses, despite the Boolean
satisfiability problem being NP-complete and widely regarded as intractable in general. At the
same time, very small crafted or randomly generated formulas are often infeasible for CDCL
solvers. A commonly proposed explanation is that these solvers somehow exploit the underlying
structure inherent in industrial instances. A better understanding of the structure of Boolean
formulas not only enables improvements to modern SAT solvers, but also lends insight as to why
solvers perform well or poorly on certain types of instances. Even further, examining solvers
through the lens of these underlying structures can help to distinguish the behavior of different
solving heuristics, both in theory and practice.
The first issue we address relates to the representation of SAT formulas. A given Boolean
satisfiability problem can be represented in arbitrarily many ways, and the type of encoding can
have significant effects on SAT solver performance. Further, in some cases, a direct encoding
to SAT may not be the best choice. We introduce a new system that integrates SAT solving
with computer algebra systems (CAS) to address representation issues for several graph-theoretic
problems. We use this system to improve the bounds on several finitely-verified conjectures
related to graph-theoretic problems. We demonstrate how our approach is more appropriate for
these problems than other off-the-shelf SAT-based tools.
For more typical SAT formulas, a better understanding of their underlying structural properties,
and how they relate to SAT solving, can deepen our understanding of SAT. We perform a largescale
evaluation of many of the popular structural measures of formulas, such as community
structure, treewidth, and backdoors. We investigate how these parameters correlate with CDCL
solving time, and whether they can effectively be used to distinguish formulas from different
domains. We demonstrate how these measures can be used as a means to understand the behavior
of solvers during search. A common theme is that the solver exhibits locality during search
through the lens of these underlying structures, and that the choice of solving heuristic can greatly
influence this locality. We posit that this local behavior of modern SAT solvers is crucial to their
performance.
The remaining contributions dive deeper into two new measures of SAT formulas. We first
consider a simple measure, denoted “mergeability,” which characterizes the proportion of input
clauses pairs that can resolve and merge. We develop a formula generator that takes as input a seed
formula, and creates a sequence of increasingly more mergeable formulas, while maintaining many
of the properties of the original formula. Experiments over randomly-generated industrial-like
instances suggest that mergeability strongly negatively correlates with CDCL solving time, i.e., as
the mergeability of formulas increases, the solving time decreases, particularly for unsatisfiable
instances.
Our final contribution considers whether one of the aforementioned measures, namely backdoor
size, is influenced by solver heuristics in theory. Starting from the notion of learning-sensitive
(LS) backdoors, we consider various extensions of LS backdoors by incorporating different branching
heuristics and restart policies. We introduce learning-sensitive with restarts (LSR) backdoors
and show that, when backjumping is disallowed, LSR backdoors may be exponentially smaller
than LS backdoors. We further demonstrate that the size of LSR backdoors are dependent on the
learning scheme used during search. Finally, we present new algorithms to compute upper-bounds
on LSR backdoors that intrinsically rely upon restarts, and can be computed with a single run of
a SAT solver. We empirically demonstrate that this can often produce smaller backdoors than
previous approaches to computing LS backdoors
Constraint Satisfaction Techniques for Combinatorial Problems
The last two decades have seen extraordinary advances in tools and techniques for constraint satisfaction. These advances have in turn created great interest in their industrial applications. As a result, tools and techniques are often tailored to meet the needs of industrial applications out of the box. We claim that in the case of abstract combinatorial problems in discrete mathematics, the standard tools and techniques require special considerations in order to be applied effectively. The main objective of this thesis is to help researchers in discrete mathematics weave through the landscape of constraint satisfaction techniques in order to pick the right tool for the job. We consider constraint satisfaction paradigms like satisfiability of Boolean formulas and answer set programming, and techniques like symmetry breaking. Our contributions range from theoretical results to practical issues regarding tool applications to combinatorial problems.
We prove search-versus-decision complexity results for problems about backbones and backdoors of Boolean formulas.
We consider applications of constraint satisfaction techniques to problems in graph arrowing (specifically in Ramsey and Folkman theory) and computational social choice. Our contributions show how applying constraint satisfaction techniques to abstract combinatorial problems poses additional challenges. We show how these challenges can be addressed. Additionally, we consider the issue of trusting the results of applying constraint satisfaction techniques to combinatorial problems by relying on verified computations
Backdoors in Satisfiability Problems
Although satisfiability problems (SAT) are NP-complete, state-of-the-art SAT solvers are able to solve large practical instances. The notion of backdoors has been introduced to capture structural properties of instances. Backdoors are a set of variables for which there exists some value assignment that leads to a polynomial-time solvable sub-problem. I address in this thesis the problem of finding all minimal backdoors, which is essential for studying value and variable ordering mistakes. I discuss our definition of sub-solvers and propose algorithms for finding backdoors. I implement our proposed algorithms by modifying a state-of-the-art SAT solver, Minisat. I analyze experimental results comparing our proposed algorithms to previous algorithms applied to random 3SAT, structured, and real-world instances. Our proposed algorithms improve over previous algorithms for finding backdoors in two ways. First, our algorithms often find smaller backdoors. Second, our algorithms often find a much larger number of backdoors
Harnessing tractability in constraint satisfaction problems
The Constraint Satisfaction Problem (CSP) is a fundamental NP-complete problem with many applications in artificial intelligence. This problem has enjoyed considerable scientific attention in the past decades due to its practical usefulness and the deep theoretical questions it relates to. However, there is a wide gap between practitioners, who develop solving techniques that are efficient for industrial instances but exponential in the worst case, and theorists who design sophisticated polynomial-time algorithms for restrictions of CSP defined by certain algebraic properties. In this thesis we attempt to bridge this gap by providing polynomial-time algorithms to test for membership in a selection of major tractable classes. Even if the instance does not belong to one of these classes, we investigate the possibility of decomposing efficiently a CSP instance into tractable subproblems through the lens of parameterized complexity. Finally, we propose a general framework to adapt the concept of kernelization, central to parameterized complexity but hitherto rarely used in practice, to the context of constraint reasoning. Preliminary experiments on this last contribution show promising results
Exploiting Structure In Combinatorial Problems With Applications In Computational Sustainability
Combinatorial decision and optimization problems are at the core of many tasks with practical importance in areas as diverse as planning and scheduling, supply chain management, hardware and software verification, electronic commerce, and computational biology. Another important source of combinatorial problems is the newly emerging field of computational sustainability, which addresses decision-making aimed at balancing social, economic and environmental needs to guarantee the long-term prosperity of life on our planet. This dissertation studies different forms of problem structure that can be exploited in developing scalable algorithmic techniques capable of addressing large real-world combinatorial problems. There are three major contributions in this work: 1) We study a form of hidden problem structure called a backdoor, a set of key decision variables that captures the combinatorics of the problem, and reveal that many real-world problems encoded as Boolean satisfiability or mixed-integer linear programs contain small backdoors. We study backdoors both theoretically and empirically and characterize important tradeoffs between the computational complexity of finding backdoors and their effectiveness in capturing problem structure succinctly. 2) We contribute several domain-specific mathematical formulations and algorithmic techniques that exploit specific aspects of problem structure arising in budget-constrained conservation planning for wildlife habitat connectivity. Our solution approaches scale to real-world conservation settings and provide important decision-support tools for cost-benefit analysis. 3) We propose a new survey-planning methodology to assist in the construction of accurate predictive models, which are especially relevant in sustainability areas such as species- distribution prediction and climate-change impact studies. In particular, we design a technique that takes advantage of submodularity, a structural property of the function to be optimized, and results in a polynomial-time procedure with approximation guarantees