12 research outputs found
Solving hard industrial combinatorial problems with SAT
The topic of this thesis is the development of SAT-based techniques and tools for solving industrial combinatorial problems. First, it describes the architecture of state-of-the-art SAT and SMT Solvers based on the classical DPLL procedure. These systems can be used as black boxes for solving combinatorial problems. However, sometimes we can increase their efficiency with slight modifications of the basic algorithm. Therefore, the study and development of techniques for adjusting SAT Solvers to specific combinatorial problems is the first goal of this thesis.
Namely, SAT Solvers can only deal with propositional logic. For solving general combinatorial problems, two different approaches are possible:
- Reducing the complex constraints into propositional clauses.
- Enriching the SAT Solver language.
The first approach corresponds to encoding the constraint into SAT. The second one corresponds to using propagators, the basis for SMT Solvers. Regarding the first approach, in this document we improve the encoding of two of the most important combinatorial constraints: cardinality constraints and pseudo-Boolean constraints. After that, we present a new mixed approach, called lazy decomposition, which combines the advantages of encodings and propagators.
The other part of the thesis uses these theoretical improvements in industrial combinatorial problems. We give a method for efficiently scheduling some professional sport leagues with SAT. The results are promising and show that a SAT approach is valid for these problems.
However, the chaotical behavior of CDCL-based SAT Solvers due to VSIDS heuristics makes it difficult to obtain a similar solution for two similar problems. This may be inconvenient in real-world problems, since a user expects similar solutions when it makes slight modifications to the problem specification. In order to overcome this limitation, we have studied and solved the close solution problem, i.e., the problem of quickly finding a close solution when a similar problem is considered
Classification of plane germs: metric and valorative properties
In this memory we follow the geometric approach of Casasâ boof [1] for studying of the singularities of plane germs of curves, which updates Enriquesâ works to modern standards and reviews the modern development of the theorey from the point of view of infinitely near points.
This memory has a two sided goal: on one hand, we want to acquire skills with the tools and concepts of the singularity theory and the valuative theory, both the classical ones and the more recent ones. On the other hand, we want to study in depth the different implicit concepts and notions involved in the Favre and Jonssonâs new approach, such as the ultrametric space structure of the set of irreducible germs of plane curves and the tree structure of the valuations
A parametric approach for smaller and better encodings of cardinality constraints
Adequate encodings for high-level constraints are a key ingredient for the application of SAT technology. In particular, cardinality constraints state that at most (at least, or exactly) k out of n propositional variables can be true. They are crucial in many applications. Although sophisticated encodings for cardinality constraints exist, it is well known that for small n and k straightforward encodings without auxiliary variables sometimes behave better, and that the choice of the right trade-off between minimizing either the number of variables or the number of clauses is highly application-dependent. Here we build upon previous work on Cardinality Networks to get the best of several worlds: we develop an arc-consistent encoding that, by recursively decomposing the constraint into smaller ones, allows one to decide which encoding to apply to each sub-constraint. This process minimizes a function λ·num- vars + num-clauses, where λ is a parameter that can be tuned by the user. Our careful experimental evaluation shows that (e.g., for λ = 5) this new technique produces much smaller encodings in variables and clauses, and indeed strongly improves SAT solvers' performance.Postprint (authorâs final draft
Aspectes geomĂštrics i dinĂ mics de transformacions racionals planes
Donada una aplicaciĂł racional en una varietat complexa, Bellon i Viallet van definit lâentropia algebraica dâaquesta aplicaciĂł i van provar que aquest valor Ă©s un invariant biracional. Un invariant biracional equivalent Ă©s el grau asimptĂČtic,
grau dinĂ mic o complexitat, definit per Boukraa i Maillard. Aquesta nociĂł Ă©s propera a la complexitat definida per Arnold.
Conjecturalment, el grau asimptĂČtic satisfĂ una recurrĂšncia lineal amb coeficients enters. Aquesta conjectura ha estat provada en el cas polinĂČmic en el pla afĂ complex per Favre i Jonsson i resta oberta en per al cas projectiu global i per al cas local. Lâestudi de lâarbre valoratiu de Favre i Jonsson ha resultat clau per resoldre la conjectura en el cas polinĂČmic en el pla afĂ complex.
El beneficiari ha estudiat lâarbre valoratiu global de Favre i Jonsson i ha reinterpretat algunes nocions i resultats des dâun punt de vista mĂ©s geomĂštric. AixĂ mateix, ha estudiat la demostraciĂł de la conjectura de Bellon â Viallet en el cas polinĂČmic en el pla afĂ complex com a primer pas per trobar una demostraciĂł en el cas local i projectiu global en estudis futurs.
El projecte inclou un estudi detallat de l'arbre valoratiu global des d'un punt de vista geomĂštric i els primers passos de la
demostraciĂł de la conjectura de Bellon - Viallet en el cas polinĂČmic en el pla afĂ complex que van efectuar Favre i Jonsson.Given a rational application on a complex variety, Bellon and Viallet defined the algebraic entropy of the application and
proved that it is a birational invariant. An equivalent birational invariant is the assymptotic degree, dynamical degree or
complexity, defined by Boukraa and Maillard. This notion is very close to Arnold's complexity.
The asymptotic degree satisfies conjecturally a linear recursion formula with integer coefficients. This conjecture has been proved in the polynomial case on the affine complex plane by Favre and Jonsson, but it remains open in the projective global case and in the local case. The valorative tree theory developed by Favre and Jonsson is a key factor for solving the conjecture in the polynomial case on the complex affine plane.
The student studied Favre and Jonsson's global valorative tree and he reinterpreted some notions and results from a more geometrical point of view. Moreover, he studied the proof of Bellon-Viallet's Conjecture in the polynomial case on the complex affine plane as a first step for solving the conjecture in the local case and in the global projective case in further studies.
The project includes a detailed study of the global valorative tree from a geometrical point of view and some notions of Favre and Jonsson's proof of Bellon-Viallet conjecture in the polynomial case on the affine complex plane
Classification of plane germs: metric and valorative properties
In this memory we follow the geometric approach of Casasâ boof [1] for studying of the singularities of plane germs of curves, which updates Enriquesâ works to modern standards and reviews the modern development of the theorey from the point of view of infinitely near points.
This memory has a two sided goal: on one hand, we want to acquire skills with the tools and concepts of the singularity theory and the valuative theory, both the classical ones and the more recent ones. On the other hand, we want to study in depth the different implicit concepts and notions involved in the Favre and Jonssonâs new approach, such as the ultrametric space structure of the set of irreducible germs of plane curves and the tree structure of the valuations
The ultrametric space of plane branches
We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] â© is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.Peer ReviewedPostprint (published version
The ultrametric space of plane branches
We study properties of the space of irreducible germs of plane curves
(branches), seen as an ultrametric space. We provide various geometrical
methods to measure the distance between two branches and to compare
distances between branches, in terms of topological invariants of the singularity
which comprises some of the branches. We show that, in spite
of being very close to the notion of intersection multiplicity between two
germs, this notion of distance behaves very differently. For instance, any
value in [0, 1] \cap Q is attained as the distance between a fixed branch and
some other branch, in contrast with the fact that the semigroup of the
fixed branch has gaps. We also present results that lead to interpret this
distance as a sort of geometric distance between the topological equivalence
or equisingularity classes of branche
The ultrametric space of plane branches
We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] â© is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.Peer Reviewe
Reducing chaos in SAT-like search: finding solutions close to a given one
Motivated by our own industrial users, we attack the following
challenge that is crucial in many practical planning, scheduling
or timetabling applications. Assume that a solver has found a solution
for a given hard problem and, due to unforeseen circumstances (e.g., rescheduling),
or after an analysis by a committee, a few more constraints
have to be added and the solver has to be re-run. Then it is almost always
important that the new solution is âcloseâ to the original one.
The activity-based variable selection heuristics used by SAT solvers
make search chaotic, i.e., extremely sensitive to the initial conditions.
Therefore, re-running with just one additional clause added at the end
of the input usually gives a completely different solution. We show that
naive approaches for finding close solutions do not work at all, and that
solving the Boolean optimization problem is far too inefficient: to find
a reasonably close solution, state-of-the-art tools typically require much
more time than was needed to solve the original problem.
Here we propose the first (to our knowledge) approach that obtains
close solutions quickly. In fact, it typically finds the optimal (i.e., closest)
solution in only 25% of the time the solver took in solving the original
problem. Our approach requires no deep theoretical or conceptual innovations.
Still, it is non-trivial to come up with and will certainly be
valuable for researchers and practitioners facing the same problem
A parametric approach for smaller and better encodings of cardinality constraints
Adequate encodings for high-level constraints are a key ingredient for the application of SAT technology. In particular, cardinality constraints state that at most (at least, or exactly) k out of n propositional variables can be true. They are crucial in many applications. Although sophisticated encodings for cardinality constraints exist, it is well known that for small n and k straightforward encodings without auxiliary variables sometimes behave better, and that the choice of the right trade-off between minimizing either the number of variables or the number of clauses is highly application-dependent. Here we build upon previous work on Cardinality Networks to get the best of several worlds: we develop an arc-consistent encoding that, by recursively decomposing the constraint into smaller ones, allows one to decide which encoding to apply to each sub-constraint. This process minimizes a function λ·num- vars + num-clauses, where λ is a parameter that can be tuned by the user. Our careful experimental evaluation shows that (e.g., for λ = 5) this new technique produces much smaller encodings in variables and clauses, and indeed strongly improves SAT solvers' performance