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Saxena A.,Axioma Inc. | Bonami P.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF) | Lee J.,IBM
Mathematical Programming | Year: 2010

This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x T . We use the non-convex constraint Y - x xT 0 to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y - x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint Y - x xT 0 to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings. © 2010 Springer and Mathematical Programming Society.


Saxena A.,Axioma Inc | Bonami P.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF) | Lee J.,IBM
Mathematical Programming | Year: 2011

A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y ij to represent each of the products xixj of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y - xxT ≥ 0 and disjunctive programming. On the other hand, the main drawback of such an extended formulation is its huge size, even for problems for which the number of xi variables is moderate. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen-reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on box-QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 s, even for large instances with 100 variables; the SDP+RLT relaxations for the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver. © 2010 Springer and Mathematical Programming Society.


Chepoi V.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF) | Felsner S.,TU Berlin
Computational Geometry: Theory and Applications | Year: 2013

In this note, we present a simple combinatorial factor 6 algorithm for approximating the minimum hitting set of a family R={R1,⋯, Rn} of axis-parallel rectangles in the plane such that there exists an axis-monotone curve γ that intersects each rectangle in the family. The quality of the hitting set is shown by comparing it to the size of a packing (set of pairwise non-intersecting rectangles) that is constructed along, hence, we also obtain a factor 6 approximation for the maximum packing of R. In cases where the axis-monotone curve γ intersects the same side (e.g. the bottom side) of each rectangle in the family the approximation factor for hitting set and packing is 3. © 2013 Elsevier B.V.


Avellaneda F.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF) | Morin R.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF)
Proceedings - International Conference on Application of Concurrency to System Design, ACSD | Year: 2013

Message Sequence Graphs (MSGs) form a popular model often used for the documentation of telecommunication protocols. They consist of typical scenarios of message exchanges depicted as partial-orders of events that lead from one control state to another. On the other hand Petri nets are a well-known formalism for distributed or parallel systems based on the notion of token game. Both approaches profit by a visual presentation and are the subject of numerous formal verification techniques and tools. In this paper we investigate a formalism which provides MSGs with the notion of token game and extends Petri nets with both control states and partial orders. Providing Petri nets with control states corresponds precisely to the model of Vector Addition Systems with States (VASSs). Thus we need to define first a partial-order semantics for VASSs which adopts the basic features of communication scenarios. To do so we extend simply the process semantics of Petri nets. We obtain a formal model that enjoys several interesting properties in terms of expressiveness and concision. The addition of control states to Petri nets under the partial-order semantics leads to undecidable problems. Similarly to MSGs, one cannot decide in particular whether two given VASSs describe the same process language. However we show that basic problems about the set of markings reached along the processes of a VASS, such as boundedness, covering and reachability, can be reduced to the analogous problems for Petri nets. This relies on a new technique that simulates all prefixes of all processes. In this way Petri net tools can be used to verify the properties of a VASS under the process semantics. We present also a technique to check effectively any MSO property of these partial orders, provided that the given system is bounded. This enables us to tackle more verification problems and subsumes known results for the model checking of MSGs. All algorithms presented in this paper have been implemented in a prototype tool available online. © 2013 IEEE.


Delacourt M.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF)
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2011

Cellular automata are a parallel and synchronous computing model, made of infinitely many finite automata updating according to the same local rule. Rice's theorem states that any nontrivial property over computable functions is undecidable. It has been adapted by Kari to limit sets of cellular automata [7], that is the set of configurations that can be reached arbitrarily late. This paper proves a new Rice theorem for μ-limit sets, which are sets of configurations often reached arbitrarily late. © 2011 Springer-Verlag.


Kaced T.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF)
IEEE International Symposium on Information Theory - Proceedings | Year: 2011

To split a secret s between several participants, we generate (for each value of s) shares for all participants. The goal: authorized groups of participants should be able to reconstruct the secret but forbidden ones get no information about it. We introduce several notions of non-perfect secret sharing, where some small information leak is permitted. We study its relation to the Kolmogorov complexity version of secret sharing (establishing some connection in both directions) and the effects of changing the secret size (showing that we can decrease the size of the secret and the information leak at the same time). © 2011 IEEE.


Santocanale L.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF) | Venema Y.,University of Amsterdam
Advances in Modal Logic 2006 | Year: 2010

We reconstruct the syntax and semantics of monotone modal logic, in the style of Moss' coalgebraic logic. To that aim, we replace the box and diamond with a modality r which takes a finite collection of finite sets of formulas as its argument. The semantics of this modality in monotone neighborhood models is defined in terms of a version of relation lifting that is appropriate for this setting. We prove that the standard modal language and our r-based one are effectively equi-expressive, meaning that there are effective translations in both directions. We prove and discuss some algebraic laws that govern the interaction of r with the Boolean operations. These laws enable us to rewrite each formula into a special kind of disjunctive normal form that we call transparent. For such transparent formulas it is relatively easy to define the bisimulation quantifiers that one may associate with our notion of relation lifting. This allows us to prove the main result of the paper, viz., that monotone modal logic enjoys the property of uniform interpolation.


Baudru N.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF)
Theoretical Computer Science | Year: 2011

Asynchronous automata are a model of communication processes with a control structure distributed on a set P of processes, global initializations and global accepting conditions. The well-known theorem of Zielonka states that they recognize exactly the class of regular Mazurkiewicz trace languages. The corresponding synthesis problem is, given a global specification A of any regular trace language L, to build an asynchronous automaton that recognizes L, automatically. Yet, all such existing constructions are quite involved and yield an explosion of the number of states in each process, which is exponential in both the sizes of A and P. In this paper, we introduce the particular case of distributed asynchronous automata, which require that the initializations and the accepting conditions are distributed as well. We present an original technique based on simple compositions/decompositions of these distributed asynchronous automata that results in the construction of smaller non-deterministic asynchronous automata: now, the number of states in each process is only polynomial in the size of A, but is still exponential in the size of P. © 2011 Elsevier B.V. All rights reserved.


Schwind C.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF)
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2010

We present a formal framework for theory revision that is based on a tableaux algorithm. We use a semantic tableaux system for generating the tableau of a belief base and of a revision formula. Specific revision operators can then be defined in terms of tableaux branches. Tableaux branches of a formula correspond to implicants of the formula. We define a new set of revision postulates for implicant sets and we show that they verify the AGM revision postulates. Then we consider specific distance based revision operators and in particular those based on the symmetrical difference between interpretations. We show that the minimal symmetric difference between the model sets of two implicants can be obtained by a simple binary operation on these implicants. Finally, we define three specific distance based revision operators. The representation of a formula by implicants is not unique (our approach is not based on prime implicants). We show that our implicant revision approach obtains the same revision result operated on different equivalent implicant sets for the three revision operators we propose. © 2010 Springer-Verlag.


Durand B.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF) | Romashchenko A.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF) | Shen A.,CNRS Laboratory of Fundamental Informatics of Marseille (LIF)
Journal of Computer and System Sciences | Year: 2012

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many fields, ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene's fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann's self-reproducing automata; similar ideas were also used by P. Gács in the context of error-correcting computations. This construction is rather flexible, so it can be used in many ways. We show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (in which any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, to characterize effectively closed one-dimensional subshifts in terms of two-dimensional subshifts of finite type (an improvement of a result by M. Hochman), to construct a tile set that has only complex tilings, and to construct a "robust" aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter, we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: There exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed. Some of these results were included in the DLT extended abstract (Durand et al., 2008 [9]) and in the ICALP extended abstract (Durand et al., 2009 [10]). © 2011 Elsevier Inc. All rights reserved.

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