The hidden matching-structure of the composition of strips: a polyhedral perspective

Stable set problems subsume matching problems since a matching is a stable set in a so- called line graph but stable set problems are hard in general while matching can be solved efficiently [11]. However, there are some classes of graphs where the stable set problem can be solved efficiently. A famous class is that of claw-free graphs; in fact, in 1980 Minty [19, 20] gave the first polynomial time algorithm for finding a maximum weighted stable set (mwss) in a claw-free graph. One of the reasons why stable set in claw-free graphs can be solved efficiently is because the so called augmenting path theorem [4] for matching generalizes to claw-free graphs [5] (this is what Minty is using). We believe that another core reason is structural and that there is a intrinsic matching structure in claw-free graphs. Indeed, recently Chudnovsky and Seymour [8] shed some light on this by proposing a decomposition theorem for claw-free graphs where they describe how to compose all claw-free graphs from building blocks. Interestingly the composition operation they defined seems to have nice consequences for the stable set problem that go much beyond claw-free graphs. Actually in a recent paper [21] Oriolo, Pietropaoli and Stauffer have revealed how one can use the structure of this composition to solve the stable set problem for composed graphs in polynomial time by reduction to matching. In this paper we are now going to reveal the nice polyhedral counterpart of this composition procedure, i.e. how one can use the structure of this composition to describe the stable set polytope from the matching one and, more importantly, how one can use it to separate over the stable set polytope in polynomial time. We will then apply those general results back to where they originated from: stable set in claw-free graphs, to show that the stable set polytope can be reduced to understanding the polytope in very basic structures (for most of which it is already known). In particular for a general claw-free graph G, we show two integral extended formulation for STAB(G) and a procedure to separate in polynomial time over STAB(G); moreover, we provide a complete characterization of STAB(G) when G is any claw-free graph with stability number at least 4 having neither homogeneous pairs nor 1-joins. We believe that the missing bricks towards the characterization of the stable set polytope of claw-free graphs are more technical than fundamentals; in particular, we have a characterization for most of the building bricks of the Chudnovsky-Seymour decomposition result and we are therefore very confident it is only a question of time before we solve the remaining case

The stable set polytope of claw-free graphs with stability number at least four. I. Fuzzy antihat graphs are W-perfect

Abstract Fuzzy antihat graphs are graphs obtained as 2-clique-bond compositions of fuzzy line graphs with three different types of three-cliqued graphs. By the decomposition theorem of Chudnovsky and Seymour [2] , fuzzy antihat graphs form a large subclass of claw-free, not quasi-line graphs with stability number at least four and with no 1-joins. A graph is W -perfect if its stable set polytope is described by: nonnegativity, rank, and lifted 5-wheel inequalities. By exploiting the polyhedral properties of the 2-clique-bond composition, we prove that fuzzy antihat graphs are W -perfect and we move a crucial step towards the solution of the longstanding open question of finding an explicit linear description of the stable set polytope of claw-free graphs

Synthesis of Switching Protocols from Temporal Logic Specifications

We propose formal means for synthesizing switching protocols that determine the sequence in which the modes of a switched system are activated to satisfy certain high-level specifications in linear temporal logic. The synthesized protocols are robust against exogenous disturbances on the continuous dynamics. Two types of finite transition systems, namely under- and over-approximations, that abstract the behavior of the underlying continuous dynamics are defined. In particular, we show that the discrete synthesis problem for an under-approximation can be formulated as a model checking problem, whereas that for an over-approximation can be transformed into a two-player game. Both of these formulations are amenable to efficient, off-the-shelf software tools. By construction, existence of a discrete switching strategy for the discrete synthesis problem guarantees the existence of a continuous switching protocol for the continuous synthesis problem, which can be implemented at the continuous level to ensure the correctness of the nonlinear switched system. Moreover, the proposed framework can be straightforwardly extended to accommodate specifications that require reacting to possibly adversarial external events. Finally, these results are illustrated using three examples from different application domains

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Experimenting with Constraint Programming Techniques in Artificial Intelligence: Automated System Design and Verification of Neural Networks

This thesis focuses on the application of Constraint Satisfaction and Optimization techniques in two Artificial Intelligence (AI) domains: automated design of elevator systems and verification of Neural Networks (NNs). The three main areas of interest for my work are (i) the languages for defining the constraints for the systems, (ii) the algorithms and encodings that enable solving the problems considered and (iii) the tools that implement such algorithms. Given the expressivity of the domain description languages and the availability of effective tools, several problems in diverse application fields have been solved successfully using constraint satisfaction techniques. The two case studies herewith presented are no exception, even if they entail different challenges in the adoption of such techniques. Automated design of elevator systems not only requires encoding of feasibility (hard) constraints, but should also take into account design preferences, which can be expressed in terms of cost functions whose optimal or near-optimal value characterizes “good” design choices versus “poor” ones. Verification of NNs (and other machine-learned implements) requires solving large-scale constraint problems which may become the main bottlenecks in the overall verification procedure. This thesis proposes some ideas for tackling such challenges, including encoding techniques for automated design problems and new algorithms for handling the optimization problems arising from verification of NNs. The proposed algorithms and techniques are evaluated experimentally by developing tools that are made available to the research community for further evaluation and improvement

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view