A method for parallel scheduling of multi-rate co-simulation on multi-core platforms

The design of cyber-physical systems is a complex process and relies on the simulation of the system behavior before its deployment. Such is the case, for instance, of joint simulation of the different subsystems that constitute a hybrid automotive powertrain. Co-simulation allows system designers to simulate a whole system composed of a number of interconnected subsystem simulators. Traditionally, these subsystems are modeled by experts of different fields using different tools, and then integrated into one tool to perform simulation at the system-level. This results in complex and compute-intensive co-simulations and requires the parallelization of these co-simulations in order to accelerate their execution. The simulators composing a co-simulation are characterized by the rates of data exchange between the simulators, defined by the engineers who set the communication steps. The RCOSIM approach allows the parallelization on multi-core processors of co-simulations using the FMI standard. This approach is based on the exploitation of the co-simulation parallelism where dependent functions perform different computation tasks. In this paper, we extend RCOSIM to handle additional co-simulation requirements. First, we extend the co-simulation to multi-rate, i.e. where simulators are assigned different communication steps. We present graph transformation rules and an algorithm that allow the execution of each simulator at its respective rate while ensuring correct and efficient data exchange between simulators. Second, we present an approach based on acyclic orientation of mixed graphs for handling mutual exclusion constraints between functions that belong to the same simulator due to the non-thread-safe implementation of FMI. We propose an exact algorithm and a heuristic for performing the acyclic orientation. The final stage of the proposed approach consists in scheduling the co-simulation on a multi-core architecture. We propose an algorithm and a heuristic for computing a schedule which minimizes the total execution time of the co-simulation. We evaluate the performance of our proposed approach in terms of the obtained execution speed. By applying our approach on an industrial use case, we obtained a maximum speedup of 2.91 on four cores.

EW-Tableaux, Le-Tableaux, Tree-Like Tableaux and the Abelian Sandpile Model

A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding uniquely to an acyclic orientation, with a unique sink, of a certain bipartite graph called a Ferrers graph. We give a bijective proof of a result of Ehrenborg and van Willigenburg showing that EW-tableaux of a given shape are equinumerous with permutations with a given set of excedances. This leads to an explicit bijection between EW-tableaux and the much studied Le-tableaux, as well as the tree-like tableaux introduced by Aval, Boussicault and Nadeau.We show that the set of EW-tableaux on a given Ferrers diagram are in 1-1 correspondence with the minimal recurrent configurations of the Abelian sandpile model on the corresponding Ferrers graph.Another bijection between EW-tableaux and tree-like tableaux, via spanning trees on the corresponding Ferrers graphs, connects the tree-like tableaux to the minimal recurrent configurations of the Abelian sandpile model on these graphs. We introduce a variation on the EW-tableaux, which we call NEW-tableaux, and present bijections from these to Le-tableaux and tree-like tableaux. We also present results on various properties of and statistics on EW-tableaux and NEW-tableaux, as well as some open problems on these.

NONCROSSING SETS AND A GRASSMANN ASSOCIAHEDRON

We study a natural generalization of the noncrossing relation between pairs of elements in$[n]$to$k$-tuples in$[n]$that was first considered by Petersenet al.[J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on$\binom{[n]}{k}$induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product$[k]\times [n-k]$of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for$k=2$. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex thenoncrossing complex, and the polytope derived from it the dualGrassmann associahedron. We extend results of Petersenet al.[J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphism$G_{k,n}\cong G_{n-k,n}$. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define aGrassmann–Tamari orderon maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersenet al.[J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part.

Some Discrete Optimization Problems With Hamming and H-Comparability Graphs

Any H-comparability graph contains a Hamming graph as spanningsubgraph. An acyclic orientation of an H-comparability graph contains an acyclic orientation of the spanning Hamming graph, called sequence graph in the classical open-shop scheduling problem. We formulate different discrete optimization problems on the Hamming graphs and on H-comparability graphs and consider their complexity and relationship. Moreover, we explore the structures of these graphs in the class of irreducible sequences for the open shop problem in this paper.

Sortable Elements for Quivers with Cycles

Each Coxeter element $c$ of a Coxeter group $W$ defines a subset of $W$ called the $c$-sortable elements. The choice of a Coxeter element of $W$ is equivalent to the choice of an acyclic orientation of the Coxeter diagram of $W$. In this paper, we define a more general notion of $\Omega$-sortable elements, where $\Omega$ is an arbitrary orientation of the diagram, and show that the key properties of $c$-sortable elements carry over to the $\Omega$-sortable elements. The proofs of these properties rely on reduction to the acyclic case, but the reductions are nontrivial; in particular, the proofs rely on a subtle combinatorial property of the weak order, as it relates to orientations of the Coxeter diagram. The $c$-sortable elements are closely tied to the combinatorics of cluster algebras with an acyclic seed; the ultimate motivation behind this paper is to extend this connection beyond the acyclic case.

Finding an Unknown Acyclic Orientation of a Given Graph

Let c(G) be the smallest number of edges we have to test in order to determine an unknown acyclic orientation of the given graph G in the worst case. For example, if G is the complete graph on n vertices, then c(G) is the smallest number of comparisons needed to sort n numbers.We prove that c(G) ≤ (1/4 + o(1))n2 for any graph G on n vertices, answering in the affirmative a question of Aigner, Triesch and Tuza [Discrete Mathematics144 (1995) 3–10]. Also, we show that, for every ϵ > 0, it is NP-hard to approximate the parameter c(G) within a multiplicative factor 74/73 − ϵ.