Crystal constructions in Number Theory

Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics

A coloring algorithm for 4K1-free line graphs

International audienc

Optimal scheduling of contract algorithms with soft deadlines

A contract algorithm is an algorithm which is given, as part of its input, a specified amount of allowable computation time. In contrast, interruptible algorithms may be interrupted throughout their execution, at which point they must report their current solution. Simulating interruptible algorithms by means of schedules of executions of contract algorithms in parallel processors is a wellstudied problem with significant applications in AI. In the classical case, the interruptions are hard deadlines in which a solution must be reported immediately at the time the interruption occurs. In this paper we study the more general setting of scheduling contract algorithms at the presence of soft deadlines. This is motivated by the observation of practitioners that soft deadlines are as common an occurrence as hard deadlines, if not more common. In our setting, at the time t of interruption the algorithm is given an additional window of time w(t) ≤ c · t to continue the contract or, indeed, start a new contract (for some fixed constant c). We explore this variation using the acceleration ratio, which is the canonical measure of performance for these schedules, and derive schedules of optimal acceleration ratio for all functions w