Extended Formulations for Packing and Partitioning Orbitopes

We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, resp. exactly, one 1-entry per row. They are important objects for symmetry reduction in certain integer programs. Using the extended formulations, we also derive a rather simple proof of the fact that basically shifted-column inequalities suffice in order to describe those orbitopes linearly.Comment: 16 page

Hidden Vertices in Extensions of Polytopes

Some widely known compact extended formulations have the property that each vertex of the corresponding extension polytope is projected onto a vertex of the target polytope. In this paper, we prove that for heptagons with vertices in general position none of the minimum size extensions has this property. Additionally, for any d >= 2 we construct a family of d-polytopes such that at least 1/9 of all vertices of any of their minimum size extensions is not projected onto vertices.Comment: 9 pages, to appear in: Operations Research Letter

Orbitopal Fixing

The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant, since this kind of symmetry unnecessarily blows up the search tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the search tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been introduced bei Kaibel and Pfetsch (Math. Programm. A, 114 (2008), 1-36). It does, however, not explicitly add inequalities to the model. Instead, it uses certain fixing rules for variables. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem.Comment: 22 pages, revised and extended version of a previous version that has appeared under the same title in Proc. IPCO 200

Streamroller : A Unified Compilation and Synthesis System for Streaming Applications.

The growing complexity of applications has increased the need for higher processing power. In the embedded domain, the convergence of audio, video, and networking on a handheld device has prompted the need for low cost, low power,and high performance implementations of these applications in the form of custom hardware. In a more mainstream domain like gaming consoles, the move towards more realism in physics simulations and graphics has forced the industry towards multicore systems. Many of the applications in these domains are streaming in nature. The key challenge is to get efficient implementations of custom hardware from these applications and map these applications efficiently onto multicore architectures. This dissertation presents a unified methodology, referred to as Streamroller, that can be applied for the problem of scheduling stream programs to multicore architectures and to the problem of automatic synthesis of custom hardware for stream applications. Firstly, a method called stream-graph modulo scheduling is presented, which maps stream programs effectively onto a multicore architecture. Many aspects of a real system, like limited memory and explicit DMAs are modeled in the scheduler. The scheduler is evaluated for a set of stream programs on IBM's Cell processor. Secondly, an automated high-level synthesis system for creating custom hardware for stream applications is presented. The template for the custom hardware is a pipeline of accelerators. The synthesis involves designing loop accelerators for individual kernels, instantiating buffers to store data passed between kernels, and linking these building blocks to form a pipeline. A unique aspect of this system is the use of multifunction accelerators, which improves cost by efficiently sharing hardware between multiple kernels. Finally, a method to improve the integer linear program formulations used in the schedulers that exploits symmetry in the solution space is presented. Symmetry-breaking constraints are added to the formulation, and the performance of the solver is evaluated.Ph.D.Computer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/61662/1/kvman_1.pd