A mathematical formulation of the loop pipelining problem

This paper presents a mathematical model for the loop pipelining problem that considers several parameters for optimization and supports any combination of resource and timing constraints. The unrolling degree of the loop is one of the variables explored by the model. By using Farey’s series, an optimal exploration of the unrolling degree is performed and optimal solutions not considered by other methods are obtained. Finding an optimal schedule that minimizes resource and register requirements is solved by using an Integer linear programming (ILP) model. A novel paradigm called branch and prune is proposed to eficiently converge towards the optimal schedule and prune the search tree for integer solutions, thus drastically reducing the running time. This is the first formulation that combines the unrolling degree of the loop with timing and resource constraints in a mathematical model that guarantees optimal solutions.Peer ReviewedPostprint (author's final draft

Using Functional Programming to recognize Named Structure in an Optimization Problem: Application to Pooling

Branch-and-cut optimization solvers typically apply generic algorithms, e.g., cutting planes or primal heuristics, to expedite performance for many mathematical optimization problems. But solver software receives an input optimization problem as vectors of equations and constraints containing no structural information. This article proposes automatically detecting named special structure using the pattern matching features of functional programming. Specifically, we deduce the industrially-relevant nonconvex nonlinear Pooling Problem within a mixed-integer nonlinear optimization problem and show that we can uncover pooling structure in optimization problems which are not pooling problems. Previous work has shown that preprocessing heuristics can find network structures; we show that we can additionally detect nonlinear pooling patterns. Finding named structures allows us to apply, to generic optimization problems, cutting planes or primal heuristics developed for the named structure. To demonstrate the recognition algorithm, we use the recognized structure to apply primal heuristics to a test set of standard pooling problems

On orbital allotments for geostationary satellites

The following satellite synthesis problem is addressed: communication satellites are to be allotted positions on the geostationary arc so that interference does not exceed a given acceptable level by enforcing conservative pairwise satellite separation. A desired location is specified for each satellite, and the objective is to minimize the sum of the deviations between the satellites' prescribed and desired locations. Two mixed integer programming models for the satellite synthesis problem are presented. Four solution strategies, branch-and-bound, Benders' decomposition, linear programming with restricted basis entry, and a switching heuristic, are used to find solutions to example synthesis problems. Computational results indicate the switching algorithm yields solutions of good quality in reasonable execution times when compared to the other solution methods. It is demonstrated that the switching algorithm can be applied to synthesis problems with the objective of minimizing the largest deviation between a prescribed location and the corresponding desired location. Furthermore, it is shown that the switching heuristic can use no conservative, location-dependent satellite separations in order to satisfy interference criteria

Integer linear programming vs. graph-based methods in code generation

A common characterictic of many applications is that they are aimed at the high-volume consumer market, which is extremely cost-sensitive. However many of them impose stringent performance demands on the underlying system. Therefore the code generation must take into account the restrictions and features given by the target architecture while satisfying these performance demands. High-level language compilers often are unable to generate code meeting these requirements. One reason is the phase coupling problem between instruction scheduling and register allocation. Many compilers perform these tasks separately with each phase ignorant of the require- ments of the other. Commonly, each task is accomplished by using heuristic methods. As the goals of the two phases often conflict, whichever phase is performed first imposes constraints on the other, sometimes producing inefficient code. Integer linear programming (ILP) provides an integrated approach to the combined instruction scheduling and register allocation problem. This way, optimal solutions can be found - albeit at the cost of high compilation times. In our experiments, we considered as target processor the 32-bit DSP ADSP-2106x. We have examined two different ILP formulations and compared them with conventional approaches including list scheduling and the critical path method. Moreover, we have investigated approximations based on the ILP formulations; this way, compilation time can be reduced considerably while still producing near-optimal results. From the results of our implementation, we have concluded that integrating ILP formulations in conventional global algorithms is a promising method for generating high-quality code

A Novel SAT-Based Approach to the Task Graph Cost-Optimal Scheduling Problem

The Task Graph Cost-Optimal Scheduling Problem consists in scheduling a certain number of interdependent tasks onto a set of heterogeneous processors (characterized by idle and running rates per time unit), minimizing the cost of the entire process. This paper provides a novel formulation for this scheduling puzzle, in which an optimal solution is computed through a sequence of Binate Covering Problems, hinged within a Bounded Model Checking paradigm. In this approach, each covering instance, providing a min-cost trace for a given schedule depth, can be solved with several strategies, resorting to Minimum-Cost Satisfiability solvers or Pseudo-Boolean Optimization tools. Unfortunately, all direct resolution methods show very low efficiency and scalability. As a consequence, we introduce a specialized method to solve the same sequence of problems, based on a traditional all-solution SAT solver. This approach follows the "circuit cofactoring" strategy, as it exploits a powerful technique to capture a large set of solutions for any new SAT counter-example. The overall method is completed with a branch-and-bound heuristic which evaluates lower and upper bounds of the schedule length, to reduce the state space that has to be visited. Our results show that the proposed strategy significantly improves the blind binate covering schema, and it outperforms general purpose state-of-the-art tool

Operations research methods in compiler backends

Operations research can be defined as the theory of numerically solving decision problems. In this context, dealing with optimization problems is a central issue. Code generation is performed by the backend phase of a compiler, a program which transforms the source code of an application into optimized machine code. Basically, code generation is an optimization problem, which can be modelled in a way similar to typical problems in the area of operations research. In this article, that similarity is demonstrated by opposing integer linear programming models for problems of the operations research and of code generation. The time frame for solving the generated integer linear programs (ILPs) as a part of the compilation process is small. As a consequence, using well-structured ILP-formulations and ILP-based approximations is necessary. The second part of the paper will give a brief survey on guidelines and techniques for both issues

Using Speculative Computation and Parallelizing Techniques to Improve Scheduling of Control based Designs

partially_open5Recent research results have seen the application of parallelizing techniques to high-level synthesis. In particular, the effect of speculative code transformations on mixed control-data flow designs has demonstrated effective results on schedule lengths. In this paper we first analyze the use of the control and data dependence graph as an intermediate representation that provides the possibility of extracting the maximum parallelism. Then we analyze the scheduling problem by formulating an approach based on Integer Linear Programming (ILP) to minimize the number of control steps given the amount of resources. We improve the already proposed ILP scheduling approaches by introducing a new conditional resource sharing constraint which is then extended to the case of speculative computation. The ILP formulation has been solved by using a Branch and Cut framework which provides better results than standard branch and bound techniquesR. Cordone; F. Ferrandi; G. Palermo; M. Santambrogio; D. SciutoR., Cordone; Ferrandi, Fabrizio; Palermo, Gianluca; Santambrogio, MARCO DOMENICO; Sciuto, Donatell