## Analyzing and Exploiting the Structure of the Constraints in the ILP Approach to the Scheduling Problem (1994)

Venue: | IEEE Transactions on VLSI Systems |

Citations: | 50 - 8 self |

### BibTeX

@ARTICLE{Chaudhuri94analyzingand,

author = {Samit Chaudhuri and Robert A. Walker and John E. Mitchell},

title = {Analyzing and Exploiting the Structure of the Constraints in the ILP Approach to the Scheduling Problem},

journal = {IEEE Transactions on VLSI Systems},

year = {1994},

volume = {2},

pages = {456--471}

}

### Years of Citing Articles

### OpenURL

### Abstract

In integer linear programming (ILP), formulating a "good" model is of crucial importance to solving that model [1]. In this paper, we begin with a mathematical analysis of the structure of the assignment, timing, and resource constraints in high-level synthesis, and then evaluate the structure of the scheduling polytope described by these constraints. We then show how the structure of the constraints can be exploited to develop a well-structured ILP formulation, which can serve as a solid theoretical foundation for future improvement. As a start in that direction, we also present two methods to further tighten the formulation. The contribution of this paper is twofold: (1) it provides the first in-depth formal analysis of the structure of the constraints, and it shows how to exploit that structure in a well-designed ILP formulation, and (2) it shows how to further improve a well-structured formulation by adding new valid inequalities. I. Introduction The scheduling problem in high-le...

### Citations

11418 |
Computers and Intractability: A Guide to the Theory of NP -completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ... time- and resourceconstrainedsscheduling (TRCS), which optimizes a given objective function when both the number of hardware resources and the number of control steps are fixed. The decision problem =-=[3]-=- corresponding to TRCS is known to be NP-complete [4]; therefore at present, no polynomial time exact algorithm exists to solve any of the scheduling problems. To solve the scheduling problem, both he... |

380 |
Discrete Cosine Transforms: Algorithms, Advantages
- Rao, Yip
- 1990
(Show Context)
Citation Context ... present results of solving the TRCS problem using formuation (ILP) on two benchmark examples: the 34-operation elliptical wave filter (EWF) [30], and the 48-operation discrete cosine transform (DCT) =-=[31]-=-. The purpose of these results is to demonstrate that the formulation (ILP) is well-behaved, and to allow us to observe the general behavior (tight bounds, and therefore small number of branches) that... |

252 |
The Lagrangian relaxation method for solving integer programming problems
- Fisher
- 1981
(Show Context)
Citation Context ...er bound, and has led to the success of Lagrangian relaxation-based branch-and-bound algorithms to solve the traveling salesman problem [27], and the minimum-tardiness-scheduling problem [28]. Fisher =-=[29]-=- has reported that the bounds produced by Lagrangian relaxation are on the average 95% within the optimum value, and that such tight bounds allow efficient pruning of the branches. The LP-relaxation o... |

200 |
Np-complete scheduling problems
- Ullman
- 1975
(Show Context)
Citation Context ...ch optimizes a given objective function when both the number of hardware resources and the number of control steps are fixed. The decision problem [3] corresponding to TRCS is known to be NP-complete =-=[4]-=-; therefore at present, no polynomial time exact algorithm exists to solve any of the scheduling problems. To solve the scheduling problem, both heuristic scheduling algorithms, which find approximate... |

183 |
The High Level Synthesis of Digital Systems
- McFarland, Parker, et al.
- 1990
(Show Context)
Citation Context ...a well-designed ILP formulation, and (2) it shows how to further improve a well-structured formulation by adding new valid inequalities. I. Introduction The scheduling problem in high-level synthesis =-=[2]-=- is concerned with sequencing the operations of a control/data flow graph (cdfg) into correct order. This is an optimization problem, and is specified in several ways depending on the goal: (1) uncons... |

113 |
A formal approach to the scheduling problem in high-level synthesis
- Hwang, Lee, et al.
- 1991
(Show Context)
Citation Context ...raints on the scheduling problem, providing a theoretical foundation for choosing an appropriate ILP formulation. With the exception of some work in OASIC [8], most work on ILP-based schedulers (ALPS =-=[9]-=-, GRAD [15]) has virtually ignored this analysis. Thus our work is the first in-depth analysis of the structure of the constraints on the scheduling problem. To date, two different types of timing con... |

61 |
Structure of a simple scheduling polyhedron
- Queyranne
- 1993
(Show Context)
Citation Context ...rk has been done on describing the scheduling polyhedra. Analyzing the scheduling polyhedra is difficult, and only very recently some work has been published on the single-machine scheduling problems =-=[20]-=-, [21]. However, the models considered in such literature are not appropriate for use in high-level synthesis. C. Motivation and Outline of the Paper As we discussed in Section I-A, to develop efficie... |

45 |
Force Directed Scheduling for the Behavioral Synthesis of ASIC’s
- Paulin, Knight
- 1989
(Show Context)
Citation Context ...r 9405512 lutions, and exact algorithms, which find optimal solutions, have been used. A wide variety of heuristic algorithms are used, including the transformational [5], list [6], and forcedirected =-=[7]-=- scheduling heuristics. In contrast, most exact algorithms employ integer linear programming (ILP) to compute the optimal solutions. Although solving an ILP formulation is NP-hard [3], significant pro... |

41 |
A time indexed formulation of non-preemptive single machine scheduling problems
- Sousa, Wolsey
- 1992
(Show Context)
Citation Context ... been done on describing the scheduling polyhedra. Analyzing the scheduling polyhedra is difficult, and only very recently some work has been published on the single-machine scheduling problems [20], =-=[21]-=-. However, the models considered in such literature are not appropriate for use in high-level synthesis. C. Motivation and Outline of the Paper As we discussed in Section I-A, to develop efficient sol... |

32 | Computing Lower Bounds on Functional Units before Scheduling
- Chaudhuri, Walker
(Show Context)
Citation Context ...e operations; their effects on the problem structure has already been mentioned in Section IV. As for the TCS and the UCS problems, the number of FU's can be found using a quick lower bounding scheme =-=[26]-=-; then the formulation (ILP) can be used to find the optimal schedule. B. Performance of the ILP Formulation As mentioned above, in order for a branch-and-bound algorithm to be successful, it is impor... |

31 |
Solving large- scale zero-one linear programming problems
- Crowder, Johnson, et al.
- 1983
(Show Context)
Citation Context ...Use of good formulations has led to significant success of the ILP approach for solving NP-complete problems in other areas, such as the traveling salesman problem (TSP) [11], 0-1 integer programming =-=[12], and mini-=-mum perfect 2-matchings [13]. It has been said in [1] that "formulating a good model is of crucial importance to solving the model". Therefore, to efficiently solve the 1063-8210/94$04.00 c ... |

24 |
Optimal solution of scheduling problems using Lagrange multipliers
- Fisher
- 1971
(Show Context)
Citation Context ...duces a tighter bound, and has led to the success of Lagrangian relaxation-based branch-and-bound algorithms to solve the traveling salesman problem [27], and the minimum-tardiness-scheduling problem =-=[28]-=-. Fisher [29] has reported that the bounds produced by Lagrangian relaxation are on the average 95% within the optimum value, and that such tight bounds allow efficient pruning of the branches. The LP... |

20 |
A Formal Method for the Specification, Analysis, and Design of Register-Transfer Level Digital Logic
- Parker
- 1981
(Show Context)
Citation Context ...op more efficient solution techniques, we must analyze the structure of the constraints of the scheduling problem, and exploit the structure in a well-designed ILP formulation. B. Previous Work Hafer =-=[14]-=- was perhaps the first to employ combinatorial optimization in the high level synthesis area. Although his work led to a formal definition of the problem, the formulation was too complex to design a c... |

19 |
The Traveling Salesman Problem and Minimal Spanning Trees
- Held, Karp
- 1970
(Show Context)
Citation Context ...f relaxation, called the Lagrangian relaxation, produces a tighter bound, and has led to the success of Lagrangian relaxation-based branch-and-bound algorithms to solve the traveling salesman problem =-=[27]-=-, and the minimum-tardiness-scheduling problem [28]. Fisher [29] has reported that the bounds produced by Lagrangian relaxation are on the average 95% within the optimum value, and that such tight bou... |

14 |
Optimal scheduling and allocation of embedded vlsi chips
- Gebotys
- 1992
(Show Context)
Citation Context ... can be produced with more efficiency. Recent work on ILP-based scheduling algorithms has instead concentrated on including more design parameters (number of busses, clock length, binding etc.) [16], =-=[17]-=-, [18] into the model in an approximate way. Because of the robustness of the general ILP model, a correct formulation 458 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 2, NO.... |

10 |
Algorithmic and Register Transfer Level Synthesis: The System Architect's Workbench
- Thomas, Lagnese, et al.
- 1990
(Show Context)
Citation Context ...FU's as indicated in Section V-A. In this section, we first present results of solving the TRCS problem using formuation (ILP) on two benchmark examples: the 34-operation elliptical wave filter (EWF) =-=[30]-=-, and the 48-operation discrete cosine transform (DCT) [31]. The purpose of these results is to demonstrate that the formulation (ILP) is well-behaved, and to allow us to observe the general behavior ... |

9 |
Eijndhoven, 'The Yorktown silicon compiler system
- Brayton, Camposano, et al.
- 1988
(Show Context)
Citation Context ...Troy, NY 12180 USA IEEE Log Number 9405512 lutions, and exact algorithms, which find optimal solutions, have been used. A wide variety of heuristic algorithms are used, including the transformational =-=[5]-=-, list [6], and forcedirected [7] scheduling heuristics. In contrast, most exact algorithms employ integer linear programming (ILP) to compute the optimal solutions. Although solving an ILP formulatio... |

8 |
A cutting plane algorithm for minimum perfect 2-matching, Computing 39
- Grötschel, Holland
- 1987
(Show Context)
Citation Context ...gnificant success of the ILP approach for solving NP-complete problems in other areas, such as the traveling salesman problem (TSP) [11], 0-1 integer programming [12], and minimum perfect 2-matchings =-=[13]. It has b-=-een said in [1] that "formulating a good model is of crucial importance to solving the model". Therefore, to efficiently solve the 1063-8210/94$04.00 c fl1994 IEEE CHAUDHURI et al.: ILP APPR... |

8 |
Golumbic, Algorithmic Graph Theory and Perfect Graphs
- Charles
- 1980
(Show Context)
Citation Context ...matrix are nothing but the rows of M a and M t , and its fractional node-packing polytope is the same as its monotone fractional timing-assignment polytope, as can be verified from (8). Definition 6: =-=[25]-=-. A graph G = fV; Eg is called transitively orientable if each edge can be assigned a one-way direction in such a way that the resulting oriented graph (V; F ) satisfies the following property: (a; b)... |

5 |
A Cost Function Based Optimization Technique for Scheduling in Data Path Synthesis
- Shin, Woo
- 1989
(Show Context)
Citation Context ...in the high level synthesis area. Although his work led to a formal definition of the problem, the formulation was too complex to design a chip of reasonable size. Other ILP-based schedulers are GRAD =-=[15]-=-, ALPS [9], and OASIC [8]. A comparison of the details of these schedulers will be presented in Section VI after our notation is introduced. At this point, it should be noted that each of the formulat... |

5 |
Optimal synthesis of multichip architectures
- GEBOTYS
- 1992
(Show Context)
Citation Context ...utions can be produced with more efficiency. Recent work on ILP-based scheduling algorithms has instead concentrated on including more design parameters (number of busses, clock length, binding etc.) =-=[16]-=-, [17], [18] into the model in an approximate way. Because of the robustness of the general ILP model, a correct formulation 458 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. ... |

5 |
An Integrated and Accelerated ILP Solution for Scheduling, Module Allocation, and Binding
- Wilson, Mukherjee, et al.
- 1993
(Show Context)
Citation Context ...e produced with more efficiency. Recent work on ILP-based scheduling algorithms has instead concentrated on including more design parameters (number of busses, clock length, binding etc.) [16], [17], =-=[18]-=- into the model in an approximate way. Because of the robustness of the general ILP model, a correct formulation 458 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 2, NO. 4, DE... |

4 | The Traveling Salesman Problem, chapter Polyhedral Theory - Lawler, Lenstra, et al. - 1985 |

3 |
Quadratic Programming with LINDO : User's Manual, Scientific
- Schrage, Linear
- 1985
(Show Context)
Citation Context ...s the number of registers. The columns labeled Branch indicate the number of nodes in the branch-and-bound tree for solving a particular problem instance. The ILP formulations were solved using LINDO =-=[32]-=- optimization software on a SPARC 2 workstation. For the EWF benchmark, we first solved the formulation (ILP) with objective function (19) (no register constraints), which attempts to minimize the num... |

2 |
et al., Algorithmic and Register Transfer Level Synthesis: The
- Thomas
- 1990
(Show Context)
Citation Context ...2180 USA IEEE Log Number 9405512 lutions, and exact algorithms, which find optimal solutions, have been used. A wide variety of heuristic algorithms are used, including the transformational [5], list =-=[6]-=-, and forcedirected [7] scheduling heuristics. In contrast, most exact algorithms employ integer linear programming (ILP) to compute the optimal solutions. Although solving an ILP formulation is NP-ha... |

2 |
The Age of Optimization-Solving Large Scale Real World Problems
- Nemhauser
- 1993
(Show Context)
Citation Context ... ILP formulation that exploits that structure. For further improvement in the solution efficiency of a well-designed ILP formulation, new ways have to be found to tighten the original formulation. In =-=[22], Nemhause-=-r said, "Preprocessing and polyhedral theory have yielded at least an order of magnitude improvement in branchand -bound algorithms for solving mixed-integer programs". This prospect of tigh... |

2 |
Topics on Tournaments, Athena
- Moon
- 1968
(Show Context)
Citation Context ... path L T ` G s such that V (L T ) = V (T ); the uniqueness and the maximality of L T will be proven in the next part of the proof. CHAUDHURI et al.: ILP APPROACH TO SCHEDULING 461 From Theorem 14 in =-=[23]-=-, we know that T has at least one spanning path L, i.e., V (L) = V (T ). Since T ` G s , the spanning path L ` G s . Furhtermore, since G s is the transitive closure of G s , G s ` G s . However, from... |