## 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 - 9 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...

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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.... |

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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 ... |

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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... |

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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)... |

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A Cost Function based Optimization Technique for Scheduling in Data Path Synthesis
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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... |

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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. ... |

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An Integrated and Accelerated ILP Solution for Scheduling, Module Allocation, and Binding
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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... |

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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... |

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et al., Algorithmic and Register Transfer Level Synthesis: The
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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... |

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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... |

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Topics on Tournaments, Athena
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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... |