## The temporal knapsack problem and its solution (2005)

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Venue: | In Proceedings of the 2 d International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Bartlett05thetemporal,

author = {Mark Bartlett and Alan M. Frisch and Youssef Hamadi and Ian Miguel and S. Armagan Tarim and Chris Unsworth},

title = {The temporal knapsack problem and its solution},

booktitle = {In Proceedings of the 2 d International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems},

year = {2005},

pages = {34--48}

}

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### Abstract

Abstract. This paper introduces a problem called the temporal knapsack problem, presents several algorithms for solving it, and compares their performance. The temporal knapsack problem is a generalisation of the knapsack problem and specialisation of the multidimensional (or multiconstraint) knapsack problem. It arises naturally in applications such as allocating communication bandwidth or CPUs in a multiprocessor to bids for the resources. The algorithms considered use and combine techniques from constraint programming, artificial intelligence and operations research. 1

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Citation Context ...multaneously require different resources to perform large-scale computations. An advanced-reservation system will be used to guarantee a timed access to resources through some service level agreement =-=[1]-=-. The agreement is reached via negotiation, where end users present reservation bids to resource providers. Each bid specifies the resource category, start time, end time and required quality of servi... |

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Citation Context ...m high-performance computing. However, the convergence between web services and grid computing, combined with the arrival of the commercial grid, make the efficient use of valuable resources critical =-=[3]-=-. Our algorithms fit well into nextgeneration grids and represent the first attempts towards efficient grid resource schedulers. 2 The Temporal Knapsack Problem A formal statement of the TKP is given ... |

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Citation Context ...of 10, which is not shown. The optimal solution to this instance is to accept bids b1,b4,b5, and b6, yielding a total price of 22. The traditional knapsack problem, as overviewed by Martello and Toth =-=[4]-=-, is a special case of TKP in which there is only a single time. Since the knapsack problem, which is NP-hard, is a special case of TKP, TKP is also NP-hard. Given: times, a finite, non-empty set tota... |

281 |
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Citation Context ... classes of cutting planes (see [13]). Using the results from the linear relaxed TKP model (0 ≤ Xb ≤ 1, ∀b ∈ bids) and an objective function value z of a known feasible integer solution, a valid GMIC =-=[14]-=- for the TKP model can be written as ⌊zU⌋−z ≥ � ⌊−ri⌋Xi + � ⌊rj⌋(1 − Xj)+ � � ⌊−ri⌋ + fi � − f0 Xi + � fj>f0 j∈N2 fi≤f0 i∈N1 � ⌊rj⌋ + fj − f0 1 − f0 fj≤f0 j∈N2 � (1 − Xj)+ � fk≤f0 k∈S fi>f0 i∈N1 ⌊−rk⌋... |

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Citation Context ...sing times t2 and t4 and bids b1, b3 and b4; and a second comprising time t8 and bids b6 and b7. Splitting is rarely used in constraint programming, though two recent exceptions are the work of Walsh =-=[6]-=- and of Marinescu and Dechter [7]. The branch operator is the familiar one from constraint programming, artificial intelligence and operations research. A bid b is selected, it is then removed from th... |

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Citation Context ...between timeefficient, space-hungry AO* and time-hungry, space-efficient depth-first search. Such an algorithm could be developed by generalising one of the memorybounded versions of A*, such as SMA* =-=[16]-=-, to operate on AND/OR search trees. It also is likely that the decomposition algorithm would benefit from a better heuristic for choosing where to force splits. We conjecture that a better heuristic ... |

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Citation Context ...should choose the next node to be expanded. To do this, we currently employ two strategies: the AO* algorithm described by Nilsson [9] (which is itself based on an algorithm of Martelli and Montanari =-=[10]-=-), and a depth-first algorithm. The AO* search strategy is an extension of the A* algorithm to AND/OR search spaces, and retains two important properties of A*: the first feasible solution found is gu... |

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Citation Context ...tential integer solutions. To enhance the given search strategy, we employ the well-known Gomory mixed-integer cut (GMIC), which is considered one of the most important classes of cutting planes (see =-=[13]-=-). Using the results from the linear relaxed TKP model (0 ≤ Xb ≤ 1, ∀b ∈ bids) and an objective function value z of a known feasible integer solution, a valid GMIC [14] for the TKP model can be writte... |

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Citation Context ...nce of the temporal knapsack problem to which the reduce operator is applied The multidimensional knapsack problem (MKP, also known as the multiconstraint knapsack problem), as overviewed by Fréville =-=[5]-=-, is a generalisation of TKP. Each time and bid in the TKP corresponds, respectively, to a dimension and an item in MKP. If t1,...,tn are the times in TKP, then every bid b corresponds to an item in M... |

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Citation Context ...rst paper published in the journal Artificial Intelligence [8].sThe Temporal Knapsack Problem and Its Solution 41 algorithm that is guaranteed to find an optimal solution expands fewer nodes than AO* =-=[11]-=-. The drawback of AO* is that it requires a large amount of memory; the number of nodes in memory is Ω(2 |bids| ). In contrast, depth-first search stores only Ω(|bids|) nodes. However, in general, dep... |

7 | AND/OR tree search for constraint optimization
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(Show Context)
Citation Context ... b3 and b4; and a second comprising time t8 and bids b6 and b7. Splitting is rarely used in constraint programming, though two recent exceptions are the work of Walsh [6] and of Marinescu and Dechter =-=[7]-=-. The branch operator is the familiar one from constraint programming, artificial intelligence and operations research. A bid b is selected, it is then removed from the problem and two branches are ge... |

5 |
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Citation Context ...guaranteed to be an optimal one, and no 1 The idea that a non-deterministic program implicitly defines an AND/OR tree was used in the very first paper published in the journal Artificial Intelligence =-=[8]-=-.sThe Temporal Knapsack Problem and Its Solution 41 algorithm that is guaranteed to find an optimal solution expands fewer nodes than AO* [11]. The drawback of AO* is that it requires a large amount o... |

4 |
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Citation Context ...tion, where end users present reservation bids to resource providers. Each bid specifies the resource category, start time, end time and required quality of service (e.g., bandwidth, number of nodes) =-=[2]-=-. If end users offer a price they are willing to pay for the resource, advanced reservation allocation in a grid infrastructure becomes equivalent to the TKP. Our algorithms are designed to be used by... |

3 |
2001, Constraint and linear programming : Using reduced costs for solving the zero/one multiple knapsack problem
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Citation Context ... none that are integer. This also helps to improve the upper bound zU. We also employ the “reduced costs constraints” (RCC ) and “reverse-reduced costs constraints” (R-RCC ) discussed by Oliva et al. =-=[15]-=-. Following their work, the “pseudo-utility criterion” is used to obtain a reasonably good feasible solution. This criterion is computationally cheap, especially once the solution to the linear relaxa... |