## Streaming Cache Placement Problems: Complexity and Algorithms (2003)

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Citations: | 5 - 5 self |

### BibTeX

@TECHREPORT{Oliveira03streamingcache,

author = {Carlos A. S. Oliveira and Panos M. Pardalos and Oleg A. Prokopyev and Mauricio and Mauricio G.C. Resende},

title = {Streaming Cache Placement Problems: Complexity and Algorithms},

institution = {},

year = {2003}

}

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### Abstract

Virtual private networks (VPN) are often used to distribute live content, such as video or audio streams, from a single source to a large number of destinations. Streaming caches or splitters are deployed in these multicast networks to allow content distribution without overloading the network. In this paper, we consider two combinatorial optimization problems that arise in multicast networks. In the Tree Cache Placement Problem (TCPP), the objective is to find a routing tree on which the number of cache nodes needed for multicasting is minimized. We also discuss a modification of this problem, called the Flow Cache Placement Problem (FCPP), where we seek any feasible flow from the source to the destinations which minimizes the number of cache nodes. We prove that these problems are NP-hard using a transformation from Satisfiability. This transformation allows us to give a proof of non-approximability by showing that it is gap-preserving. We also consider approximation algorithms for the TCPP and FCPP and special cases where these problems can be solved in polynomial time.

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Citation Context ...e y and constraints (7)-(8) are dropped. 2. Complexity of the Cache Placement Problems In this section, we prove that the TCPP and the FCPP are NP-hard, by using a reduction from Satisfiability (SAT) =-=[3]-=-. The proof is different from the one in [4] and the transformation used provides a scheme for proving non-appoximability in Section 3. 2.1. Complexity of the TCPP. We first prove that the TCPP is NP-... |

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Citation Context ...at must be exercised during the design of the network. Examples of problems occurring in multicast networks are the minimum cost multicast tree problem [11, 12, 15, 16], center based tree computation =-=[2, 4, 23]-=-, andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem [22, 27, 28]. A survey by Oliveira and Pardalos [19] gives several other examples of problems and algorithms f... |

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Citation Context ...the optimum solution. For minimization problems, A(I) must satisfy A(I) ≤ ɛ · OP T (I), for any fixed ɛ > 1.sSTREAMING CACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 9 The following theorem =-=from [2] is -=-very useful to prove hardness of approximation results. Theorem 5. There is a polynomial time reduction from SAT to Max-3Sat(5) which transforms formula φ into a formula φ ′ such that, for some fi... |

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Citation Context ...e tested empirically, in order to find cache placements under different situations. Other interesting related work includes the problem of placing replicas of objects in content distribution networks =-=[13, 14]-=-. The objective of this problem, as studied by Kangasharju et al. [13], is to minimize the average number of hops traversed by the content being downloaded. The problem has been proved NP-hard by a re... |

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Citation Context ...at must be exercised during the design of the network. Examples of problems occurring in multicast networks are the minimum cost multicast tree problem [11, 12, 15, 16], center based tree computation =-=[2, 4, 23]-=-, andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem [22, 27, 28]. A survey by Oliveira and Pardalos [19] gives several other examples of problems and algorithms f... |

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Citation Context ...ng combinatorial appeal, due to the tradeoffs that must be exercised during the design of the network. Examples of problems occurring in multicast networks are the minimum cost multicast tree problem =-=[11, 12, 15, 16]-=-, center based tree computation [2, 4, 23], andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem [22, 27, 28]. A survey by Oliveira and Pardalos [19] gives several o... |

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Citation Context ...lem [11, 12, 15, 16], center based tree computation [2, 4, 23], andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem [22, 27, 28]. A survey by Oliveira and Pardalos =-=[19]-=- gives several other examples of problems and algorithms for multicast networks. Multicast tree construction is a problem heavily studied by network engineers [11, 12, 15, 16, 27, 28]. The objective i... |

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Citation Context ...ection 6. 1.1. Literature Review. Applications of multicast routing occur in diverse areas. They range from the deployment of corporate services, such as automatic software updates [10] and groupware =-=[3]-=-, to end-user programs for video-conferencing [7] and even game communities [21]. Such problems have also a strong combinatorial appeal, due to the tradeoffs that must be exercised during the design o... |

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Citation Context ...a way similar to the one described in the present paper. The authors proposed algorithms for improved routing in this situation, using simulation methods for performance evaluation. In a second paper =-=[24]-=-, the same authors studied the minimization of servers required in a multicast network. However they did not consider capacity constraints (as we do in the present paper), since they were more interes... |

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Citation Context ...etworks are the minimum cost multicast tree problem [11, 12, 15, 16], center based tree computation [2, 4, 23], andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem =-=[22, 27, 28]-=-. A survey by Oliveira and Pardalos [19] gives several other examples of problems and algorithms for multicast networks. Multicast tree construction is a problem heavily studied by network engineers [... |

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Citation Context ...etworks are the minimum cost multicast tree problem [11, 12, 15, 16], center based tree computation [2, 4, 23], andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem =-=[22, 27, 28]-=-. A survey by Oliveira and Pardalos [19] gives several other examples of problems and algorithms for multicast networks. Multicast tree construction is a problem heavily studied by network engineers [... |

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Citation Context ...]. The objective is to find a distribution tree, such that source and destination nodes are connected at minimum cost. The problem is a generalization of the well-known Steiner tree problem in graphs =-=[6]-=-. Here, in contrast to our cache location problems, it is assumed that all nodes in the network act as caches. Solution methods for the minimum cost multicast tree include approximation algorithms [5,... |

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Citation Context ...d FCPP and special cases where these problems can be solved in polynomial time. Finally, concluding remarks are made in Section 5. The cache placement problem was introduced and studied in Mao et al. =-=[4]-=-. There, the general problem is proved to be NP-hard and algorithms and heuristics for efficient and robust cache placement under different constraints are proposed and tested. 1.1. The tree cache pla... |

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Citation Context ...marks are made in Section 6. 1.1. Literature Review. Applications of multicast routing occur in diverse areas. They range from the deployment of corporate services, such as automatic software updates =-=[10]-=- and groupware [3], to end-user programs for video-conferencing [7] and even game communities [21]. Such problems have also a strong combinatorial appeal, due to the tradeoffs that must be exercised d... |

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Citation Context ...etworks are the minimum cost multicast tree problem [11, 12, 15, 16], center based tree computation [2, 4, 23], andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem =-=[22, 27, 28]-=-. A survey by Oliveira and Pardalos [19] gives several other examples of problems and algorithms for multicast networks. Multicast tree construction is a problem heavily studied by network engineers [... |

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Citation Context ...verse areas. They range from the deployment of corporate services, such as automatic software updates [10] and groupware [3], to end-user programs for video-conferencing [7] and even game communities =-=[21]-=-. Such problems have also a strong combinatorial appeal, due to the tradeoffs that must be exercised during the design of the network. Examples of problems occurring in multicast networks are the mini... |

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Citation Context ...at must be exercised during the design of the network. Examples of problems occurring in multicast networks are the minimum cost multicast tree problem [11, 12, 15, 16], center based tree computation =-=[2, 4, 23]-=-, andsCACHE PLACEMENT PROBLEMS: COMPLEXITY AND ALGORITHMS 3 the multicast packing problem [22, 27, 28]. A survey by Oliveira and Pardalos [19] gives several other examples of problems and algorithms f... |

1 |
C.A.S. Oliveira) Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, FL 32611, USA E-mail address, C.A.S. Oliveira: oliveira@ufl.edu (P.M
- Hall
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Citation Context ...oblem Π is an algorithm that, for each ɛ > 0 and instance I ∈ Π, returns a solution A(I), such that A(I) ≤ (1+ɛ)OP T (I), and A has running time polynomial in the size of I, depending on ɛ (s=-=ee, e.g. [5], page-=- 425). Corollary 2. Unless P = NP , the TCPP cannot be approximated by (1 + ɛ2) for any ɛ2 ≤ ɛ1, where ɛ1 is given in Theorem 6, and therefore there is no polynomial time approximation scheme (P... |

1 | Optimization Problems in Telecommunications and the Internet
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Citation Context ...e placement problem, showing that there is no approximation algorithm for the problems with performance guarantee better than log k, where k is the number of destinations. The proof appeared first in =-=[18]-=-, but we extend it here for the undirected version of the problems. The technique used is based on a reduction from the Set Cover problem. Set Cover: Given a ground set T = t1, . . . , tn, with subset... |