## A constant-factor approximation algorithm for packet routing, and balancing local vs. global criteria (1997)

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Venue: | In Proceedings of the ACM Symposium on the Theory of Computing (STOC |

Citations: | 48 - 4 self |

### BibTeX

@INPROCEEDINGS{Srinivasan97aconstant-factor,

author = {Aravind Srinivasan and Chung-piaw Teo},

title = {A constant-factor approximation algorithm for packet routing, and balancing local vs. global criteria},

booktitle = {In Proceedings of the ACM Symposium on the Theory of Computing (STOC},

year = {1997},

pages = {636--643}

}

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### Abstract

Abstract. We present the first constant-factor approximation algorithm for a fundamental problem: the store-and-forward packet routing problem on arbitrary networks. Furthermore, the queue sizes required at the edges are bounded by an absolute constant. Thus, this algorithmbalances a global criterion (routing time) with a local criterion (maximum queue size) and shows how to get simultaneous good bounds for both. For this particular problem, approximating the routing time well, even without considering the queue sizes, was open. We then consider a class of such local vs. global problems in the context of covering integer programs and show how to improve the local criterion by a logarithmic factor by losing a constant factor in the global criterion.

### Citations

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Citation Context ... integer k, let [k] denote the set {1, 2, . . . , k}. We start by reviewing the Cherno#-Hoe#ding bounds in Theorem 3.1. Let G(, #) . = (exp(#)/(1 + #) (1+#) ) , H(, #) . = exp(-# 2 /2). Theorem 3.1. (=-=[23]-=-) Let X 1 , X 2 , . . . , X # be independent random variables, each taking values in [0, 1], R = # # i=1 X i , and E[R] = . Then, for any # # 0, Pr(R # (1 + #)) # G(, #). Also, if 0 # # # 1, Pr(R # (1... |

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Citation Context ...be formulated as a covering problem with side packing constraints. In CIPs, the packing constraints are upper bound constraints on the variables. Definition 1.2 (Covering Integer Programs). Given A # =-=[0, 1]-=- mn , b # [0, #) m and c # [0, 1] n , a CIP seeks to minimize c T x subject to Ax # b, x # Z n + , and 0 # x j # d j for each j (the d j # Z+ are given integers). If A # {0, 1} mn , then we assume w.l... |

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Citation Context ...t Routing. Our main result is a constant-factor approximation algorithm for store-and-forward packet routing, a fundamental routing problem in interconnection networks (see Leighton's book and survey =-=[14, 15]-=-); furthermore, the queue sizes will all be bounded by a constant. This packet routing problem has received considerable attention for more than 15 years, and is as follows: Definition 1.1 (Store-and-... |

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Citation Context ... any S # N , let #S # {0, 1} n denote the incidence vector of S. We shall consider the problem (OPT ) min{c T #F : F # F}, where c # # n + is a weight function on the elements of N . Definition 2.5. (=-=[25]-=-) The blocking clutter of F is the family B(F), whose members are precisely those H # N that satisfy: P1. Intersection: H # F #= # for all F # F , and P2. Minimality: If H # is any proper subset of H,... |

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Citation Context ... work: a distribution that is "good" in expectation on individual elements (edges, paths) is su#cient. Recall that in many "discrete ham-sandwich theorems" (Beck & Spencer [4], Rag=-=havan & 10 Thompson [27]-=-), it is easy to ensure good expectation on individual entities (e.g., the constraints of an integer program), but is much more di#cult to construct one solution that is simultaneously good on all the... |

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Citation Context ...t. However, they did not address the issue of path selection; one motivation for their work is that paths can plausibly be selected using, e.g., the well-known "random intermediate destinations&q=-=uot; idea [33, 34]-=-. However, no general results on path selection, and hence on the time needed for packet routing, were known for arbitrary networks G. We address this issue here by studying the paths selection proble... |

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Citation Context ...). No approximation algorithms with a sub-logarithmic approximation guarantee were known for this problem, to the best of our knowledge. For instance, a result from the seminal work of Leighton & Rao =-=[19]-=- leads to routing algorithms that are existentially good. Their network embedding of G ensures that there is some routing instance on G for which their routing time is to within an O(lg N) factor of o... |

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Citation Context ...t. However, they did not address the issue of path selection; one motivation for their work is that paths can plausibly be selected using, e.g., the well-known "random intermediate destinations&q=-=uot; idea [33, 34]-=-. However, no general results on path selection, and hence on the time needed for packet routing, were known for arbitrary networks G. We address this issue here by studying the paths selection proble... |

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Citation Context ...pproximation algorithms has focused a fair amount on bicriteria (or even multicriteria) minimization problems, attempting to simultaneously keep the values of two or more parameters "low" (s=-=ee, e.g., [11, 21, 22, 29, 30, 32]-=-). One motivation for this is that real-world problems often require such balancing. In this work, we consider a family of bicriteria problems that involve balancing a local capacity constraint (e.g.,... |

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Citation Context ...ven any routing problem with N packets on an N-node butterfly, there is a randomized on-line routing algorithm that, with high probability, routes the packets in O(log N) time using O(1)-sized queues =-=[28]-=-. (We let e denote the base of the natural logarithm, and, for x > 0, lg x, ln x, and ln + x respectively denote log 2 x, log e x, and max{log e x, 1}. Also, Z+ will denote the set of non-negative int... |

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Citation Context ...pproximation algorithms has focused a fair amount on bicriteria (or even multicriteria) minimization problems, attempting to simultaneously keep the values of two or more parameters "low" (s=-=ee, e.g., [11, 21, 22, 29, 30, 32]-=-). One motivation for this is that real-world problems often require such balancing. In this work, we consider a family of bicriteria problems that involve balancing a local capacity constraint (e.g.,... |

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Citation Context ...sky and Rabani [26] for good on-line packet scheduling algorithms, given the path to be traversed for each packet. 3 By combining some new ideas with certain powerful results of Leighton, Maggs & Rao =-=[16]-=-, Leighton, Maggs & Richa [17], Karp, Leighton, Rivest, Thompson, Vazirani & Vazirani [13] and Lin & Vitter [20], we present the first polynomial-time o#- line constant-factor approximation algorithm ... |

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Citation Context ...pproximation algorithms has focused a fair amount on bicriteria (or even multicriteria) minimization problems, attempting to simultaneously keep the values of two or more parameters "low" (s=-=ee, e.g., [11, 21, 22, 29, 30, 32]-=-). One motivation for this is that real-world problems often require such balancing. In this work, we consider a family of bicriteria problems that involve balancing a local capacity constraint (e.g.,... |

76 |
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Citation Context ...t. 3 By combining some new ideas with certain powerful results of Leighton, Maggs & Rao [16], Leighton, Maggs & Richa [17], Karp, Leighton, Rivest, Thompson, Vazirani & Vazirani [13] and Lin & Vitter =-=[20]-=-, we present the first polynomial-time o#- line constant-factor approximation algorithm for the store-and-forward packet routing problem. Furthermore, the queue sizes of the edges are bounded by O(1).... |

65 | Fast al- gorithms for finding O(congestion + dilation) packet routing schedules - Leighton, Maggs, et al. - 1996 |

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Citation Context |

52 |
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Citation Context ...t Routing. Our main result is a constant-factor approximation algorithm for store-and-forward packet routing, a fundamental routing problem in interconnection networks (see Leighton's book and survey =-=[14, 15]-=-); furthermore, the queue sizes will all be bounded by a constant. This packet routing problem has received considerable attention for more than 15 years, and is as follows: Definition 1.1 (Store-and-... |

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Citation Context ...roach of appropriately using the rounding theorem of [13] has subsequently been applied by Bar-Noy, Guha, Naor & Schieber to develop approximation algorithms for a family of multi-casting 17 problems =-=[3]-=-. It has also been applied for a family of routing problems by Andrews & Zhang [2]. The second major result in the paper improves upon a class of results in multicriteria covering integer programs. We... |

46 | On the fault tolerance of some popular bounded-degree networks
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Citation Context ...lt-tolerance: the possibility of G being a reasonable routing network when its nodes are subject to (e.g., random or worst-case) faults. See, e.g., Kaklamanis et al. [12], Leighton, Maggs & Sitaraman =-=[18]-=-, and Cole, Maggs & Sitaraman [6]. In this case, we do not expect good on-line algorithms, since the fault-free subgraph G of G has an unpredictable structure. Indeed, a fair amount of research in thi... |

45 | Improved approximation guarantees for packing and covering integer programs
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Citation Context ...ra [5] conduct a detailed study of approximating CIPs and present an approximation algorithm which finds a feasible solution whose value is O(y # lg m) [5]. Previous work of this paper's first author =-=[31] pres-=-ents an algorithm that computes an x # Z n + such that Ax # b and c T x # a 0 y # max{ln + (mB/y # )/B, # ln + (mB/y # )/B} (1.1) for some absolute constant a 0 > 0. 1 The bound "x j # d ## j may... |

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Citation Context ...e global criterion 4 of total construction cost. For similar reasons, CIPs typically include the constraints {x j # d j : 1 # j # n}. In fact, the case where d j = 1 for all j is quite common. Dobson =-=[7]-=- and Fisher &Wolsey [8] study a natural greedy algorithm GA for CIPs. For a given CIP, let OPT denote the value of its optimal integral solution. We define # 1 . = min i,j {A i,j /c j : A i,j #= 0} an... |

32 | Global wire routing in two-dimensional arrays
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Citation Context ...aversed for each packet. 3 By combining some new ideas with certain powerful results of Leighton, Maggs & Rao [16], Leighton, Maggs & Richa [17], Karp, Leighton, Rivest, Thompson, Vazirani & Vazirani =-=[13]-=- and Lin & Vitter [20], we present the first polynomial-time o#- line constant-factor approximation algorithm for the store-and-forward packet routing problem. Furthermore, the queue sizes of the edge... |

31 | Asymptotically tight bounds for computing with faulty arrays of processors
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Citation Context ...h in routing is concerned with fault-tolerance: the possibility of G being a reasonable routing network when its nodes are subject to (e.g., random or worst-case) faults. See, e.g., Kaklamanis et al. =-=[12]-=-, Leighton, Maggs & Sitaraman [18], and Cole, Maggs & Sitaraman [6]. In this case, we do not expect good on-line algorithms, since the fault-free subgraph G of G has an unpredictable structure. Indeed... |

29 | Anti-blocking polyhedra - Fulkerson - 1972 |

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Citation Context ...failure at a node), we may wish to upper bound the number of facilities that can be placed at individual sites. The more general problem of "file sharing" in a network has been studied by Na=-=or & Roth [24]-=-, where again, the maximum load (number of facilities) per node is balanced with the global criterion 4 of total construction cost. For similar reasons, CIPs typically include the constraints {x j # d... |

22 |
Integral approximation sequences
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Citation Context ...ore relaxed approaches could work: a distribution that is "good" in expectation on individual elements (edges, paths) is su#cient. Recall that in many "discrete ham-sandwich theorems&qu=-=ot; (Beck & Spencer [4]-=-, Raghavan & 10 Thompson [27]), it is easy to ensure good expectation on individual entities (e.g., the constraints of an integer program), but is much more di#cult to construct one solution that is s... |

18 | Rounding algorithms for covering problems
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Citation Context ...elaxation, wherein each x j is allowed to be a real in the range [0, d j ]. Throughout, we shall let y # denote the LP optimum of a given CIP. Clearly, y # is a lower bound on OPT . Bertsimas & Vohra =-=[5]-=- conduct a detailed study of approximating CIPs and present an approximation algorithm which finds a feasible solution whose value is O(y # lg m) [5]. Previous work of this paper's first author [31] p... |

18 |
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Citation Context ...n + (mB/y∗ ) ≤ B is handled well in [31]; thus we shall assume ln + (mB/y∗ ) > B. We now summarize the main result of [31] for CIPs as a theorem. A key ingredient of Theorem 3.2 is the FKG inequality =-=[9]-=-. Theorem 3.2 (see [31]). For any given CIP, suppose we are given any 1 ≤ β< α<κsuch that (3.2) (1 − H(Bα,1 − β/α)) m >G(y ∗ α, κ/α − 1) holds. Then we can find in deterministic polynomial time a vect... |

15 | Fast algorithms for finding O(congestion+dilation) packet routing schedules - Leighton, Maggs, et al. |

13 | Routing on butterfly networks with random faults
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(Show Context)
Citation Context ... being a reasonable routing network when its nodes are subject to (e.g., random or worst-case) faults. See, e.g., Kaklamanis et al. [12], Leighton, Maggs & Sitaraman [18], and Cole, Maggs & Sitaraman =-=[6]-=-. In this case, we do not expect good on-line algorithms, since the fault-free subgraph G of G has an unpredictable structure. Indeed, a fair amount of research in this area, e.g., [6, 18], focuses on... |

10 |
On the greedy heuristic for continuous covering and packing problems
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(Show Context)
Citation Context ... total construction cost. For similar reasons, CIPs typically include the constraints {x j # d j : 1 # j # n}. In fact, the case where d j = 1 for all j is quite common. Dobson [7] and Fisher &Wolsey =-=[8]-=- study a natural greedy algorithm GA for CIPs. For a given CIP, let OPT denote the value of its optimal integral solution. We define # 1 . = min i,j {A i,j /c j : A i,j #= 0} and # 2 . = max j ( # m i... |

10 |
Scheduling n independent jobs on m uniform machines with both flow time and makespan objectives: A parametric approach
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(Show Context)
Citation Context |

8 |
Universal O(congestion � dilation � log 1�� N) local control packet switching algorithms
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(Show Context)
Citation Context ...ratio. Furthermore, it seems like a di#cult problem to construct on-line routing algorithms for arbitrary networks, even with, say, a polylogarithmic approximation guarantee. See Ostrovsky and Rabani =-=[26]-=- for good on-line packet scheduling algorithms, given the path to be traversed for each packet. 3 By combining some new ideas with certain powerful results of Leighton, Maggs & Rao [16], Leighton, Mag... |

5 | Packet routing with arbitrary end-to-end delay requirements
- Andrews, Zhang
- 1999
(Show Context)
Citation Context ...plied by Bar-Noy, Guha, Naor & Schieber to develop approximation algorithms for a family of multi-casting 17 problems [3]. It has also been applied for a family of routing problems by Andrews & Zhang =-=[2]-=-. The second major result in the paper improves upon a class of results in multicriteria covering integer programs. We show that the local criterion of unweighted covering integer programs can be impr... |

5 |
Blocking polyhedra, Graph Theory and its Applications
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(Show Context)
Citation Context ...uestion, we need to present an alternative (polyhedral) perspective of our (path) selection routine. First we recall some standard definitions from polyhedral combinatorics. The reader is referred to =-=[10]-=- for related concepts. Suppose we are given a finite set N = {1, 2,...,n} and a family F of subsets of N . For any S ⊆N, let χS ∈{0, 1} n denote the incidence vector of S. We shall consider the proble... |

5 |
Universal O(congestion + dilation + log 1+ɛ n) local control packet switching algorithms
- Ostrovsky, Rabani
- 1997
(Show Context)
Citation Context ...tio. Furthermore, it seems like a difficult problem to construct on-line routing algorithms for arbitrary networks, even with, say, a polylogarithmic approximation guarantee. See Ostrovsky and Rabani =-=[26]-=- for good on-line packet scheduling algorithms, given the path to be traversed for each packet. By combining some new ideas with certain powerful results of Leighton, Maggs, and Rao [16], Leighton, Ma... |

4 | Service-constrained network design problems
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- 1996
(Show Context)
Citation Context ...pproximation algorithms has focused a fair amount on bicriteria (or even multicriteria) minimization problems, attempting to simultaneously keep the values of two or more parameters “low” (see, e.g., =-=[11, 21, 22, 29, 30, 32]-=-). One motivation for this is that real-world problems often require such balancing. In this work, we consider a family of bicriteria problems that involve balancing a local capacity constraint (e.g.,... |