Minimum Birkhoff-von Neumann Decomposition

Kulkarni, Janardhan; Lee, Euiwoong; Singh, Mohit

doi:10.1007/978-3-319-59250-3_28

Cited by 11 publications

(8 citation statements)

References 16 publications

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“…From Lemma 4 in [19] we can obtain that the pyramidal width of the Birkhoff polytope is lower bounded by 1/ √ d = 1/n. 23 Hence, (λ/M ) 2 ≥ 1/(2n 3 ) as claimed.…”

Section: Proof Of Lemmamentioning

confidence: 66%

See 1 more Smart Citation

Birkhoff’s Decomposition Revisited: Sparse Scheduling for High-Speed Circuit Switches

Valls

Iosifidis

2021

IEEE/ACM Trans. Networking

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Data centers are increasingly using high-speed circuit switches to cope with the growing demand and reduce operational costs. One of the fundamental tasks of circuit switches is to compute a sparse collection of switching configurations to support a traffic demand matrix. Such a problem has been addressed in the literature with variations of the approach proposed by Birkhoff in 1946 to decompose a doubly stochastic matrix exactly. However, the existing methods are heuristic and do not have theoretical guarantees on how well a collection of switching configurations (i.e., permutations) can approximate a traffic matrix (i.e., a scaled doubly stochastic matrix). In this paper, we revisit Birkhoff's approach and make three contributions. First, we establish the first theoretical bound on the sparsity of Birkhoff's algorithm (i.e., the number of switching configurations necessary to approximate a traffic matrix). In particular, we show that by using a subset of the admissible permutation matrices, Birkhoff's algorithm obtains an -approximate decomposition with at most O(log(1/)) permutations. Second, we propose a new algorithm, Birkhoff+, which combines the wealth of Frank-Wolfe with Birkhoff's approach to obtain sparse decompositions in a fast manner. And third, we evaluate the performance of the proposed algorithm numerically and study how this affects the performance of a circuit switch. Our results show that Birkhoff+ is superior to previous algorithms in terms of throughput, running time, and number of switching configurations.

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“…From Lemma 4 in [19] we can obtain that the pyramidal width of the Birkhoff polytope is lower bounded by 1/ √ d = 1/n. 23 Hence, (λ/M ) 2 ≥ 1/(2n 3 ) as claimed.…”

Section: Proof Of Lemmamentioning

confidence: 66%

“…X = k i=1 θ i P i ) was addressed in [15] and shown to be NP-hard. In [23], the authors also show that the problem is not tractable when the minimal decomposition can be expressed with k ≥ 4 permutations. The work in [24] formulates a similar problem.…”

Section: B Related Mathematical Problemsmentioning

confidence: 99%

Birkhoff’s Decomposition Revisited: Sparse Scheduling for High-Speed Circuit Switches

Valls

Iosifidis

2021

IEEE/ACM Trans. Networking

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“…Some other works assume moderate configuration delay in that they allow ( ) configurations, where is the number of nodes/flows, and with that constraint, focus on other optimization objectives such as higher link utilization [34], bounded packet delay [39]. Some works have also addressed minimizing the number of reconfigurations [22] without regard to the total time window.…”

Section: Related Workmentioning

confidence: 99%

Near-optimal multihop scheduling in general circuit-switched networks

Gupta

Curran

Zhan

2020

Proceedings of the 16th International Conference on Emerging Networking EXperiments and Technologies

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Circuit switched networks with high-bandwidth links are essential to handling ever increasing traffic demands in today's data centers. As these networks incur a non-trivial reconfiguration delay, they are mainly suited for bursty traffic or large flows. To address the reconfiguration delay vs. high-bandwidth tradeoff in circuit networks, an essential traffic scheduling problem is to determine a sequence of network configurations to optimally serve a given traffic. Recent works have addressed this scheduling problem for one-hop traffic in fully-connected circuit networks. In this work, we consider the traffic scheduling problem in general circuit networks with multi-hop traffic load. Such a general model is essential for networks with indirect routes between some nodes, e.g., for recently proposed wireless optical (FSO-based) networks, or to allow multi-hop routes for load balancing. In this context, we develop an efficient algorithm that empirically delivers high network throughput, while also guaranteeing a constant-factor approximation with respect to an objective closely related to network throughput. We generalize our technique and approximation result to more general settings, including to the joint optimization problem of determining flow routes as well as a sequence of network configurations. We demonstrate the effectiveness of our techniques via extensive simulations on synthetic traffic loads based on published traffic characteristics as well as publicly available real traffic loads; we observe significant performance gains in terms of network throughput when compared to approaches based on prior work, and very similar performance to an appropriate upper bound.

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“…The number of stages correspond to the number of (sub-)permutation matrices. A recent variation of this problem appears in routing in data centers [6]. BvN decompositions are also used to build preconditioners for solving sparse linear systems [7].…”

Section: Introductionmentioning

confidence: 99%

“…Second, we investigate the worst case performance guarantee of the original Birkhoff heuristic and show that it can be very bad. To do so, we describe a family of matrices where the smallest number of permutation matrices required is 3 while the Birkhoff heuristic finds decompositions with n permutation matrices, for a matrix of size n × n. Kulkarni et al [6,Theorem 7] also present examples to show that the worst case performance of Birkhoff heuristic can be very bad. Our construction is more explicit and seems simpler.…”

Section: Introductionmentioning

confidence: 99%

Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

Dufossé

Kaya

Panagiotas

et al. 2018

Linear Algebra and its Applications

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The well-known Birkhoff-von Neumann (BvN) decomposition expresses a doubly stochastic matrix as a convex combination of a number of permutation matrices. For a given doubly stochastic matrix, there are many BvN decompositions, and finding the one with the minimum number of permutation matrices is NP-hard. There are heuristics to obtain BvN decompositions for a given doubly stochastic matrix. A family of heuristics is based on the original proof of Birkhoff and proceeds step by step by subtracting a scalar multiple of a permutation matrix at each step from the current matrix, starting from the given matrix. At every step, the subtracted matrix contains nonzeros at the positions of some nonzero entries of the current matrix and annihilates at least one entry, while keeping the current matrix nonnegative. Our first result, which supports a claim of Brualdi [Canad. Math. Bull. 25 (1982), pp. 191-199], shows that this family of heuristics can miss optimal decompositions. We also investigate the performance of two heuristics from this family theoretically.

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Minimum Birkhoff-von Neumann Decomposition

Cited by 11 publications

References 16 publications

Birkhoff’s Decomposition Revisited: Sparse Scheduling for High-Speed Circuit Switches

Birkhoff’s Decomposition Revisited: Sparse Scheduling for High-Speed Circuit Switches

Near-optimal multihop scheduling in general circuit-switched networks

Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

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