Birkhoff-von Neumann Research Articles

Approximating Weighted and Priced Bribery in Scoring Rules

The classic Bribery problem is to find a minimal subset of voters who need to change their vote to make some preferred candidate win. Its important generalizations consider voters who are weighted and also have different prices. We provide an approximate solution for these problems for a broad family of scoring rules (which includes Borda and t-approval), in the following sense: for constant weights and prices, if there exists a strategy which costs k, we efficiently find a strategy which costs at most k+\widetilde{O}(sqrt(k)). An extension for non-constant weights and prices is also given.
 Our algorithm is based on a randomized reduction from these Bribery generalizations to weighted coalitional manipulation (WCM). To solve this WCM instance, we apply the Birkhoff-von Neumann (BvN) decomposition to a fractional manipulation matrix. This allows us to limit the size of the possible ballot search space reducing it from exponential to polynomial, while still obtaining good approximation guarantees. Finding a solution in the truncated search space yields a new algorithm for WCM, which is of independent interest.

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

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 (1982) [3], shows that this family of heuristics can miss optimal decompositions. We also investigate the performance of two heuristics from this family theoretically.

Approximating Bribery in Scoring Rules

The classic bribery problem is to find a minimal subset of voters who need to change their vote to make some preferred candidate win.We find an approximate solution for this problem for a broad family of scoring rules (which includes Borda and t-approval), in the following sense: if there is a strategy which requires bribing k voters, we efficiently find a strategy which requires bribing at most k + Õ(√k) voters. Our algorithm is based on a randomized reduction from bribery to coalitional manipulation (UCM). To solve the UCM problem, we apply the Birkhoff-von Neumann (BvN) decomposition to a fractional manipulation matrix. This allows us to limit the size of the possible ballot search space reducing it from exponential to polynomial, while still obtaining good approximation guarantees. Finding the optimal solution in the truncated search space yields a new algorithm for UCM, which is of independent interest.

Deflection-Compensated Birkhoff–von-Neumann Switches

Despite the high throughput and low complexity achieved by input scheduling based on Birkhoff-von-Neumann (BvN) decomposition, the performance of the BvN switch becomes less predictable when the input traffic is bursty. In this paper, we propose a deflection-compensated BvN (D-BvN) switch architecture to enhance the quasistatic scheduling based on BvN decomposition. D-BvN switches provide capacity guarantee for virtual circuits (VCs) and deflect bursty traffic when overflow occurs. The deflection scheme is devised to offset the excessive buffer requirement of each VC when input traffic is bursty. The design of our conditional deflection mechanism is based on the fact that it is unlikely that the traffic input to VCs is all bursty at the same time; most likely, some starving VCs have spare capacities when some other VCs are in the overflow state. The proposed algorithm makes full use of the spare capacities of those starving VCs to deflect the overflow traffic to other inputs and provide bandwidth for the deflected traffic to re-access the desired VC. Our analysis and simulation results show that this deflection-compensated mechanism can support BvN switches to achieve close to 100% throughput of offered load even with bursty input traffic, and reduces the average end-to-end delay and delay jitter. Also, our result indicates that the packet out-of-sequence probability due to deflection of overflow traffic is negligible, and thus, only a small re-sequencing buffer is needed at each output port. We also compare D-BvN with the well-established online scheduling algorithm iSLIP, and the result demonstrates that D-BvN outperforms iSLIP in terms of the throughput of offered load when the traffic is non-uniform or the traffic load is not very high.

Notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

Birkhoff–von Neumann (BvN) decomposition of doubly stochastic matrices expresses a double stochastic matrix as a convex combination of a number of permutation matrices. There are known upper and lower bounds for the number of permutation matrices that take part in the BvN decomposition of a given doubly stochastic matrix. We investigate the problem of computing a decomposition with the minimum number of permutation matrices and show that the associated decision problem is strongly NP-complete. We propose a heuristic and investigate it theoretically and experimentally on a set of real world sparse matrices and random matrices.

EBvN: efficient BvN in multi-fiber/multi-wavelength overlaid-star optical networks

Overlaid-star networks with reservation-based scheduling could be appropriate networks for metro areas. The Birkhoff–von Neumann (BvN) scheduling could be used at the core nodes of an overlaid network to schedule lossless traffic transmission among edge nodes. The common method is to schedule traffic separately for each wavelength channel, called separated BvN (SBvN) scheduling in this paper. However, SBvN cannot schedule all traffic demands, especially at high-traffic loads. In this paper, the BvN scheduling procedure is modified to efficiently schedule traffic in overlaid-star networks with multi-fiber/multi-wavelength architecture, called efficient BvN (EBvN). Instead of using one processor to schedule traffic on one wavelength channel in each core node, the proposed EBvN technique uses only one processor to schedule all traffic demands on all fibers/wavelength channels at the same time. Performance evaluation results under both uniform and non-uniform traffic distributions show that more traffic demands can be scheduled under EBvN compared with SBvN. In addition, the scheduling speed of EBvN is mostly faster than SBvN. Finally, EBvN can provide bound on the maximum scheduling time of EBvN. As a trade-off between scheduling time and residual traffic, EBvN with filling empty cells (EBvN_FEC) is proposed that can reduce residual traffic, but at the expense of slightly increasing scheduling time. EBvN_FEC is more effective than EBvN under non-uniform traffic distribution.

Agile bandwidth management techniques in slotted all-optical packet switched networks

The next generation metro network is most likely to be based on high-capacity agile all-optical networks. We consider a metro network architecture that consists of a number of buffer-less all-optical core switches, where a number of edge switches are connected to each core switch in an overlaid star topology. We would like to design potential candidates of resource-sharing schemes in such a network and evaluate their performances, where they guarantee to deliver optical packets to their destinations. The first scheme is reservation-based, in which decisions are made at each core switch so that collisions (and therefore loss) can be avoided through reservations. For this scheme, we have re-designed the BvN (Birkhoff von Neumann) scheduling algorithm to accommodate the asymmetric multi-fiber/multi-wavelength feature in star OPS. In the second scheme, distributed and independent decisions are made at edge switches, but we need to retransmit the traffic dropped at core switches in the optical domain. We also implement a hybrid case as the third scheme which combines the good attributes of both previous schemes. We have characterized and evaluated the performance of these resource-sharing schemes by various QoS measures such as delay and loss probabilities in edge switches.

Extending the Birkhoff-von Neumann switching strategy for multicast - On the use of optical splitting in switches

The Birkhoff-von Neumann (BVN) strategy for single-stage input-queued crossbar switches does not support multicast, as it considers only permutation-based switch configurations. This paper extends the BVN strategy to multicast switching, where an input can simultaneously transmit to multiple outputs. Knowledge of the average rates of flows is used to compute an offline schedule. We begin by considering a system in which the fanout of each flow is split in a predecided manner. We call this static splitting (as opposed to dynamic splitting where no such constraint is imposed), and we study the rate region of the switch under this restriction. We provide a graph-theoretic formulation of the rate region.

Timeslot Weighted Fair Scheduling in EPFTS

Based on the technology of EPFTS (ethernet-like physical frame timeslot switching), this paper introduces a new scheduling algorithm call TWFS (timeslot weighted fair scheduling) which could be implemented in EPFTS nodes to fit the requirement of fast data switching and QoS (quality of service) guarantee in SUPANET (single physical layer user-data platform architecture network). By analyzing merits and shortcomings of the typical scheduling techniques such as iSlip (iteration round-robin match with slip) and BvN-switch (Birkhoff-von neumann switch), TWFS utilizes the same iteration mechanism as iSlip and takes the total reserved timeslots of an I/O port pair as the weight (priority) for data switching, thus overcomes the shortcoming of the slow response to traffic fluctuation of BvN-switch, while keeping the same calculation complexity in iteration to O(log2N) as with iSlip. Simulation results have shown that TWFS can made an optimal balance among effectiveness、fairness, and complexity and is most suitable for future high-speed EPFTS nodes in SUPANET.

An algorithmic proof of a generalization of the birkhoff-von neumann theorem

An algorithmic proof of a generalization of the birkhoff-von neumann theorem