Birkhoff-von Neumann Decomposition Research Articles

On the finite representation of linear group equivariant operators via permutant measures

Recent advances in machine learning have highlighted the importance of using group equivariant non-expansive operators for building neural networks in a more transparent and interpretable way. An operator is called equivariant with respect to a group if the action of the group commutes with the operator. Group equivariant non-expansive operators can be seen as multi-level components that can be joined and connected in order to form neural networks by applying the operations of chaining, convex combination and direct product. In this paper we prove that each linear G-equivariant non-expansive operator (GENEO) can be produced by a weighted summation associated with a suitable permutant measure, provided that the group G transitively acts on a finite signal domain. This result is based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices and some well known facts in group theory. Our theorem makes available a new method to build all linear GENEOs with respect to a transitively acting group in the finite setting. This work is part of the research devoted to develop a good mathematical theory of GENEOs, seen as relevant components in machine learning.

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.

Perfect Matchings in Õ (n1.5) Time in Regular Bipartite Graphs

In this paper we consider the well-studied problem of finding a perfect matching in a d-regular bipartite graph on 2n nodes with m=nd edges. The best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes time O(m\sqrt{n}). In regular bipartite graphs, however, a matching is known to be computable in O(m) time (due to Cole, Ost and Schirra). In a recent line of work by Goel, Kapralov and Khanna the O(m) time algorithm was improved first to \tilde O(min{m, n^{2.5}/d}) and then to \tilde O(min{m, n^2/d}). It was also shown that the latter algorithm is optimal up to polylogarithmic factors among all algorithms that use non-adaptive uniform sampling to reduce the size of the graph as a first step. In this paper, we give a randomized algorithm that finds a perfect matching in a d-regular graph and runs in O(n\log n) time (both in expectation and with high probability). The algorithm performs an appropriately truncated random walk on a modified graph to successively find augmenting paths. Our algorithm may be viewed as using adaptive uniform sampling, and is thus able to bypass the limitations of (non-adaptive) uniform sampling established in earlier work. We also show that randomization is crucial for obtaining o(nd) time algorithms by establishing an \Omega(nd) lower bound for any deterministic algorithm. Our techniques also give an algorithm that successively finds a matching in the support of a doubly stochastic matrix in expected time O(n\log^2 n) time, with O(m) pre-processing time; this gives a simple O(m+mn\log^2 n) time algorithm for finding the Birkhoff-von Neumann decomposition of a doubly stochastic matrix.

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.

Preconditioning Techniques Based on the Birkhoff–von Neumann Decomposition

Abstract We introduce a class of preconditioners for general sparse matrices based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices. These preconditioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.

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.

Birkhoff-von Neumann switching with statistical traffic profiles

Offline scheduling is attractive when the traffic profile is known a priori since the complexity of online scheduling is largely removed. Given any admissible rate matrix Λ = ( λ ij ) , Birkhoff-von Neumann decomposition offers a quasi-offline scheduling strategy that can provide a service rate of at least λ ij for each flow ( i , j ) without internal speedup. This paper extends the Birkhoff-von Neumann switching strategy to consider traffic profiles in which the traffic demands are specified as statistical distributions rather than fixed rates.

On guaranteed smooth scheduling for input-queued switches

Input-queued switches are used extensively in the design of high-speed routers. As switch speeds and sizes increase, the design of the switch scheduler becomes a primary challenge, because the time interval for the matching computations needed for determining switch configurations becomes very small. Possible alternatives in scheduler design include increasing the scheduling interval by using envelopes , and using a frame-based scheduler that guarantees fixed rates between input-output pairs. However, both these alternatives have significant jitter drawbacks: the jitter increases with the envelope size in the first alternative, and previously-known methods do not guarantee tight jitter bounds in the second. In this paper, we propose a hybrid approach to switch scheduling. Traffic with tight jitter constraints is first scheduled using a frame-based scheduler that achieves low jitter bounds. Jitter-insensitive traffic is later scheduled using an envelope-based scheduler. The main contribution of this paper is a scheduler design for generating low-jitter schedules. The scheduler uses a rate matrix decomposition designed for low jitter and different from the minimum-bandwidth Birkhoff-Von Neumann (BV) decomposition. In addition to generating low-jitter schedules, this decomposition in the worst case yields fewer switch configuration matrices (O(n)) than the BV decomposition (O(n/sup 2/)), and so requires far less high-speed switch memory. We develop an efficient algorithm for decomposing the rate matrix and for scheduling the permutation matrices. We prove that our low-jitter algorithm has an O(logn) factor bound on its bandwidth consumption in comparison to the minimum-bandwidth BV decomposition. Experimentally, we find that the bandwidth increase in practice is much lower than the theoretical bound. We also prove several related performance bounds for our scheduler. Finally, we propose a practical algorithm for bandwidth-guaranteed algorithm, and show how our findings could even be extended to systems with large tuning time.

Load balanced Birkhoff-von Neumann switches with resequencing

Load balanced Birkhoff-von Neumann switches with resequencing

Enhanced Birkhoff–von Neumann decomposition algorithm for input queued switches

The enhanced Birkhoff–von Neumann decomposition (EBVND) algorithm, a new class of scheduling arbitrators for input queued (IQ) crossbar switches that is based on the Birkhoff–von Neumann decomposition algorithm, is introduced. Theoretical analysis shows that the performance of EBVND is better than the Birkhoff–von Neumann decomposition algorithm in terms of throughput and cell delay, and can also provide rate and cell delay guarantees. Also, the weighted rate filling algorithm (WRFA), a new algorithm that can be ueed to construct doubly stochastic matrices from doubly substochastic matrices with less complexity and better fairness, is proposed. The wave front Birkhoff–von Neumann decomposition (WFBVND) algorithm and its simplified version WFBVND with logN iterations (WFBVND-logN), the special cases of EBVND, are also introduced and evaluated. Simulations show that the WFBVND and WFBVND-logN algorithms have much lower average cell delay as compared to the Birkhoff–von Neumann decomposition algorithm.