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The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.

The question is to point out different generalizations of this theorem, different "non-generalizations" namely cases where an expected generalization is false, and to briefly describe the context of these generalizations.

A related MO question: Sampling from the Birkhoff polytope

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These are all very nice answers. To encourage more I start a little bounty where as before I will "accept" one useful answer. –  Gil Kalai Dec 4 '09 at 9:54
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12 Answers 12

up vote 9 down vote accepted

I am cheating a little to give this answer, because I am fairly sure that it is part of Gil's motivation in asking the question. The most natural generalization of the Birkhoff hypothesis to quantum probability is only true for qubits. (It might also be true for a qubit tensor a classical system; I did not check that case.)

A quantum measurable space is a von Neumann algebra. We are most interested in the finite-dimensional case, where classically "measurable space" is just a fancy name for the random variables on a finite set. A finite-dimensional von Neumann algebra is a direct sum of matrix algebras. In particular, $M_2$ is called a qubit and $M_d$ is called a qudit.

To make a long story short, the Birkhoff hypothesis can be stated for a direct sum of $a$ copies of $M_b$, or $aM_b$. In this setting, a doubly stochastic map $E$ is a linear map from $aM_b$ to itself that preserves trace, that preserves the identity element, and that is completely positive. In this setting, $E$ is completely positive if it takes positive semidefinite elements of $aM_b$ to positive semidefinite elements, and if $E \otimes I$ also has that property on the algebra $aM_b \otimes N$ for another von Neumann or $C^*$-algebra $N$. The natural analogue of permutation matrices are the *-algebra automorphisms of $aM_b$. These are permutations of the matrix blocks, composed with maps of the form $E(x) = uxu^*$, where $u$ is a unitary element of $aM_b$. The question as before is whether the doubly stochastic maps are the convex hull of the automorphisms.

This Birkhoff hypothesis is true for $M_2$, false for $M_d$ for $d \ge 3$, and I should check it for $nM_2$. It is true for $aM_1 = a\mathbb{C}$, because then it is the usual Birkhoff-von Neumann theorem.

I am left wondering about two infinite classical versions of Birkhoff's theorem, for the algebras $\ell^\infty(\mathbb{N})$ and $L^\infty([0,1])$. In the former case, one would ask whether any stochastic map that preserves counting measure (even though counting measure is not normalized) is an infinite convex sum of permutations of $\mathbb{N}$. In the latter case, whether any stochastic map that preserve Lebesgue measure is a convex integral of measure-preserving permutations of $[0,1]$. Addendum: At least the discrete infinite case is addressed, with generally positive results, in this review and in this older review. The older paper also raises the continuous question but with no results. However, with some more Googling I found this counterexample paper.


Since Gil asks for a reference, a recent one is Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem, by Mendl and Wolf.


Here also is a more orthodox combinatorial generalization of the Birkhoff theorem, and also another case that I once encountered that is between a generalization and a non-generalization. Since Gil now offers a bounty, maybe it's better to merge this answer with the other one.

A doubly stochastic matrix can be interpreted as a flow through a directed graph, with unit capacities. (See Unimodular matrix in Wikipedia; I learned about this long ago from Jesus de Loera.) Any such graph has a polytope of flows, called a network flow polytope. Any network flow polytope has integer vertices, because it is a totally unimodular polytope.

A totally unimodular polytope is a polytope whose facets have integer equations, and with the property that any maximal, linearly independent collection of facets intersects in an integer point because their matrix has determinant $\pm 1$. In particular the vertices are such intersections, so the vertices are all integral. This is a vast generalization of Birkhoff's theorem that comes from generalizing one of the proofs of Birkhoff's theorem.

Example: An alternating-sign matrix is equivalent to a square ice orientation of a square grid. The square ice orientations can be defined by a network flow, so you obtain an alternating-sign-matrix polytope. The generalized Birkhoff theorem in this case says that every vertex of the polytope is an alternating-sign matrix, in fact that every integer point of the $n$-dilated polytope is a sum of $n$ alternating-sign matrices.


The other case that I encountered was the polytope of fractional perfect matchings of a non-bipartite set with $2n$ elements. By contrast, the Birkhoff polytope is the case of a bipartite set with $n$ elements of each type. By definition, it is the polytope of non-negative weights assigned to the edges of the complete graph on $2n$ vertices, such that the total weight at each vertex is 1. Strictly speaking, the Birkhoff theorem is false; not every vertex is a perfect matching. Instead, all of the vertices are combinations of matched pairs, and odd cycles with weight $\frac12$.

At first glance this looks like bad news for the application of computing a perfect matching or the optimum perfect matching of a graph. Indeed, if instead you take the convex hull of the perfect matchings, the result is a polytope with exponentially many facets. However, a good algorithm exists anyway; there is a version of the simplex algorithm that only ever uses polynomially many of the facets.

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Actually the motivation for the question is a recent extension of the Birkhoff-von Neumann theorem by Ellis, Friedgut and Pilpel that I heard yeasterday. I thought it can be useful to try to collect various such generalizations fron various directions and it is also true that Greg and I discussed over the years no less than three different directions where this theorem is extended on of which is the "non-generalization" Greg's mention. (Greg, is there any reference or link?) (In any case, what is "cheating" about it?) –  Gil Kalai Nov 27 '09 at 11:59
    
It is cheating in the sense that you already knew this answer. –  Greg Kuperberg Nov 27 '09 at 17:15
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Here is a problem that contains interesting links imaph.tu-bs.de/qi/problems/30.html –  Guillaume Aubrun Dec 6 '09 at 19:17
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Not sure if this is what you're looking for, but the statement for symmetric doubly stochastic matrices is that every such one can be written as a convex combination of $(\sigma + \sigma^t)/2$ where $\sigma$ is a permutation matrix. Not all of these are necessarily vertices, but it definitely is not an integral polytope, so maybe this would be a "non-generalization."

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I have found a paper on a generalization of the Birkhoff-von Neumann theorem here:

http://cowles.econ.yale.edu/conferences/2009/sum-09/theory/che.pdf

The authors are Eric Budish, Yeon-Koo Che, Fuhito Kojima, and Paul Milgram.

Here is the Abstract:

The Birkhoff-von Neumann Theorem shows that any bistochastic matrix can be written as a convex combination of permutation matrices. In particular, in a setting where n objects must be assigned to n agents, one object per agent, any random assignment matrix can be resolved into a deterministic assignment in accordance with the specified probability matrix. We generalize the theorem to accommodate a complex set of constraints encountered in many real-life market design problems. Specifically, the theorem can be extended to any environment in which the set of constraints can be partitioned into two hierarchies. Further, we show that this bihierarchy structure constitutes a maximal domain for the theorem, and we provide a constructive algorithm for implementing a random assignment under bihierarchical constraints. We provide several applications, including (i) single-unit random assignment, such as school choice; (ii) multi-unit random assignment, such as course allocation and fair division; and (iii) two- sided matching problems, such as the scheduling of inter-league sports matchups. The same method also finds applications beyond economics, generalizing previous results on the minimize makespan problem in the computer science literature

I have also found a master's thesis that involved generalization from a matrix to a hyper matrix, a matrix in higher dimensions. So one example would be a cubic array of numbers instead of a square. He proves a generalization to the three dimensional matrices which are called blocks. There are some open questions there as well. I found it interesting as I have wondered about extending the two dimensions of matrices to three coordinates and seeing what happened. It is available here:

https://ritdml.rit.edu/dspace/bitstream/1850/5967/1/NReffThesis05-18-2007.pdf

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It would have been nice if you quoted a sentence or two from the abstract, or gave us a notion what the paper is about. Life is too short to download PDFs just in case there might be something of interest there. –  Harald Hanche-Olsen Nov 26 '09 at 20:28
    
I have added the abstract –  Kristal Cantwell Nov 27 '09 at 2:27
    
Dear Kristal, the link is broken; can you add the name of the author? –  Gil Kalai Feb 11 '11 at 15:15
    
I have replaced the URL and added the names of the authors. –  Kristal Cantwell Feb 11 '11 at 19:55
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There exists a "true" generalization within quantum theory given by John Watrous in arXiv 0807.2668v1, where it is shown that a mixture of a double stochastic channel with a completely depolarizing channel satisfies the "quantum version" of the Birkhoff's theorem (convex combination of unitary channels).

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This is a weaker result: The convex hull of the unitary channels contains a homothetically shrunken copy of all of the doubly stochastic channels. –  Greg Kuperberg Nov 28 '09 at 16:08
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On recent extension of Birkhoff's theorem is by Ellis, Friedgut and Pilpel. The context is Erdos-Ko-Rado theorems for permutations. See Section 5 and especially Definition 17 and Theorem 29 of this paper.

Let me also mention this paper by Jessica Striker regarding the alternating-sign-matrix polytope.

A conjecture extending Birkoff von Neumann theorem to finite Coxeter groups (The A_n family gives the original case) which is proved in all cases but one can be found in this paper by N. McCarthy, D. Ogilvie, I. Spitkovsky, and N. Zobin.

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There is a relevant paper by Gromova regarding high dimensional matrices:

Gromova, M. B., The Birkhoff-von Neumann theorem for polystochastic matrices. (Russian) Operations research and statistical simulation, No. 2 (Russian), pp. 3–15, 149. Izdat. Leningrad. Univ., Leningrad, 1974. It was translated to English in the early 90s.

The MR review by by George P. Barker reads: In Section 1 the author gives the basic definitions and introduces a notion of the spectrum of a multidimensional matrix as a vector. The basic theorem which gives necessary and sufficient conditions for a vector to be the spectrum of an extremal matrix is then formulated. The necessary condition of the basic theorem consists of a direct generalization of the Birkhoff-von Neumann theorem.

A related paper is: Brualdi, R. A.; Csima, J. Extremal plane stochastic matrices of dimension three. Linear Algebra and Appl. 11 (1975), no. 2, 105–133.

and an older paper with some relevance is: Jurkat, W. B.; Ryser, H. J. Extremal configurations and decomposition theorems. I. J. Algebra 8 1968 194–222.

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The remaining case is actually H4. For this case the conjecture is false, since I found 1063 orbits of facets. In the same paper, it is claimed that for F4 the convex hull is given by the Birkhoff tensors. But this is false since I found another orbit defined by a matrix of rank 3. See details there

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This is remarkable. Thanks, Mathieu! –  Gil Kalai Feb 11 '11 at 15:12
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Dave Perkinson and coauthors have studied sub-polytopes of the Birkhoff-von Neumann polytope, in the sense that they consider convex hulls of the permutation matrices corresponding to certain subgroups of $S_n$. See, e.g., his paper with Jeff Hood for the case $A_n$.

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The paper DESIGNING RANDOM ALLOCATION MECHANISMS: THEORY AND APPLICATIONS by ERIC BUDISH, YEON-KOO CHE, FUHITO KOJIMA, AND PAUL MILGROM Also describes an extension of the Birkhoff-von Neumann theorem.

This is related to the following: There is some relation between the Birkhoff-von Neumann theorem and Scarf's theorem that asserts that balanced games have non empty core. The fact that that the houses allocation game is balanced follows directly from B-vN theorem. The fact that the game describing the stable-marriage problem requires a certain generalization.

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The book Combinatorial Matrix Classes by Richard Brualdi has a number of generalizations of a combinatorial nature, around Chapter 8. Also, the least common denominator of the vertices of a rational polytope is related to the period of the Ehrhart quasipolynomial, see for example Exercise 3.25 in the book of Beck and Robins, http://math.sfsu.edu/beck/papers/noprint.pdf .

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Thanks, Brendan, best --Gil –  Gil Kalai Aug 28 '11 at 17:54
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The asymptotic quantum Birkhoff conjecture by Smolin, Verstraete, and Winter was disproved by Haagerup and Musat. The following paper http://www.iro.umontreal.ca/~qip2012/SUBMISSIONS/short/qip2012_submission_105.pdf presents the state of the art and gives some references.

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A non-combinatorial non-generalization occurs in the Monge-Kantorovich optimal transport problem. In the finite, combinatorial, version, there are $N$ sources and $N$ destinations. One wants to transport one object from each source to a different destination. Different source-destination combinations have different costs, represented by an $N\times N$-matrix $C$. The problem is now to find the cheapest (in sum) way to get the objects from the sources to the destinations. Formally, one wants to find a cost-minimizing permutation matrix. One can solve the problem as a linear programming problem over the set of doubly stochastic matrices. A linear programming problem has a solution at extreme points, so the Birkhoff-von Neumann theorem guarantees that this approach works.

In the measure theoretic optimal transport problem (I'm not using the most general formulation), one is given two nonempty compact metric spaces $S\times T$, a continuous function $c:S\times T\to\mathbb{R}$ and two Borel probability measures $\mu_S$ and $\mu_T$ on $S$ and $T$ respectively. Let $P$ be the set of all Borel probability measures on $S\times T$ with $S$-marginal $\mu_S$ and $T$-marginal $\mu_T$. The set $P$ generalizes the doubly stochastic matrices and is a convex and weak*-compact set. The optimal transport problem is now solving the linear programming problem $$\min_{\mu\in P}\int c~d\mu.$$ The problem has a solution and there is a solution concentrated at an extreme point of $P$. But the extreme points of $P$ are in general not measures concentrated on the graph of a measure-preserving function, and so there is a natural generalization of the Birkhoff-von Neumann theorem that fails

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