In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan’s question, in this paper we define a pattern-avoiding Birkhoff polytope called a c -Birkhoff polytope for each Coxeter element c of the symmetric group. We then show that the c -Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the c -sorting word of the longest permutation. When c = s 1 s 2 ⋯ s n , this result recovers an affirmative answer to Davis and Sagan’s question. Another consequence of this result is that the normalized volume of the c -Birkhoff polytope is the number of the longest chains in the (type A) c -Cambrian lattice.

Ten Proofs of the Birkhoff-von Neumann Theorem

Abstract The Chan–Robbins–Yuen polytope ( $CRY_n$ ) of order n is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph $K_{n+1}$ with netflow $(1,0,0, \ldots , 0, -1)$ . The volume and lattice points of this polytope have been actively studied; however, its face structure has received less attention. We give generating functions and explicit formulas for computing the f -vector by using Hille’s (2003) result bijecting faces of a flow polytope to certain graphs, as well as Andresen–Kjeldsen’s (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes of the complete graph having arbitrary (non-negative) netflow vectors and recover the f -vector of the Tesler polytope of Mészáros–Morales–Rhoades (2017).

Abstract We compute the norm on $$l^p$$ l p of some special kind of matrices. The sharp computation (not just an estimate) of a matrix norm is a hard problem in general (see, e.g., [20]), but it turns out to be solvable for matrices having a suitable special form. This is analogous to what happens for some linear operators studied in analysis. Here we describe some families of matrices for which the solution can be found in a reasonably simple way. In particular, we focus on circulant matrices, on matrices having constant line sums, and on special kinds of block matrices. As a related result we can prove that any doubly stochastic matrix has p-norm equal to 1, which can be interpreted as an additional characterization of the Birkhoff polytope. This also leads to the sharp computation of the p-norm of special non-symmetric matrices where positive, zero or even negative entries appear.

We study faces of the signed Birkhoff polytopes, denoted by $\Omega_n^{\pm}$. We describe its nonempty faces, $1$-dimensional faces, $2$-dimensional faces, and facets. Moreover, we study the diameter and Hamiltonian connectivity of the graph of $\Omega_n^{\pm}$. In the end, we show that the reduced Gröbner basis of the toric ideal of the signed Birkhoff polytope $\Omega_n^{\pm}$ with respect to the graded reverse lexicographic order induced by rank orders has square-free initial monomials of degree $\leq n$.

Abstract A quantum measurement, often referred to as positive operator-valued measurement, is a set of positive operators P j = P j † ⩾ 0 summing to identity, ∑ j P j = 𝟙 . This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe transformations in the set of quantum measurements by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares). Analyzing their dynamical properties, we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in the set of quantum measurements.

On the geometry of the Birkhoff polytope I: the operator $$\ell ^p_n$$-norms

The k-matching polytope of a graph is the convex hull of all its matchings of a given size k when they are considered as indicator vectors. In this paper, we prove that the k-matching polytope of a bipartite graph is normal, that is, every integer point in its t-dilate is the sum of t integers points of the original polytope. This generalizes the known fact that Birkhoff polytopes are normal. As a preliminary result, we prove that for bipartite graphs the k-matching polytope is equal to the fractional k-matching polytope, having thus the H-representation of the polytope. This generalizes the Birkhoff-Von Neumann Theorem which establish that every doubly stochastic matrix can be written as a convex combination of permutation matrices.

On the geometry of the Birkhoff polytope II: the Schatten p-norms

Generalized unistochastic matrices

Abstract The geometry of the compact convex set of alln×nn\times ndoubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms fromℓnp{\ell }_{n}^{p}toℓnp{\ell }_{n}^{p}and the Schattenpp-norms, both for the range1≤p≤∞1\le p\le \infty, have only recently been studied in depth. In this article, we continue in this vein by determining the diameter of the Birkhoff polytope with respect to the metrics induced by the aforementioned matrix norms.

Several central problems in quantum information theory (such as measurement compatibility and quantum steering) can be rephrased as membership in the minimal matrix convex set corresponding to special polytopes (such as the hypercube or its dual). In this article, we generalize this idea and introduce the notion of polytope compatibility, by considering arbitrary polytopes. We find that semiclassical magic squares correspond to Birkhoff polytope compatibility. In general, we prove that polytope compatibility is in one-to-one correspondence with measurement compatibility, when the measurements have some elements in common and the post-processing of the joint measurement is restricted. Finally, we consider how much tuples of operators with appropriate joint numerical range have to be scaled in the worst case in order to become polytope compatible and give both analytical sufficient conditions and numerical ones based on linear programming.

We tackle the problem of efficiently approximating the volume of convex polytopes, when these are given in three different representations: H-polytopes, which have been studied extensively, V-polytopes, and zonotopes (Z-polytopes). We design a novel practical Multiphase Monte Carlo algorithm that leverages random walks based on billiard trajectories, as well as a new empirical convergence test and a simulated annealing schedule of adaptive convex bodies. After tuning several parameters of our proposed method, we present a detailed experimental evaluation of our tuned algorithm using a rich dataset containing Birkhoff polytopes and polytopes from structural biology. Our open-source implementation tackles problems that have been intractable so far, offering the first software to scale up in thousands of dimensions for H-polytopes and in the hundreds for V- and Z-polytopes on moderate hardware. Last, we illustrate our software in evaluating Z-polytope approximations.

The primal-dual hybrid gradient (PDHG) algorithm is popular in solving min-max problems which are being widely used in a variety of areas. To improve the applicability and efficiency of PDHG for different application scenarios, we focus on the preconditioned PDHG (PrePDHG) algorithm, which is a framework covering PDHG, alternating direction method of multipliers (ADMM), and other methods. We give the optimal convergence condition of PrePDHG in the sense that the key parameters in the condition can not be further improved, which fills the theoretical gap in the-state-of-art convergence results of PrePDHG, and obtain the ergodic and non-ergodic sublinear convergence rates of PrePDHG. The theoretical analysis is achieved by establishing the equivalence between PrePDHG and indefinite proximal ADMM. Besides, we discuss various choices of the proximal matrices in PrePDHG and derive some interesting results. For example, the convergence condition of diagonal PrePDHG is improved to be tight, the dual stepsize of the balanced augmented Lagrangian method can be enlarged to 4/3 from 1, and a balanced augmented Lagrangian method with symmetric Gauss-Seidel iterations is also explored. Numerical results on the matrix game, projection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction verify the effectiveness and superiority of PrePDHG.

Low dimensional flow polytopes and their toric ideals

Finding the dimension of a non-empty orthogonal array polytope

Scholars have recently become interested in the importance of the reflexive and Birkhoff polytopes in a variety of applications in our daily lives. Unanswered queries and educated guesses abound in reflexive polytopes. We use the free sum and product for reflexives polytopes, as well as the product for two Birkhoff polytopes, and the proven theorem to get specific unimodality results. The computations are acquired via algorithms.

The Birkhoff polytope Bd consisting of all bistochastic matrices of order d assists researchers from many areas, including combinatorics, statistical physics, and quantum information. Its subset Ud of unistochastic matrices, determined by squared moduli of unitary matrices, is of particular importance for quantum theory as classical dynamical systems described by unistochastic transition matrices can be quantized. In order to investigate the problem of unistochasticity, we introduce the set Ld of bracelet matrices that forms a subset of Bd, but a superset of Ud. We prove that for every dimension d, this set contains the set of factorizable bistochastic matrices Fd and is closed under matrix multiplication by elements of Fd. Moreover, we prove that both Ld and Fd are star-shaped with respect to the flat matrix. We also analyze the set of d × d unistochastic matrices arising from circulant unitary matrices and show that their spectra lie inside d-hypocycloids on the complex plane. Finally, applying our results to small dimensions, we fully characterize the set of circulant unistochastic matrices of order d ≤ 4 and prove that such matrices form a monoid for d = 3.

We study discrete-time quantum walks on generalized Birkhoff polytope graphs (GBPGs), which arise in the solution-set to certain transportation linear programming problems (TLPs). It is known that quantum walks mix at most quadratically faster than random walks on cycles, two-dimensional lattices, hypercubes, and bounded-degree graphs. In contrast, our numerical results show that it is possible to achieve a greater than quadratic quantum speedup for the mixing time on a subclass of GBPG (TLP with two consumers and m suppliers). We analyze two types of initial states. If the walker starts on a single node, the quantum mixing time does not depend on m, even though the graph diameter increases with it. To the best of our knowledge, this is the first example of its kind. If the walker is initially spread over a maximal clique, the quantum mixing time is , where ϵ is the threshold used in the mixing times. This result is better than the classical mixing time, which is .

Complexity of linear minimization and projection on some sets