Permutahedron Research Articles

From Hopf--Lax Formula to Optimal Weak Transfer Plan

We study the properties of the Hopf--Lax formula restricted to convex functions and provide a characterization of the optimal transfer plan for weak transport problems on the real line. On $n$ dimensional real space, we also provide a sufficient condition on the potential function such that the optimal plan of the classical Monge--Kantorovich problem does not depend on the cost function. As a byproduct, we establish a link between the combinatorial object (the permutation polytope) and the Hamilton--Jacobi equation.

An exact completely positive programming formulation for the discrete ordered median problem: an extended version

This paper presents a first continuous, linear, conic formulation for the discrete ordered median problem (DOMP). Starting from a binary, quadratic formulation in the original space of location and allocation variables that are common in location analysis (L.A.), we prove that there exists a transformation of that formulation, using the same space of variables, that allows us to cast DOMP as a continuous, linear programming problem in the space of completely positive matrices. This is the first positive result that states equivalence between the family of continuous, convex problems and some hard combinatorial problems in L.A. The result is of theoretical interest because it allows us to share the tools from continuous optimization to shed new light into the difficult combinatorial structure of the class of ordered median problems that combines elements of the p-median, quadratic assignment and permutation polytopes.

Equivalent of permutation polytopes with applications

An open conjecture that relates to the equivalents between the permutation polytopes associated to their groups is proved and also the proof for two permutation groups which are effectively equivalent implies the permutation polytopes associated to these groups is combinatorial equivalent is given.

Affine symmetries of orbit polytopes

An orbit polytope is the convex hull of an orbit under a finite group G⩽GL(d,R). We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups. We prove that the symmetry group of a generic orbit polytope is again G if G is itself the affine symmetry group of some orbit polytope, or if G is absolutely irreducible. On the other hand, we describe some general cases where the affine symmetry group grows.We apply our theory to representation polytopes (the convex hull of a finite matrix group) and show that their affine symmetries can be computed effectively from a certain character. We use this to construct counterexamples to a conjecture of Baumeister et al. on permutation polytopes (Baumeister et al., 2009 [4, Conjecture 5.4]).

On permutation polytopes: notions of equivalence

By assigning a permutation polytope to a group, we produce new interesting polytopes. For the effective use of this construction method, it is desirable to understand which groups are leading to affine equivalent polytopes. Therefore, the notion of effective equivalence has been introduced (Baumeister et al. in Adv Math 222(2):431---452, 2009). In this note, we clarify the notion of effective equivalence and characterize it geometrically. Moreover, we present examples showing that the effective equivalent permutation groups do not correspond to affinely equivalent polytopes. We also apply the characterization to the examples. In our approach we provide a framework for the use of representation theoretic methods in the study of permutation polytopes.

Permutation Polytopes of Cyclic Groups

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets. Nous ètudions les propriètès combinatoires et gèomètriques des polytopes de permutations pour des groupes cycliques. C'est à dire, donnè un groupe cyclique de matrices de permutations, nous considèrons son enveloppe convexe. Si le gènèrateur du groupe possède un ou deux orbites il y a une dèscription simple du polytope. Par contre, le cas de trois (ou plus) orbites est beaucoup plus compliquè. Pour une classe ample d'examples nous construisons un nombre exponentiel de faces de co-dimension un.

Quantum sign permutation polytopes

Convex polytopes are convex hulls of point sets in the n-dimensional space that generalize two-dimensional convex polygons and three-dimensional convex polyhedra. We concentrate on the class of n-dimensional polytopes in called sign permutation polytopes. We characterize sign permutation polytopes before relating their construction to constructions over the space of quantum density matrices. Finally, we consider the problem of state identification and show how sign permutation polytopes may be useful in addressing issues of robustness.

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices. We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ⩽4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online. We conclude with several questions suggested by a general finiteness result.

Equivalence of permutation polytopes corresponding to strictly supermodular functions

This paper demonstrates a strong equivalence of all permutation polytopes corresponding to strictly supermodular functions.

Multidimensional Unfolding by Nonmetric Multidimensional Scaling of Spearman Distances in the Extended Permutation Polytope

A multidimensional unfolding technique that is not prone to degenerate solutions and is based on multidimensional scaling of a complete data matrix is proposed: distance information about the unfolding data and about the distances both among judges and among objects is included in the complete matrix. The latter information is derived from the permutation polytope supplemented with the objects, called the preference sphere. In this sphere, distances are measured that are closely related to Spearman's rank correlation and that are comparable among each other so that an unconditional approach is reasonable. In two simulation studies, it is shown that the proposed technique leads to acceptable recovery of given preference structures. A major practical advantage of this unfolding technique is its relatively easy implementation in existing software for multidimensional scaling.

For All Possible Distances Look to the Permutohedron

For All Possible Distances Look to the Permutohedron

A criterion for the adjacency of vertices of polytopes generated by subsets of symmetric groups

We propose a characterization of the adjacency of vertices in the class of permutation polytopes generated by arbitrary subsets of symmetric groups. In particular, this class contains polytopes for the well-known classical problems, such as the assignment problem, 2-and 3-combinations, the traveling salesman problem and their various modifications. Up to now, the problem of vertex adjacency has been studied for a single polytope only. In the present paper, we obtain, for general permutation polytopes, necessary and sufficient conditions that guarantee that two given vertices are adjacent (or not) to each other. The conditions are formulated in terms of permutations and of the solvability of certain special systems of linear equations. The presently known adjacency criteria for vertices of polytopes for the assignment problem are simple corollaries of our conditions. The latter allow us to develop a general algorithmic scheme for recognizing vertex adjacency of a general permutation polytope and estimate its complexity.

Permutation polytopes and indecomposable elements in permutation groups

Each group G of n × n permutation matrices has a corresponding permutation polytope, P ( G ) : = conv ( G ) ⊂ R n × n . We relate the structure of P ( G ) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min { 2 t , ⌊ n / 2 ⌋ } is a sharp upper bound on the diameter of the graph of P ( G ) . We also show that P ( G ) achieves its maximal dimension of ( n − 1 ) 2 precisely when G is 2-transitive. We then extend the results of Pak [I. Pak, Four questions on Birkhoff polytope, Ann. Comb. 4 (1) (2000) 83–90] on mixing times for a random walk on P ( G ) . Our work depends on a new result for permutation groups involving writing permutations as products of indecomposable permutations.

Supermodularity in Mean-Partition Problems*

Supermodularity of the ? function which defines a permutation polytope has proved to be crucial for the polytope to have some nice fundamental properties. Supermodularity has been established for the ? function for the sum-partition problem under various models. On the other hand, supermodularity has not been established for the mean-partition problem even for the most basic labeled single-shape model. In this paper, we fill this gap and also settle for all other models except one. We further extend our results to other types of supermodularity.

Geometric representation of association between categories

Categories can be counted, rated, or ranked, but they cannot be measured. Likewise, persons or individuals can be counted, rated, or ranked, but they cannot be measured either. Nevertheless, psychology has realized early on that it can take an indirect road to measurement: What can be measured is the strength of association between categories in samples or populations, and what can be quantitatively compared are counts, ratings, or rankings made under different circumstances, or originating from different persons. The strong demand for quantitative analysis of categorical data has thus created a variety of statistical methods, with substantial contributions from psychometrics and sociometrics. What is the common basis of these methods dealing with categories? The basic element they share is that the sample space has a special geometry, in which categories (or persons) are point masses forming a simplex, while distributions of counts or profiles of ratings are centers of gravity, which are also point masses. Rankings form a discrete subset in the interior of the simplex, known as the permutation polytope, and paired comparisons form another subset on the edges of the simplex. Distances between point masses form the basic tool of analysis. The paper gives some history of major concepts, which naturally leads to a new concept: the shadow point. It is then shown how loglinear models, Luce and Rasch models, unfolding models, correspondence analysis and homogeneity analysis, forced classification and classification trees, as well as other models and methods, fit into this particular geometrical framework.

On the even permutation polytope

We consider the convex hull of the even permutations on a set of n elements. We define a class of valid inequalities and prove that they induce a large class of distinct facets of the polytope. Using the inequalities, we characterize the polytope for n=4, and we confirm a conjecture of Brualdi and Liu that, unlike the convex hull of all permutations, this polytope cannot be described as the solution set of polynomially many linear inequalities. We also discuss the difficulty of determining whether a given point is in the polytope.

On the mean radius of permutation polytopes

After calculating the mean value of the support function for the simplex of the coordinate vectors over the unit sphere, we find the mean for a number of polytopes, including the Birkhoff and Asymmetric Traveling Salesman polytopes, and discuss what the latter means for the corresponding Asymmetric Traveling Salesman Problem. We also discuss how to apply these results to efficiently count the vertices of certain other permutation polytopes.

Permutation polytopes corresponding to strongly supermodular functions

Throughout, let p be a positive integer and let Σ be the set of permutations over {1,…, p}. A real-valued function λ over subsets of {1,…, p}, with λ(∅)=0, defines a mapping of Σ into R p where σ∈ Σ is mapped into the vector λ σ whose kth coordinate ( λ σ ) k is the augmented λ-value obtained from adding k to the coordinates that precede it, according to the ranking induced by σ. The permutation polytope corresponding to λ is then the convex hull of the vectors corresponding to all permutations. We introduce a new class of strongly supermodular functions and for such functions we derive an isomorphic representation for the face-lattices of the corresponding permutation polytope.

New Facets of the Linear Ordering Polytope

The linear ordering problem has many applications and was studied by many authors (for a survey, see [P. Fishburn, SIAM J. Discrete Math., 4 (1990), pp. 478--488; M. Grotschel, M. Junger, and G. Reinelt, Math. Programming, 33 (1985), pp. 43--60; G. Reinelt, The Linear Ordering Problem: Algorithm and Applications, Heldermann Verlag, 1985]). One approach to solving this problem, the so-called cutting plane method [M. Grotschel, M. Junger, and G. Reinelt, Oper. Res., 32 (1984), pp. 1195--1220; V. A. Yemelichev, M. M. Kovalev, and M. K. Kravtsov, Polytopes, Graphs, Optimization, Cambridge University Press, 1984], derives facet-defining inequalities which are violated by current nonfeasible solution and adds them to the system of inequalities of current linear programming problems. We present a method (rotation method) for generating new facets of polyhedra by using previously known ones. The rotation method for the linear ordering polytope generalizes facets induced by subgraphs called $m$-fences, Mobius ladders, and Zm-facets introduced by Reinelt, (m,k)-fences introduced by Bolotashvili [G. C. Bolotashvili, A Class of Facets of the Permutation Polytope and a Method for Constructing Facets of the Permutation Polytope, preprint VINITI N3403-B87, Moscow, 1987 (in Russian)]; and t-reinforced m-fences introduced by Leung and Lee [J. Leung and J. Lee, Discrete Appl. Math., 50 (1984), pp. 185--200]. We introduce 10 collections of inequalities representing facets of the linear ordering polytope. Among them are three that coincide with earlier known ones: m-wheel-facets introduced by Reinelt, augmented m-fences introduced by McLennan [A. McLennan, in Preferences, Uncertainty and Optimality, West View Press, 1990, pp. 187--202]; and augmented t-reinforced m-fences introduced by Leung and Lee.

All 0–1 polytopes are traveling salesman polytopes

We study the facial structure of two important permutation polytopes in\(\mathbb{R}^{n^2 } \), theBirkhoff orassignment polytopeB n , defined as the convex hull of alln×n permutation matrices, and theasymmetric traveling salesman polytopeT n , defined as the convex hull of thosen×n permutation matrices corresponding ton-cycles. Using an isomorphism between the face lattice ofB n and the lattice of elementary bipartite graphs, we show, for example, that every pair of vertices ofB n is contained in a cubical face, showing faces ofB n to be fairly special 0–1 polytopes. On the other hand, we show thatevery 0–1d-polytope is affinely equivalent to a face ofT n , ford∼logn, by showing that every 0–1d-polytope is affinely equivalent to the asymmetric traveling salesman polytope of some directed graph withn nodes. The latter class of polytopes is shown to have maximum diameter [n/3].