Base Polytope Research Articles

Valuative invariants for large classes of matroids

AbstractWe study an operation in matroid theory that allows one to transition a given matroid into another with more bases via relaxing a stressed subset. This framework provides a new combinatorial characterization of the class of (elementary) split matroids. Moreover, it permits to describe an explicit matroid subdivision of a hypersimplex, which, in turn, can be used to write down concrete formulas for the evaluations of any valuative invariant on these matroids. This shows that evaluations on these matroids depend solely on the behavior of the invariant on a tractable subclass of Schubert matroids. We address systematically the consequences of our approach for several invariants. They include the volume and Ehrhart polynomial of base polytopes, the Tutte polynomial, Kazhdan–Lusztig polynomials, the Whitney numbers of the first and second kinds, spectrum polynomials and a generalization of these by Denham, chain polynomials and Speyer's ‐polynomials, as well as Chow rings of matroids and their Hilbert–Poincaré series. The flexibility of this setting allows us to give a unified explanation for several recent results regarding the listed invariants; furthermore, we emphasize it as a powerful computational tool to produce explicit data and concrete examples.

Ehrhart theory of paving and panhandle matroids

Abstract We show that the base polytope P M of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of P M , starting with Katzman’s formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni’s work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.

Gorenstein Matroids

Abstract Matroid base polytope and independence polytope are Cohen–Macaulay. There are plenty of examples when they are also Gorenstein. We provide a full classification of matroids whose independence polytope or base polytope is Gorenstein. This answers to a question raised by Herzog and Hibi.

Slack matrices, [formula omitted]-products, and 2-level polytopes

In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra.The 1-product of matrices S1∈Rm1×n1,S2∈Rm2×n2 is a matrix in R(m1+m2)×(n1n2) whose columns are the concatenation of each column of S1 with each column of S2. The k-product generalizes the 1-product, by taking as input two matrices S1,S2 together with k−1 special rows of each of those matrices, and outputting a certain composition of S1,S2. Our first result is a polynomial time algorithm for the following problem: given a matrix S, is S a k-product of some matrices, up to permutation of rows and columns? Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information from information theory.Our study is motivated by a close link between the 1-product of matrices and the Cartesian product of polytopes, and more generally between the k-product of matrices and the glued product of polytopes. These connections rely on the concept of a slack matrix, which gives an algebraic representation of classes of affinely equivalent polytopes. The slack matrix recognition problem is the problem of determining whether a given matrix is a slack matrix. This is an intriguing problem whose complexity is unknown. Our algorithm reduces the problem to instances which cannot be expressed as k-products of smaller matrices.In the second part of the paper, we give a combinatorial interpretation of k-products for two well-known classes of polytopes: 2-level matroid base polytopes and stable set polytopes of perfect graphs. We also show that the slack matrix recognition problem is polynomial-time solvable for such polytopes. Those two classes are special cases of 2-level polytopes, for which we conjecture that the slack matrix recognition problem is polynomial-time solvable.

Extended formulations for matroid polytopes through randomized protocols

The hitting number of a polytope P is the smallest size of a subset of vertices of P such that every facet of P has a vertex in the subset. We show that, if P is the base polytope of any matroid, then P admits an extended formulation of linear size on the hitting number of P. Our results generalize those of the spanning tree polytope given by Martin and Wong, and extend to polymatroids.

The Tutte polynomial via lattice point counting

We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of alternating sign with a combinatorial interpretation closely tied to the Dawson partition. Our definition extends in a straightforward way to polymatroids, and in this setting our polynomial has Kálmán's internal and external activity polynomials as its univariate specialisations.

Projection-Free Bandit Optimization with Privacy Guarantees

We design differentially private algorithms for the bandit convex optimization problem in the projection-free setting. This setting is important whenever the decision set has a complex geometry, and access to it is done efficiently only through a linear optimization oracle, hence Euclidean projections are unavailable (e.g. matroid polytope, submodular base polytope). This is the first differentially-private algorithm for projection-free bandit optimization, and in fact our bound matches the best known non-private projection-free algorithm and the best known private algorithm, even for the weaker setting when projections are available.

Flip Distances Between Graph Orientations

Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called alpha-orientations of a graph G, in which every vertex v has a specified outdegree alpha (v), and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two alpha-orientations of a planar graph G is at most two is NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569–576, 2019).

A lattice point counting generalisation of the Tutte polynomial

The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.

Facial structures of lattice path matroid polytopes

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.

Weighted quasisymmetric enumerator for generalized permutohedra

We introduce a weighted quasisymmetric enumerator function associated with generalized permutohedra. It refines the Billera, Jia and Reiner quasisymmetric function which also includes the Stanley chromatic symmetric function. Besides that, it carries information of face numbers of generalized permutohedra. We consider more systematically the cases of nestohedra and matroid base polytopes.

Polyhedra and parameter spaces for matroids over valuation rings

In this paper we address two of the major foundational questions in the theory of matroids over rings. First, we provide a cryptomorphic axiomatisation, by introducing an analogue of the base polytope for matroids. Second, we describe a parameter space for matroids over a valuation ring, which turns out to be a tropical version of the Bott–Samelson varieties for the full flag variety. Thus a matroid over a valuation ring is a sequence of flags of tropical linear spaces a.k.a. valuated matroids.

Classifying Fano complexity-one T-varieties via divisorial polytopes

The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and S\"u{\ss} to a correspondence between Gorenstein Fano complexity-one $T$-varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger's classification of Gorenstein del Pezzo $\mathbb{K}^*$-surfaces. Furthermore, we show that any three-dimensional Fano divisorial polytope is equivalent to one involving only eight functions.

Sensitivity Analysis for Convex Separable Optimization Over Integral Polymatroids

We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each eleme...

On 2-Level Polytopes Arising in Combinatorial Settings

2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that v(P)*f(P) is upper bounded by d*2^(d+1), for a large collection of families of such polytopes P. Here v(P) (resp. f(P)) is the number of vertices (resp. facets) of P, and d is its dimension. Whether this holds for all 2-level polytopes was asked in [Bohn et al., ESA 2015], and experimental results from [Fiorini et al., ISCO 2016] showed it true up to dimension 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.

Identification of unidentified equality constraints for integer programming problems

Characterising the smallest dimension polytope containing all integer solutions of an integer programming problem can be a very challenging task. Frequently, this task is facilitated by identifying linear equality constraints that all integer solutions must satisfy. Typically, some of these constraints are readily available but others need to be discovered by more technical means. This paper develops a method to assist modellers to obtain such equality constraints. Note that the set of new equality constraints is not unique, and the proposed method generates a set of these new equality constraints for a sufficiently large dimension of the underlying problem. These generated constraints may be of a form that is easily extended for general case of the underlying problem, or they may be in a more complicated form where a generalisable pattern is difficult to identify. For the latter case, a new mixed-integer program is developed to detect a pattern-recognisable constraints. Furthermore, this mixed-integer program allows modellers to check if there is a new constraint satisfying specific criteria, such as only permitting coefficients to be 1, 0, and −1, or placing a limit on the number of non-zero coefficients. In order to illustrate the proposed method, a set of new equality constraints to supplement a previously published “Base polytope” are derived. Subsequently, exploiting these results, some techniques are proposed to tighten integer programming problems. Finally, relaxations of widely used TSP formulations are compared against one another and strengthened with help of the newly discovered equality constraints.

Theta rank, levelness, and matroid minors

The Theta rank of a finite point configuration V is the maximal degree necessary for a sum-of-squares representation of a non-negative affine function on V. This is an important invariant for polynomial optimization that is in general hard to determine. We study the Theta rank and levelness, a related discrete-geometric invariant, for matroid base configurations. It is shown that the class of matroids with bounded Theta rank or levelness is closed under taking minors. This allows for a characterization of matroids with bounded Theta rank or levelness in terms of forbidden minors. We give the complete (finite) list of excluded minors for Theta-1 matroids which generalizes the well-known series-parallel graphs. Moreover, the class of Theta-1 matroids can be characterized in terms of the degree of generation of the vanishing ideal and in terms of the psd rank for the associated matroid base polytope. We further give a finite list of excluded minors for k-level graphs and matroids and we investigate the graphs of Theta rank 2.

Many 2-Level Polytopes from Matroids

The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-Level matroids generalize series-parallel graphs, which have been already successfully analyzed from the enumerative perspective. We bring to light some structural properties of 2-level matroids and exploit them for enumerative purposes. Moreover, the counting results are used to show that the number of combinatorially non-equivalent $$(n-1)$$(n-1)-dimensional 2-level polytopes is bounded from below by $$c \cdot n^{-5/2} \cdot \rho ^{-n}$$c·n-5/2·?-n, where $$c\approx 0.03791727 $$c?0.03791727 and $$\rho ^{-1} \approx 4.88052854$$?-1?4.88052854.

Extended formulations for sparsity matroids

We show the existence of a polynomial-size extended formulation for the base polytope of a $$(k,\ell )$$(k,l)-sparsity matroid. For an undirected graph $$G=(V,E)$$G=(V,E), the size of the formulation is $$O(|V|\cdot |E|)$$O(|V|·|E|) when $$k \ge \ell $$kźl and $$O(|V|^2 |E|)$$O(|V|2|E|) when $$k \le \ell $$k≤l. To this end, we employ the technique developed by Faenza et al. recently that uses a randomized communication protocol.

Matroid base polytope decomposition II: Sequences of hyperplane splits

This is a continuation of an early paper Chatelain et al. (2011) [3] about matroid base polytope decomposition. We will present sufficient conditions on a matroid M so its base polytope P(M) has a sequence of hyperplane splits. These yield to decompositions of P(M) with two or more pieces for infinitely many matroids M. We also present necessary conditions on the Euclidean representation of rank three matroids M for the existence of decompositions of P(M) into 2 or 3 pieces. Finally, we prove that P(M1⊕M2) has a sequence of hyperplane splits if either P(M1) or P(M2) also has a sequence of hyperplane splits.