Flow Polytopes Research Articles

A Subdivision Algebra for a Product of Two Simplices via Flow Polytopes

A Subdivision Algebra for a Product of Two Simplices via Flow Polytopes

Triangulations of flow polytopes, ample framings, and gentle algebras

Triangulations of flow polytopes, ample framings, and gentle algebras

Efficient Learning in Polyhedral Games via Best-Response Oracles

We study online learning and equilibrium computation in games with polyhedral decision sets, a property shared by normal-form games (NFGs) and extensive-form games (EFGs), when the learning agent is restricted to utilizing a best-response oracle. We show how to achieve constant regret in zero-sum games and O(T^0.25) regret in general-sum games while using only O(log t) best-response queries at a given iteration t, thus improving over the best prior result, which required O(T) queries per iteration. Moreover, our framework yields the first last-iterate convergence guarantees for self-play with best-response oracles in zero-sum games. This convergence occurs at a linear rate, though with a condition-number dependence. We go on to show a O(T^(-0.5)) best-iterate convergence rate without such a dependence. Our results build on linear-rate convergence results for variants of the Frank-Wolfe (FW) algorithm for strongly convex and smooth minimization problems over polyhedral domains. These FW results depend on a condition number of the polytope, known as facial distance. In order to enable application to settings such as EFGs, we show two broad new results: 1) the facial distance for polytopes in standard form is at least γ/k where γ is the minimum value of a nonzero coordinate of a vertex of the polytope and k≤n is the number of tight inequality constraints in the optimal face, and 2) the facial distance for polytopes of the form Ax=b, Cx≤d, x≥0 where x∈R^n, C≥0 is a nonzero integral matrix, and d≥0, is at least 1/(c√n), where c is the infinity norm of C. This yields the first such results for several problems such as sequence-form polytopes, flow polytopes, and matching polytopes.

Column-Convex Matrices, G-Cyclic Orders, and Flow Polytopes

Column-Convex Matrices, G-Cyclic Orders, and Flow Polytopes

Adjacencies on random ordering polytopes and flow polytopes

The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak(1960). Fishburn(1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). To derive a more enlightening proof of Falmagne Theorem, Fiorini(2004) assimilates the MCP with the flow polytope of some acyclic network. However, apart from a recognition of the facets by Suck(2002), the geometric structure of the MCP was apparently not much investigated. We characterize the adjacency of vertices and the adjacency of facets. Our characterization of the edges of the MCP helps understand recent findings in economics papers such as Chang, Narita and Saito(2022) and Turansick(2022). Moreover, our results on adjacencies also hold for the flow polytope of any acyclic network. In particular, they apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of other models, see for instance Fishburn and Falmagne, 1989, and Marley and Regenwetter, 2017).

Flow-oriented perturbation theory

We introduce a new diagrammatic approach to perturbative quantum field theory, which we call flow-oriented perturbation theory (FOPT). Within it, Feynman graphs are replaced by strongly connected directed graphs (digraphs). FOPT is a coordinate space analogue of time-ordered perturbation theory and loop-tree duality, but it has the advantage of having combinatorial and canonical Feynman rules, combined with a simplified iε dependence of the resulting integrals. Moreover, we introduce a novel digraph-based representation for the S-matrix. The associated integrals involve the Fourier transform of the flow polytope. Due to this polytope’s properties, our S-matrix representation exhibits manifest infrared singularity factorization on a per-diagram level. Our findings reveal an interesting interplay between spurious singularities and Fourier transforms of polytopes.

Triangulations, Order Polytopes, and Generalized Snake Posets

This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in related order polytopes and then conclude that all of their triangulations are unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of upper order ideals comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.Mathematics Subject Classifications: 52B20, 52B05, 52B12, 06A07Keywords: Order polytopes, triangulations, flow polytopes, circuits

Schubert polynomials as projections of Minkowski sums of Gelfand-Tsetlin polytopes

Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of \(\mathfrak{gl}_n(\mathbb{C})\). The integer point transform of the Gelfand-Tsetlin polytope \(\mathrm{GT}(\lambda)\) projects to the Schur function \(s_{\lambda}\). Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials \(\mathfrak{S}_{w}\) corresponding to Grassmannian permutations. For any permutation \(w \in S_n\) with column-convex Rothe diagram, we construct a polytope \(\mathcal{P}_{w}\) whose integer point transform projects to the Schubert polynomial \(\mathfrak{S}_{w}\). Such a construction has been sought after at least since the construction of twisted cubes by Grossberg and Karshon in 1994, whose integer point transforms project to Schubert polynomials \(\mathfrak{S}_{w}\) for all \(w \in S_n\). However, twisted cubes are not honest polytopes; rather one can think of them as signed polytopal complexes. Our polytope \(\mathcal{P}_{w}\) is a convex polytope, namely it is a Minkowski sum of Gelfand-Tsetlin polytopes of varying sizes. When the permutation \(w\) is Grassmannian, the Gelfand-Tsetlin polytope is recovered. We conclude by showing that the Gelfand-Tsetlin polytope is a flow polytope.Mathematics Subject Classifications: 05E05Keywords: Schubert polynomials, Gelfand-Tsetlin polytopes, flow polytopes

Low dimensional flow polytopes and their toric ideals

The toric ideal of a d-dimensional flow polytope has an initial ideal generated by square-free monomials of degree at most d. The toric ideal of a flow polytope of dimension at most four has an initial ideal generated by square-free monomials of degree at most two, with the only exception of the four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal generated by a square-free cubic monomial. The proof is based on a method to classify certain compressed flow polytopes, and a construction of a quadratic pulling triangulation of them. Along the way compressed flow polytopes are classified up to dimension four, and their Ehrhart polynomials are computed.

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which M\'esz\'aros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize M\'esz\'aros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-M\'esz\'aros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity. Lastly, we study a symmetry of the Morris identity bijectively using the Danilov-Karzanov-Koshevoy triangulation of flow polytopes and a bijection of M\'esz\'aros-Morales-Striker.

Lorentzian polynomials from polytope projections

Lorentzian polynomials, recently introduced by Brändén and Huh, generalize the notion of log-concavity of sequences to homogeneous polynomials whose supports are integer points of generalized permutahedra. Brändén and Huh show that normalizations of integer point transforms of generalized permutahedra are Lorentzian. Moreover, normalizations of certain projections of integer point transforms of generalized permutahedra with zero-one vertices are also Lorentzian. Taking this polytopal perspective further, we show that normalizations of certain projections of integer point transforms of flow polytopes are Lorentzian.

Counting Integer Points of Flow Polytopes

The Baldoni–Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function, fundamental in representation theory. On the other hand, the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these formulas, the second author and Morales gave a fully geometric proof for the volume formula and a partial generating function proof for the Ehrhart polynomial formula. The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula for flow polytopes.

Feasible Bases for a Polytope Related to the Hamilton Cycle Problem

We study a certain polytope depending on a graph G and a parameter β ∈ (0,1) that arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Literature suggests a conjecture a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress toward a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter β when the parameter is close to one, (2) the polytope can be interpreted as a generalized network flow polytope, and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.

On Volume Functions of Special Flow Polytopes Associated to the Root System of Type $A$

In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type $A$.

Volumes of Flow Polytopes Related to Caracol Graphs

Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.

From generalized permutahedra to Grothendieck polynomials via flow polytopes

We study a family of dissections of flow polytopes arising from the subdivision algebra. To each dissection of a flow polytope, we associate a polynomial, called the left-degree polynomial, which we show is invariant of the dissection considered (proven independently by Grinberg). We prove that left-degree polynomials encode integer points of generalized permutahedra. Using that certain left-degree polynomials are related to Grothendieck polynomials, we resolve special cases of conjectures by Monical, Tokcan, and Yong regarding the saturated Newton polytope property of Grothendieck polynomials.

Relating single-scenario facets to the convex hull of the extensive form of a stochastic single-node flow polytope

We study when a facet-defining inequality for a deterministic, single-scenario subproblem is also facet-defining for the extensive form of a two-stage stochastic mixed-integer linear program (SMIP). To answer this question, we introduce a novel stochastic variant of the well-known single-node flow (SNF) polytope, and present necessary and sufficient conditions for single-scenario facet-defining inequalities to be facet-defining for the extensive form. We further demonstrate that our stochastic SNF polytope is a relaxation of a broad subclass of SMIPs, illustrating its generality.

New valid inequalities for the fixed-charge and single-node flow polytopes

The most effective software packages for solving mixed 0–1linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge and single-node flow polytopes. The resulting inequalities are very different from the previously known inequalities (such as flow cover and flow pack inequalities), and define facets under certain conditions.

A combinatorial model for computing volumes of flow polytopes

We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne’s generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph. As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan numbers and the number of parking functions, we prove the log-concavity of rows of this triangle along with other sequences derived from volume computations, and we introduce a new Ehrhart-like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider.

Volumes and Ehrhart polynomials of flow polytopes

The Lidskii formula for the type $$A_n$$ root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant partition functions. For every integer polytope the volume is the leading coefficient of the Ehrhart polynomial. The beauty of the Lidskii formula is the revelation that for these polytopes their Ehrhart polynomial function can be deduced from their volume function! Baldoni and Vergne generalized Lidskii’s result for flow polytopes of arbitrary graphs G. While their formulas are combinatorial in nature, their proofs are based on residue computations. In this paper we construct canonical polytopal subdivisions of flow polytopes which we use to prove the Baldoni–Vergne–Lidskii formulas. In contrast with the original computational proof of these formulas, our proof reveal their geometry and combinatorics. We conclude by exhibiting enumerative properties of the Lidskii formulas via our canonical polytopal subdivisions.