The main theme of this dissertation is the study of the lattice points in a rational convex polyhedron and their encoding in terms of Barvinok's short rational functions. The first part of this thesis looks into theoretical applications of these rational functions to Optimization, Statistics, and Computational Algebra. The main theorem on Chapter 2 concerns the computation of the \emph{toric ideal} $I_A$ of an integral $n \times d$ matrix $A$. We encode the binomials belonging to the toric ideal $I_A$ associated with $A$ using Barvinok's rational functions. If we fix $d$ and $n$, this representation allows us to compute a universal Gröbner basis and the reduced Gröbner basis of the ideal $I_A$, with respect to any term order, in polynomial time. We derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions of the division algorithm. Chapter 3 presents three ways to use Barvinok's rational functions to solve Integer Programs.
The second part of the thesis is experimental and consists mainly of the software package {\tt LattE}, the first implementation of Barvinok's algorithm. We report on experiments with families of well-known rational polytopes: multiway contingency tables, knapsack type problems, and rational polygons. We also developed a new algorithm, {\em the homogenized Barvinok's algorithm} to compute the generating function for a rational polytope. We showed that it runs in polynomial time in fixed dimension. With the homogenized Barvinok's algorithm, we obtained new combinatorial formulas: the generating function for the number of $5\times 5$ magic squares and the generating function for the number of $3\times 3 \times 3 \times 3$ magic cubes as rational functions.
We extend to Barvinok's valuations the Euler-Maclaurin expansion formula which we obtained previously for the sum of values of a polynomial over the integral points of a rational polytope. This leads to an improvement of Barvinok's polynomial type algorithm for computing the highest coefficients of the corresponding Ehrhart quasi-polynomial.
Computations with Barvinok's short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion--exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion--exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using half-open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric…
We introduce variants of Barvinok's algorithm for counting lattice points in polyhedra. The new algorithms are based on irrational signed decomposition in the primal space and the construction of rational generating functions for cones with low index. We give computational results that show that the new algorithms are faster than the existing algorithms by a large factor.
A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which this retraction is a rational function. This is a contribution via real algebraic geometry which rests on results due to Huh and Kapranov on Horn uniformization. We present an algorithm for constructing models with rational MLE, and we demonstrate it on a range of instances. Our focus lies on models familiar to statisticians, like Bayesian networks, decomposable graphical models, and staged trees.
In this article we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by applying Garcia-Puente and Sottile's theorem that establishes a correspondence between polytopes with rational linear precision and log-linear models with rational MLE. Throughout this article we also study the interplay…
We classify the two-way independence quasi-independence models (or independence models with structural zeros) that have rational maximum likelihood estimators, or MLEs. We give a necessary and sufficient condition on the bipartite graph associated to the model for the MLE to be rational. In this case, we give an explicit formula for the MLE in terms of combinatorial features of this graph. We also use the Horn uniformization to show that for general log-linear models $\mathcal{M}$ with rational MLE, any model obtained by restricting to a face of the cone of sufficient statistics of $\mathcal{M…
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the u…
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an `algebraic exponential family'. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and ill…
We consider the problem of computing the joint distribution of order statistics of stochastically independent random variables in one- and two-group models. While recursive formulas for evaluating the joint cumulative distribution function of such order statistics exist in the literature for a longer time, their numerical implementation remains a challenging task. We tackle this task by presenting novel generalizations of known recursions which we utilize to obtain exact results (calculated in rational arithmetic) as well as faithfully rounded results. Finally, some applications in stepwise mu…
We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where s is an arbitrary real vertex and c is a rational polyhedral cone. For a given rational subspace L, we integrate a given polynomial function h over all lattice slices of the affine cone s + c parallel to the subspace L and sum up the integrals. We study these …
We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two representations. Examples of such counting functions include Ehrhart quasi-polynomials, vector partition functions, integer points in parametric polytopes, and projections of the integer points in parametric polytopes. For this last example, th…
We explore the computational complexity of computing pure Nash equilibria for a new class of strategic games called integer programming games with difference of piecewise linear convex payoffs. Integer programming games are games where players' action sets are integer points inside of polytopes. Using recent results from the study of short rational generating functions for encoding sets of integer points pioneered by Alexander Barvinok, we present efficient algorithms for enumerating all pure Nash equilibria, and other computations of interest, such as the pure price of anarchy, and pure threa…
We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed, this representation allows us to compute the Graver basis and the reduced Gröbner basis of the ideal $I_A$, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computation which replaces in this new encoding the usual reductions typical of the division algorithm. We describe o…
We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok's algorithm. We report polyhedral and computational results for two special cases: counting contingency tables and Kostant's partition function.
Solutions to a linear Diophantine system, or lattice points in a rational convex polytope, are important concepts in algebraic combinatorics and computational geometry. The enumeration problem is fundamental and has been well studied, because it has many applications in various fields of mathematics. In algebraic combinatorics, MacMahon's partition analysis has become a general approach for linear Diophantine system related problems. Many algorithms have been developed, but "bottlenecks" always arise when dealing with complex problems. While in computational geometry, Barvinok's important resu…
We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the res…
A Rational Analysis of the Speech-to-Song Illusion
Princeton University（普林斯顿大学）；Max Planck Institute for Empirical Aesthetics（马克斯·普朗克经验美学研究所）；Max Planck Institute for Human Cognitive and Brain Sciences（马克斯·普朗克人类认知与脑科学研究所）
Let $P(b)\subset R^d$ be a semi-rational parametric polytope, where $b=(b_j)\in R^N$ is a real multi-parameter. We study intermediate sums of polynomial functions $h(x)$ on $P(b)$, $$ S^L (P(b),h)=\sum_{y}\int_{P(b)\cap (y+L)} h(x) \mathrm dx, $$ where we integrate over the intersections of $P(b)$ with the subspaces parallel to a fixed rational subspace $L$ through all lattice points, and sum the integrals. The purely discrete sum is of course a particular case ($L=0$), so $S^0(P(b), 1)$ counts the integer points in the parametric polytopes. The chambers are the open conical subsets of $R^N$ s…
College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；Key Laboratory of Nonlinear Analysis and its Applications (Chongqing University), Ministry of Education（非线性分析及其应用教育部重点实验室（重庆大学））；Center for Combinatorics, LPMC, Nankai University（南开大学组合数学中心、理论力学与系统科学研究所）
For a given sequence $\mathbfα = [α_1,α_2,\dots,α_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\mathbfα)(t)$ that counts the nonnegative integer solutions of the equation $α_1x_1+α_2 x_2+\cdots+α_{N} x_{N}+α_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\mathbfα)(t)$ is a quasi-polynomial function in the variable $t$ of degree $N$. In combinatorial number theory this function is known as Sylvester's denumerant. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial …
We define two refinements of the skew length statistic on simultaneous core partitions. The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core. The second one is equivalent to a generalisation of Shi tableaux to the rational level of Catalan combinatorics. These rational Shi tableaux encode dominant $p$-stable elements in the affine symmetric group. We prove that the rational Shi tableau is injective, that is, each dominant $p$-stable affine permutation is determined uniquely by its Shi …
We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area,skip,dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q,t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area,skips,dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q,t$-symmetry of $(m…
Given two relatively prime positive integers $α$ and $β$, we consider simple lattice paths (with unit East and unit North steps) from $(0,0)$ to $(αk,βk)$, and enumerate them by their left and right bounces with respect to the line $y=\fracβα x$. We give the corresponding multivariate generating functions for all such paths as well as for subclasses of paths that start and end with a prescribed step. For illustration purposes, we discuss the case $β=1$ and express some of our functions in terms of the Fuss-Catalan generating function $c_α(x)$.
For $m,n$ coprime we introduce a new statistic skip on $(m,n)$-rational Dyck paths and give a fast way to compute dinv and skip statistics. We also introduce $(m,n)$-rank words, which are in one-to-one correspondence with $(m,n)$-Dyck paths. Defining an equivalence relation on pairs of certain ranks in a rank word, we prove that the number of equivalence classes is the skips of the rank word, and the skips of the corresponding Dyck path. We construct a homogeneous generating function $W_{m,n}(q,t,b)$ using statistics area, dinv and skip, where $W_{m,n}(q,t,1)=C_{m,n}(q,t)$, the rational Catala…
We consider the set of affine permutations that avoid a fixed permutation pattern. Crites has given a simple characterization for when this set is infinite. We find the generating series for this set using the Coxeter length statistic and prove that it can always be represented as a rational function. We also give a characterization of the patterns for which the coefficients of the generating series are periodic. The proofs exploit a new polyhedral encoding for the affine symmetric group.
We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund's bijection exchanging the pairs of statistics (area,dinv) and (bounce,area) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constru…
We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters $β$ (the interaction) and $λ$ (the external field), except for the case $\vertλ\vert=1$ (the "zero-field" case). A randomized algorithm (FPRAS) for all graphs, and all $β,λ$, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our al…
We introduce the higher rank $(q,t)$-Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a dinv statistic on rank $r$ semistandard $(m,n)$-parking functions and prove codinv counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Cata…
We observe that for a and b relatively prime, the "abacus construction" identifies the set of simultaneous (a,b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson's theorem that there are (a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24. Our methods also give new derivations …
Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $\mathbb{F}_q[x_1, x_2, \ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined rational function $M_{d,\infty}(q)$. Furthermore we show that the limit $M_{d,\infty}(q)$ is related to the classic necklace polynomial $M_{d,1}(q)$ by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of s…
We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an …
