The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of $P$, and the edges are given the 1-dimensional faces of $P$. The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard.
In this paper, we show that the problem of computing the diameter is strongly NP-hard even for a polytope with a very simple structure: namely, the \emph{fractional matching} polytope. We also show that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytope is an APX-hard problem. We prove these results by giving an \emph{exact characterization} of the diameter of the fractional matching polytope, that is of independent interest.
The combinatorial diameter of a polytope $P$ is the maximum value of a shortest path between two vertices of $P$, where the path uses the edges of $P$ only. In contrast to the combinatorial diameter, the circuit diameter of $P$ is defined as the maximum value of a shortest path between two vertices of $P$, where the path uses potential edge directions of $P$ i.e., all edge directions that can arise by translating some of the facets of $P$. In this paper, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the…
The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch conjecture is true for general $m{\times}n$--transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. The key reason was that this hierarchy provides some interesting lower bounds for the usual graph diameter. This paper has three contributions: First, we compare the hierarc…
In 1992, Kalai and Kleitman proved a quasipolynomial upper bound on the diameters of convex polyhedra. Todd and Sukegawa-Kitahara proved tail-quasipolynomial bounds on the diameters of polyhedra. These tail bounds apply when the number of facets is greater than a certain function of the dimension. We prove tail-quasipolynomial bounds on the diameters of polytopes and normal simplicial complexes. We also prove tail-polynomial upper bounds on the diameters of polyhedra.
Feasible binary programs often have multiple optimal solutions, which is of interest in applications as they allow the user to choose between alternative optima without deteriorating the objective function. In this article, we present the optimal diameter of a feasible binary program as a metric for measuring the diversity among all optimal solutions. In addition, we present the diameter binary program whose optima contains two optimal solutions of the given feasible binary program that are as diverse as possible with respect to the optimal diameter. Our primary interest is in the study of the…
This short note extends a recent result (Bonifas et al, On sub-determinants and the diameter of polyhedra, Discrete Computational Geometry, 52, 2014) of an upper bound of the diameter of a convex polytope defined by an integer matrix to a similar upper bound of the diameter of a convex polytope defined by a real matrix. It also shows, by an example, that the new bound may be better than the ones of Bonifas et al.
We determine the diameter of the 1-skeleton and the combinatorial automorphism group of any Gelfand-Tsetlin polytope GT$_λ$ associated to an integer partition $λ.$
It has been shown that the edge structure of the characteristic imset polytope is closely connected to the question of causal discovery. The diameter of a polytope is an indicator of how connected the polytope is and moreover gives us a hypothetical worst case scenario for an edge-walk over the polytope. We present low-degree polynomial bounds on the diameter of $\operatorname{CIM}_n$ and, for any given undirected graph $G$, the face $\operatorname{CIM}_G$.
The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but s…
We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with $n$ nodes and $m$ arcs is never more than $m+n-1$. A key step to prove this is to show the same result for classical transportation polytopes.
Given an $\mathcal{H}$-polytope $P$ and a $\mathcal{V}$-polytope $Q$, the decision problem whether $P$ is contained in $Q$ is co-NP-complete. This hardness remains if $P$ is restricted to be a standard cube and $Q$ is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orlin dates back to 1985, for general dimension there seems to be only limited progress on that problem so far. Based on a formulation of the problem in terms of a bilinear feasibility problem, we study sum of squares certificates to decide the containment problem. These certif…
We show that the largest possible diameter $δ(d,k)$ of a $d$-dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rceil$ when $k\geq3$. In addition, we show that $δ(4,3)=8$. This substantiates the conjecture whereby $δ(d,k)$ is at most $\lfloor(k+1)d/2\rfloor$ and is achieved by a Minkowski sum of lattice vectors.
The diameter of the graph of a $d$-dimensional lattice polytope $P \subseteq [0,k]^{n}$ is known to be at most $dk$ due to work by Kleinschmidt and Onn. However, it is an open question whether the monotone diameter, the shortest guaranteed length of a monotone path, of a $d$-dimensional lattice polytope $P = \{\mathbf{x}: A\mathbf{x} \leq \mathbf{b}\} \subseteq [0,k]^{n}$ is bounded by a polynomial in $d$ and $k$. This question is of particular interest in linear optimization, since paths traced by the Simplex method must be monotone. We introduce partial results in this direction including a …
In this paper we show that the diameter of a d-dimensional lattice polytope in [0,k]^n is at most (k - 1/2) d. This result implies that the diameter of a d-dimensional half-integral polytope is at most 3/2 d. We also show that for half-integral polytopes the latter bound is tight for any d.
The stable marriage problem with ties is a well-studied and interesting problem in game theory. We are given a set of men and a set of women. Each individual has a preference ordering on the opposite group, which can possibly contain ties. A stable marriage is given by a matching between men and women for which there is no blocking pair, i.e., a men and a women who strictly prefer each other to their current partner in the matching. In this paper, we study the diameter of the polytope given by the convex hull of characteristic vectors of stable marriages, in the setting with ties. We prove an …
We improve Larman's bound on the diameter of a polytope by showing that if $Δ$ is a normal simplicial complex, all of whose missing faces have size at most $r$, then the diameter of the facet-ridge graph of $Δ$ is not larger than $2^{r-2}n$, where $n$ is the number of vertices of $Δ$. We then use this result to provide new upper bounds on the diameters of the facet-ridge graphs of $k$-neighborly spheres, $r$-stacked spheres, and polytopes with small $g_r$. Specifically, our bounds imply that $r$-stacked spheres with $r=O(\log n)$ satisfy the polynomial Hirsch conjecture.
Brightwell, van den Heuvel and Stougie proved that the diameter of an $m \times n$ transportation polytope is at most $8(m+n-2)$, a factor of eight away from the Hirsch Conjecture. This bound was improved to $3(m+n-1)$ by Hurkens. We investigate diameters for certain classes of transportation polytopes. Note: After the completion of this note, we discovered that the class of transportation polytopes studied in this note was already considered in Michel L. Balinski. On two special classes of transportation polytopes. Math. Programming Stud., 1:43-58, 1974. Michel L. Balinski and Fred J. Rispoli…
We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes P and Q, we show that P can be written as a Minkowski sum with a summand homothetic to Q if and only if P has the same number of vertices as it…
We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \leq 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n-d \leq 6. We show this result by showing this bound for a more general structure -- so-called matroid polytopes -- by reduction to a small number of satisfiability problems.
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high pr…
This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of non-negative real numbers satisfying certain summation conditions. Transportation problems are, in many ways, the simplest kind of linear programs and thus have a rich combinatorial structure. First, we give new results on the diameters of certain classes of transportation polytopes and their relation to the Hirsch Conjecture, which asserts that the diamete…
The combinatorial diameter $\operatorname{diam}(P)$ of a polytope $P$ is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an $n$-dimensional polytope $P$ defined by the intersection of $m$ i.i.d.\ half-spaces whose normals are chosen uniformly from the sphere, we show that $\operatorname{diam}(P)$ is $Ω(n m^{\frac{1}{n-1}})$ and $O(…
Finding a good bound on the maximal edge diameter $Δ(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known, the behaviour of the function $Δ(d,n)$ is largely unknown. The Hirsch conjecture, formulated in 1957 and reported in \cite{GD}, states that $Δ(d,n)$ is linear in $n$ and $d$: $Δ(d,n) \leq n-d$. The conjecture is known to hold in small dimensions, i.e., for $d \leq 3$ \cite{VK}, along with other specific pairs of $d$ and $n$ (Table \ref{before}). However, the asympt…
The aim of this paper is the determination of the largest $n$-dimensional polytope with $n+3$ vertices of unit diameter. This is a special case of a more general problem proposed by Graham.
This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a mu…
Let $R$ be a Noetherian commutative ring of positive dimension. The Hochster-Huneke graph of $R$ (sometimes called the dual graph of Spec $R$ and denoted by $\mathcal{G} (R)$) is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height $(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of Hochster-Huneke graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of var…
We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The regime where we prove this computational difference (conditioned on standard conjectures on the density of primes and on $P\neq NP$) is when the dimension $d$ tends to infinity and the number of facets is linear in $d$.
The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy …
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conje…
The purpose of this paper is the formal verification of a counterexample of Santos et al. to the so-called Hirsch Conjecture on the diameter of polytopes (bounded convex polyhedra). In contrast with the pen-and-paper proof, our approach is entirely computational: we implement in Coq and prove correct an algorithm that explicitly computes, within the proof assistant, vertex-edge graphs of polytopes as well as their diameter. The originality of this certificate-based algorithm is to achieve a tradeoff between simplicity and efficiency. Simplicity is crucial in obtaining the proof of correctness …
Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the shortest and the longest monotone paths and estimate the monotone diameter and height of polyhedra. Our analysis applies to transportation polytopes, matroid polytopes, matching polytopes, shortest-path polytopes, and the TSP, among others. We begin by showing that combinatorial cubes have monotone and Bland pivot height bounde…
Every generic linear functional $f$ on a convex polytope $P$ induces an orientation on the graph of $P$. From the resulting directed graph one can define a notion of $f$-arborescence and $f$-monotone path on $P$, as well as a natural graph structure on the vertex set of $f$-monotone paths. These concepts are important in geometric combinatorics and optimization. This paper bounds the number of $f$-arborescences, the number of $f$-monotone paths, and the diameter of the graph of $f$-monotone paths for polytopes $P$ in terms of their dimension and number of vertices or facets.
Let V be a finite set of points in Euclidean d-space (d >= 2). The intersection of all unit balls B(v,1) centered at v, where v ranges over V, henceforth denoted by B(V) is the ball polytope associated with V. Note that B(V) is non-empty iff the circumradius of V is <= 1. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. [a] Define the boundary complex of B(V) (assuming it is non-empty, of course), i.e., define its vertices, edges and facets in dimension 3 (in dimension 2 this complex is just a circuit), and investigate its…
Each group G of nxn permutation matrices has a corresponding permutation polytope, P(G):=conv(G) in R^{nxn}. 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{2t,floor(n/2)} is a sharp upper bound on the diameter of the graph of P(G); so if G is transitive, the diameter is at most 2. We also show that P(G) achieves its maximal dimension of (n-1)^2 precisely when G is 2-transitive. We then extend results of I. Pak on mixing times for a random walk on P(G). Our work depends on a new result for permutation groups involving…
Circuit-augmentation algorithms are generalizations of the Simplex method, where in each step one is allowed to move along a fixed set of directions, called circuits, that is a superset of the edges of a polytope. We show that in the circuit-augmentation framework the greatest-improvement and Dantzig pivot rules are NP-hard, already for 0/1-LPs. Differently, the steepest-descent pivot rule can be carried out in polynomial time in the 0/1 setting, and the number of circuit augmentations required to reach an optimal solution according to this rule is strongly-polynomial for 0/1-LPs. The number o…
The Minkowski length of a lattice polytope $P$ is a natural generalization of the lattice diameter of $P$. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in $P$. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates $tP$ of a lattice polytope $P$ behaves polynomially in $t\in\mathbb{N}$. In this paper we prove that for any lattice polytope $P$, the Minkowski length of $tP$ for $t\in\mathbb{N}$ is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of c…
The groundbreaking work of Rothvoß [arxiv:1311.2369] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the polyhedral inapproximability of the matching polytope: for fixed $0 < \varepsilon < 1$, every polyhedral $(1 + \varepsilon / n)$-approximation requires an exponential number of inequalities, where $n$ is the number of vertices. This is sharp given the well-known $ρ$-approximation of size $O(\binom{n}{ρ…
We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n-2. The proof avoids the ball polytopes used in the original proofs by Grunbaum, Heppes and Straszewicz. As a corollary we obtain that any three-dimensional diameter graph can be embedded in the projective plane.
Institute of Computing, UFMS（乌法联邦大学计算研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学系）；School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）
The NP-hard 2-Club problem is, given an undirected graph G=(V,E) and l\in N, to decide whether there is a vertex set S\subseteq V of size at least l such that the induced subgraph G[S] has diameter at most two. We make progress towards a systematic classification of the complexity of 2-Club with respect to a hierarchy of prominent structural graph parameters. First, we present the following tight NP-hardness results: 2-Club is NP-hard on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter thr…
