We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a polynomial time algorithm for the combinatorial polytope isomorphism problem in bounded dimensions. Furthermore, we derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic are graph isomorphism hard.
The original version of the paper (June 2001, 11 pages) had the title ``On the Complexity of Isomorphism Problems Related to Polytopes''. The main difference between the current and the former version is a new polynomial time algorithm for polytope isomorphism in bounded dimension that does not rely on Luks polynomial time algorithm for checking two graphs of bounded valence for isomorphism. Furthermore, the treatment of geometric isomorphism problems was extended.
We investigate the Sherali-Adams lift & project hierarchy applied to a graph isomorphism polytope whose integer points encode the isomorphisms between two graphs. In particular, the Sherali-Adams relaxations characterize a new vertex classification algorithm for graph isomorphism, which we call the generalized vertex classification algorithm. This algorithm generalizes the classic vertex classification algorithm and generalizes the work of Tinhofer on polyhedral methods for graph automorphism testing. We establish that the Sherali-Adams lift & project hierarchy when applied to a graph isomorph…
We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano d-polytopes for d<=7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The problems are grouped into the sections ``Coordinate Descriptions'', ``Combinatorial Structure'', ``Isomorphism'', ``Optimization'', ``Realizability'', and ``Beyond Polytopes''.
The convex hull $ψ_{n,n}$ of certain $(n!)^2$ tensors was considered recently in connection with graph isomorphism. We consider the convex hull $ψ_n$ of the $n!$ diagonals among these tensors. We show: 1. The polytope $ψ_n$ is a face of $ψ_{n,n}$. 2. Deciding if a graph $G$ has a subgraph isomorphic to $H$ reduces to optimization over $ψ_n$. 3. Optimization over $ψ_n$ reduces to optimization over $ψ_{n,n}$. In particular, this implies that the subgraph isomorphism problem reduces to optimization over $ψ_{n,n}$.
We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The Young polytope $P((n-2,2))$ is the convex hull of all such matrices. In 2009, S. Onn showed that the subgraph isomorphism problem can be reduced to optimization both over $QAP(n)$ and over $P(…
Up to isomorphism and duality, there are exactly two non-degenerate abstract regular polytopes of rank greater than $n-3$, one of rank $n-1$ and one of rank $n-2$, with automorphism groups that are transitive permutation groups of degree $n\geq 7$. In this paper we extend this classification of high rank regular polytopes to include the ranks $n-3$ and $n-4$. The result is, up to a isomorphism and duality, seven abstract regular polytopes of rank $n-3$ for each $n\geq 9$, and nine abstract regular polytopes of rank $n-4$ for each $n \geq 11$. Moreover we show that if a transitive permutation g…
The problem of finding a triangulation of a convex three-dimensional polytope with few tetrahedra is proved to be NP-hard. We discuss other related complexity results.
We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining 4 are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all smooth embeddings of toric threefolds in $\mathbb{P}^N$ where $N\le 15$. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in $\mathbb{P}^N$ and the remaining 4 are blow-ups of such tor…
There are (at least) two reasons to study random polytopes. The first is to understand the combinatorics and geometry of random polytopes especially as compared to other classes of polytopes, and the second is to analyze average-case complexity for algorithms which take polytopal data as input. However, establishing results in either of these directions often requires quite technical methods. Here we seek to give an elementary introduction to random polytopes avoiding these technicalities. In particular we explore the general paradigm that polytopes obtained from the convex hull of random poin…
We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the super-exponentially many $d$-polytopes with $d+4$ vertices, all have extension complexity $d+4$ except for some families of size $θ(d^2)$. On the other hand, we show that generic realizations of simplicial/simple $d$-polytopes with $d+1+α$ vertices/facets have extension complexity at least $2 \sqrt{d(d+α)} -d + 1$, which shows that for all $d>(\frac{α-1}{2})^2$ there are…
We consider geometric and computational measures of complexity for sets of integer vectors, asking for a qualitative difference between $f$-vectors of simplicial and general $d$-polytopes, as well as flag $f$-vectors of $d$-polytopes and regular CW $(d-1)$-spheres, for $d\ge 4$.
We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behaviour of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph $G$ depending linearly on the size of $G$ and involving the depth o…
The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß 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-dimensi…
In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cub…
In Rothvoß it was shown that there exists a 0/1 polytope (a polytope whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope projecting to it must have 2^{Ω(n)} facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension~2^{Ω(n)} and an affine space. Our proof relies on a new technique to…
Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more spe…
We exhibit an $n$-node graph whose independent set polytope requires extended formulations of size exponential in $Ω(n/\log n)$. Previously, no explicit examples of $n$-dimensional $0/1$-polytopes were known with extension complexity larger than exponential in $Θ(\sqrt{n})$. Our construction is inspired by a relatively little-known connection between extended formulations and (monotone) circuit depth.
A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a polynomial. After two decades of standstill, recent years have brought amazing progress in showing lower bounds for the so called extension complexity, which for a polytope P denotes the smallest number of inequalities necessary to describe a higher dimensional polytope Q that can be linearly projected on P. However, the central question in this field remained wi…
We illustrate the limitations of the hyperplane separation bound, a non-combinatorial lower bound on the extension complexity of a polytope. Most notably, this bounding technique is used by Rothvoß (J ACM 64.6:41, 2017) to establish an exponential lower bound for the perfect matching polytope. We point out that the technique is sensitive to the particular choice of slack matrix. For the canonical slack matrices of the spanning tree polytope and the completion time polytope, we show that the lower bounds produced by the hyperplane separation method are trivial. These bounds may, however, be str…
We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on the maximum semidefinite extension complexity of 0/1-polytopes. We further show how our result can be used to improve on the corresponding bounds known for polygons with integer vertices. Our geometric proof builds upon nothing else than a simple well-known property of maximum volume inscribed ellipsoids of convex bodies. In particular, it does not rely on fac…
We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite quadratic forms. We illustrate some of the results on the Traveling Salesman Polytope, an example of a complicated convex body studied in combinato…
We obtain 866 isomorphism classes of five-dimensional nonsingular toric Fano varieties using a computer program and the database of four-dimensional reflexive polytopes. The algorithm is based on the existence of facets of Fano polytopes having small integral distance from any vertex.
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\text{MP}_β$ obtained from the Mermin scenario, parametrized by a function $β$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\text{MP}_0$ and $\text{MP}_1$ depending on the parity of $β$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\text{MP}_0$ turn out to be deterministic. This result provides a new topological p…
We establish that the extension complexity of the nXn correlation polytope is at least 1.5^n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least 1.5^n, and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least .58n. We thereby slightly improve on the previously best known lower bounds 1.24…
The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $\mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n \leqslant \mathsf{xc} (\mathsf{STAB}(G)) \leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $\mathsf{xc} (\mathsf{STAB}(G))$ is $O(\frac{n^2}{\log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that …
We consider maximizing a continuous convex function over the assignment polytope. Such problems arise in Graph Matching (the optimization version of Graph Isomorphism) and Quadratic Assignment problems. In the typical case of maximizing a convex function over a polytope the problem can be solved by using a simplicial algorithm such as Tuy's method or the Falk-Hoffman method, but these algorithms require that the underlying polytope be non-degenerate, which is not the case for the assignment polytope. In this note we show how a simple perturbation scheme can be used to create a "surrogate probl…
The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last of these, for which the least is known, and in particular on understanding which polytopes are complex psd-minimal. We prove the existence of an obstruction to complex psd-minimality which is efficiently computable via lattice membership problems. Using this tool, we complete the classification of complex psd-minimal polygons (geometrically as well as combina…
We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and half-integral polytopes.
We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace representation (H-representation). Depending on the polytope, the Z-representation can be more compact than the V-representation and the H-representation. In addition, the Z-representation enables the computation of linear maps, Minkowski addition, and convex hull with a computational complexity that is polynomial in the representation size. The usefulness of the new r…
Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of matrices, new classes of such polytopes are obtained from known small polytopes. This concept, called "consolidation", is applied to find a new compact graph which is known as an approach for the graph isomorphism problem. Encoding and decoding algorithms for our new permutation codes are obtained from existing algorithms for small polytopes. The minimum distan…
The complexity of the matching polytope of graphs may be measured with the maximum length $β$ of a starting sequence of odd ears in an ear-decomposition. Indeed, a theorem of Edmonds and Pulleyblank shows that its facets are defined by 2-connected factor-critical graphs, which have an odd ear-decomposition (according to a theorem of Lovász). In particular, $β(G) \leq 1$ if and only if the matching polytope of the graph $G$ is completely described by non-negativity, star and odd-circuit inequalities. This is essentially equivalent to the h-perfection of the line-graph of $G$, as observed by Cao…
A two-step model for generating random polytopes is considered. For parameters $d$, $m$, and $p$, the first step is to generate a simple polytope $P$ whose facets are given by $m$ uniform random hyperplanes tangent to the unit sphere in $\mathbb{R}^d$, and the second step is to sample each vertex of $P$ independently with probability $p$ and let $Q$ be the convex hull of the sampled vertices. We establish results on how well $Q$ approximates the unit sphere in terms of $m$ and $p$ as well as asymptotics on the combinatorial complexity of $Q$ for certain regimes of $p$.
We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P. We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These examples include both the spanning tree and the perfect matching polytopes of complete graphs, uncapacitated flow polytopes for non-trivially decomposable directed acyclic graphs, hypersimplices, and random 0/1-…
We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J.D. Hamkins, and R. Miller. A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarit…
The nonvanishing problem asks if a coefficient of a polynomial is nonzero. Many families of polynomials in algebraic combinatorics admit combinatorial counting rules and simultaneously enjoy having saturated Newton polytopes (SNP). Thereby, in amenable cases, nonvanishing is in the complexity class $NP\cap coNP$ of problems with "good characterizations". This suggests a new algebraic combinatorics viewpoint on complexity theory. This report discusses the case of Schubert polynomials. These form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. We give a …
Bowdoin College（博多因学院）；University of Technology Nuremberg（纽伦堡工业大学）；London School of Economics and Political Science（伦敦政治经济学院）；Freie Universität Berlin（柏林自由大学）
MPI-SWS（马克斯·普朗克软件系统研究所）；Caltech（加州理工学院）；University of Chicago（芝加哥大学）
We examine the task of locating a target region among those induced by intersections of $n$ halfspaces in $\mathbb{R}^d$. This generic task connects to fundamental machine learning problems, such as training a perceptron and learning a $ϕ$-separable dichotomy. We investigate the average teaching complexity of the task, i.e., the minimal number of samples (halfspace queries) required by a teacher to help a version-space learner in locating a randomly selected target. As our main result, we show that the average-case teaching complexity is $Θ(d)$, which is in sharp contrast to the worst-case tea…
We study the computational question whether a given polytope or spectrahedron $S_A$ (as given by the positive semidefiniteness region of a linear matrix pencil $A(x)$) is contained in another one $S_B$. First we classify the computational complexity, extending results on the polytope/polytope-case by Gritzmann and Klee to the polytope/spectrahedron-case. For various restricted containment problems, NP-hardness is shown. We then study in detail semidefinite conditions to certify containment, building upon work by Ben-Tal, Nemirovski and Helton, Klep, McCullough. In particular, we discuss variat…
We prove that, for every $\ell\geq 4$, there exists an $\ell$-vertex graph and a first order sentence having a quantifier depth at most $\ell-1$ defining the property of having an induced subgraph isomorphic to the given one. We prove that a first order sentence defining the property of containing an induced subgraph on $\ell$ vertices isomorphic to a given disjoint union of isomorphic complete multipartite graphs has a quantifier depth at least $\ell$. Finally, we prove that, for every graph on $\ell\leq 5$ vertices a sentence defining the property of containing an induced subgraph isomorphic…
We completely solve the problem of enumerating combinatorially inequivalent $d$-dimensional polytopes with $d+3$ vertices. A first solution of this problem, by Lloyd, was published in 1970. But the obtained counting formula was not correct, as pointed out in the new edition of Grünbaum's book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution, relying on similar preliminaries but using then a different enumeration method involving automata. In addition, we introduce and solve the problem of counting oriented and achiral (i.e. stable under reflection) …
We prove that the extension complexity of the independence polytope of every regular matroid on $n$ elements is $O(n^6)$. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a $O(n^2)$ bound for the special case of (co)graphic matroids. However, the case of a general regular matroid was open, despite recent attempts. We also consider the extension complexity of circuit dominants of regular matroids, for which we give a $O(n^2)$ bound.
In this paper we study lift-and-project polyhedral operators defined by Lov?asz and Schrijver and Balas, Ceria and Cornu?ejols on the clique relaxation of the stable set polytope of web graphs. We compute the disjunctive rank of all webs and consequently of antiweb graphs. We also obtain the disjunctive rank of the antiweb constraints for which the complexity of the separation problem is still unknown. Finally, we use our results to provide bounds of the disjunctive rank of larger classes of graphs as joined a-perfect graphs, where near-bipartite graphs belong.
Given a set $Σ$ of spheres in $\mathbb{E}^d$, with $d\ge{}3$ and $d$ odd, having a fixed number of $m$ distinct radii $ρ_1,ρ_2,...,ρ_m$, we show that the worst-case combinatorial complexity of the convex hull $CH_d(Σ)$ of $Σ$ is $Θ(\sum_{1\le{}i\ne{}j\le{}m}n_in_j^{\lfloor\frac{d}{2}\rfloor})$, where $n_i$ is the number of spheres in $Σ$ with radius $ρ_i$. To prove the lower bound, we construct a set of $Θ(n_1+n_2)$ spheres in $\mathbb{E}^d$, with $d\ge{}3$ odd, where $n_i$ spheres have radius $ρ_i$, $i=1,2$, and $ρ_2\neρ_1$, such that their convex hull has combinatorial complexity $Ω(n_1n_2^{…
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…
In this paper, we study the parallel and the space complexity of the graph isomorphism problem (\GI{}) for several parameterizations. Let $\mathcal{H}=\{H_1,H_2,\cdots,H_l\}$ be a finite set of graphs where $|V(H_i)|\leq d$ for all $i$ and for some constant $d$. Let $\mathcal{G}$ be an $\mathcal{H}$-free graph class i.e., none of the graphs $G\in \mathcal{G}$ contain any $H \in \mathcal{H}$ as an induced subgraph. We show that \GI{} parameterized by vertex deletion distance to $\mathcal{G}$ is in a parameterized version of $\AC^1$, denoted $\PL$-$\AC^1$, provided the colored graph isomorphism …
In this paper, we show that Graph Isomorphism (GI) is not $\textsf{AC}^{0}$-reducible to several problems, including the Latin Square Isotopy problem, isomorphism testing of several families of Steiner designs, and isomorphism testing of conference graphs. As a corollary, we obtain that GI is not $\textsf{AC}^{0}$-reducible to isomorphism testing of Latin square graphs and strongly regular graphs arising from special cases of Steiner $2$-designs. We accomplish this by showing that the generator-enumeration technique for each of these problems can be implemented in $β_{2}\textsf{FOLL}$, which c…
As it follows from Gödel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in computational mathematics and complexity theory concerning the computational complexity of problems from the class NP, particularly and especially the NP-complete ones. While there is a wide set of algorithms for these problems that we call heuristic, the correctness or/and complexity of each concrete algorithm (or the probability of its correct and polynomial-time work) …
Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $β$ with $0<β<1$, if $\mathcal G$ is a grap…
