Combinatorial Criterion Research Articles

The Jordan-Hölder property and Grothendieck monoids of exact categories

We investigate the Jordan-Hölder property (JHP) in exact categories. First, we show that (JHP) holds in an exact category if and only if the Grothendieck monoid introduced by Berenstein and Greenstein is free. Moreover, we give a criterion for this which only uses the Grothendieck group and the number of simple objects. Next, we apply these results to the representation theory of artin algebras. For a large class of exact categories including functorially finite torsion(-free) classes, (JHP) holds precisely when the number of indecomposable projectives is equal to that of simples. We study torsion-free classes in a quiver of type A in detail using the combinatorics of symmetric groups. We introduce Bruhat inversions of permutations and show that simples in a torsion-free class are in bijection with Bruhat inversions of the corresponding c-sortable element. We use this to give a combinatorial criterion for (JHP).

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph Qn with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph Fk and the class of circulant graph Cn(1,2).

Analysis and assessment of weighted combinatorial criterion for test suite reduction

AbstractSoftware plays an important role in modern society. New software features have to be completed and tested rapidly due to business and user needs and goals. To ensure the quality of modified software components, new test cases must be generated and included in the current test pool. Furthermore, test pool size often grows too large to test all cases. A considerably large test pool typically consumes an exorbitant quantity of time, leading to inefficient regression testing. Test suite reduction is often used to solve this problem by removing redundant test cases. Thus, time and resource constraints make it necessary to minimize the test suite. That is, test pool size should be efficiently reduced, while the remaining test cases should be adequate to provide the same coverage as the pool. The main purpose of this paper is to propose four weighted combinatorial algorithms and a three‐objective binary integer linear programming (ILP) model to improve fault detection effectiveness. We first consider both the control flow and data flow as our testing criteria and combine them by using a weighted means with four existing test suite reduction algorithms. A genetic algorithm is used to find the best combination of weighting factors to assign to the two testing criteria. Experiments based on real subject programs are presented. Experimental results demonstrate that 77.5% of our proposed weighted combinatorial algorithms have an even or better performance in suite size reduction, 83.75% of our approaches perform even or better in fault detection effectiveness, and 82.5% of our approaches have even or better effectiveness in fault‐to‐test ratio. For the three‐objective binary ILP model, test suite size is significantly reduced without any accompanying loss in fault detection ability.

Compactifications of Cluster Varieties and Convexity

Abstract Gross–Hacking–Keel–Kontsevich [13] discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let ${\mathfrak {D}}$ be the scattering diagram of a cluster variety $V$ (of either type– ${\mathcal {A}}$ or ${\mathcal {X}}$), and let $S$ be a closed subset of $\left (V^\vee \right )^{\textrm {trop}} \left ({\mathbb {R}}\right )$—the ambient space of ${\mathfrak {D}}$. The set $S$ is positive if the theta functions corresponding to the integral points of $S$ and its ${\mathbb {N}}$-dilations define an ${\mathbb {N}}$-graded subalgebra of $\Gamma (V, \mathcal {O}_V){ [x]}$. In particular, a positive set $S$ defines a compactification of $V$ through a Proj construction applied to the corresponding ${\mathbb {N}}$-graded algebra. In this paper, we give a natural convexity notion for subsets of $\left (V^\vee \right )^{\textrm {trop}} \left ({\mathbb {R}}\right )$, called broken line convexity, and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of $\left (V^\vee \right )^{\textrm {trop}} \left ({\mathbb {R}}\right )$ or to check positivity of a given subset.

Slice‐torus link invariants, combinatorial invariants and positivity conditions

We prove some necessary conditions for a link to be either concordant to a quasi-positive link, quasi-positive, positive or the closure of a positive braid. The main applications of our results are a characterisation of positive links with unlinking numbers 1 and 2, and a combinatorial criterion to test if a positive link is the closure of a positive braid. Finally, we compile a table of all positive and positive-braid prime links with less than 8 crossings.

Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

For an element w of a simply-laced Weyl group, Buan-Iyama-Reiten-Scott defined a subcategory of the module category over the preprojective algebra of Dynkin type. This paper studies categorical properties of using the root system. We show that simple objects in bijectively correspond to Bruhat inversion roots of w, and obtain a combinatorial criterion for to satisfy the Jordan-Hölder property (JHP). For type A case, we give a diagrammatic construction of simple objects, and show that (JHP) can be characterized via a forest-like permutation, introduced by Bousquet-Mélou and Butler in the study of Schubert varieties.

Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One

Symmetric varieties are normal equivarient open embeddings of symmetric homogeneous spaces, and they are interesting examples of spherical varieties. We prove that all smooth Fano symmetric varieties with Picard number one admit Kähler–Einstein metrics by using a combinatorial criterion for K-stability of Fano spherical varieties obtained by Delcroix. For this purpose, we present their algebraic moment polytopes and compute the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure.

Hard Lefschetz properties and distribution of spectra in singularity theory and Ehrhart theory

We discuss the distribution of the spectrum at infinity of a convenient and nondegenerate Laurent polynomial (singularity side) and the distribution of the Newton spectrum of a polytope (Ehrhart theory side). To this end, we study a hard Lefschetz property for Laurent polynomials and for polytopes and we give combinatorial criteria for this property to be true. This provides informations about a conjecture by Katzarkov-Kontsevitch-Pantev.

Floer homology, group orderability, and taut foliations of hyperbolic 3–manifolds

This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In particular, it adds further evidence in favor of this conjecture by studying these three properties for more than 300,000 hyperbolic rational homology 3-spheres. New or much improved methods for studying each of these properties form the bulk of the paper, including a new combinatorial criterion, called a foliar orientation, for showing that a 3-manifold has a taut foliation.

From Hierarchical to Relative Hyperbolicity

Abstract We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, this criteria characterizes relative hyperbolicity. We apply our criteria to graphs associated to surfaces and prove that the separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures. We also recover a celebrated theorem of Brock and Masur on the relative hyperbolicity of the Weil–Petersson metric on Teichmüller space for surfaces with complexity three.

Ehrhart functions and symplectic embeddings of ellipsoids

McDuff has previously shown that one four-dimensional symplectic ellipsoid can be symplectically embedded into another if and only if a certain combinatorial criteria holds. We reinterpret this combinatorial criteria using the theory of Ehrhart quasipolynomials, and we use this to give purely combinatorial proofs of theorems of McDuff–Schlenk and Frenkel–Müller, concerning the existence of ‘infinite staircases’ in symplectic embedding problems. We then find a third, new, staircase and conjecture that these are the only three staircases for embeddings into rational ellipsoids. Several other applications are also discussed; for example, we give new examples of triangles whose Ehrhart function exhibits a period collapse.

Weighted Projective Lines and Rational Surface Singularities

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory.

A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors

We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map $f$ acting on the open orthant $\mathbb{R}_{\scriptscriptstyle >0}^n$. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of $f$ "at infinity". In this way, we characterize the situation in which for all $\alpha, \beta > 0$, the "slice space" $\mathcal{S}_\alpha^\beta := \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \mid \alpha x \leq f(x) \leq \beta x \}$ is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations $g$ of $f$, all the orbits of $g$ are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of $f$ near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors.

Stability of Equivariant Vector Bundles over Toric Varieties

We give a complete answer to the question of (semi)stability of tangent bundles on any nonsingular complex projective toric variety with Picard number 2 by using combinatorial criterion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundles on all toric Fano 4-folds with Picard number \leq 3 which are classified by Batyrev [1]. We construct a collection of equivariant indecomposable rank 2 vector bundles on Bott towers and pseudo-symmetric toric Fano varieties. Furthermore, in case of Bott towers, we show the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.

Semi-regular Tilings of the Hyperbolic Plane

A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons surrounding the vertex. We determine combinatorial criteria for the existence, and uniqueness, of a semi-regular tiling with a given vertex-type, and pose some open questions.

Tropical schemes, tropical cycles, and valuated matroids

We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed in [GG16], so there is a well-defined Hilbert–Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that the Giansiracusa tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d ≥ 0 . We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme structure.

Removal Lemmas with Polynomial Bounds

Abstract A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. One of the most celebrated results of this type is the Ruzsa–Szemerédi triangle removal lemma, which states that if a graph is $\varepsilon $-far from being triangle free, then most subsets of vertices of size $C(\varepsilon )$ are not triangle free. Unfortunately, the best known upper bound on $C(\varepsilon )$ is given by a tower-type function, and it is known that $C(\varepsilon )$ is not polynomial in $\varepsilon ^{-1}$. The triangle removal lemma has been extended to many other graph properties, and for some of them the corresponding function $C(\varepsilon )$ is polynomial. This raised the natural question, posed by Goldreich in 2005 and more recently by Alon and Fox, of characterizing the properties for which one can prove removal lemmas with polynomial bounds. Our main results in this paper are new sufficient and necessary criteria for guaranteeing that a graph property admits a removal lemma with a polynomial bound. Although both are simple combinatorial criteria, they imply almost all prior positive and negative results of this type. Moreover, our new sufficient conditions allow us to obtain polynomially bounded removal lemmas for many properties for which the previously known bounds were of tower type. In particular, we show that every semi-algebraic graph property admits a polynomially bounded removal lemma. This confirms a conjecture of Alon.

Commuting homogeneous locally nilpotent derivations

Let be an affine algebraic variety endowed with an action of complexity one of an algebraic torus . It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions can be described in terms of proper polyhedral divisors corresponding to the -variety . We prove that homogeneous locally nilpotent derivations commute if and only if a certain combinatorial criterion holds. These results are used to describe actions of unipotent groups of dimension two on affine -varieties. Bibliography: 10 titles.

Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of a parabolic subgroup. We show that the class of all right-angled Coxeter groups is not smooth and establish some general combinatorial criteria for such classes to be abstract elementary classes (AECs), for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translates into a machinery to build concrete examples of AECs satisfying given model-theoretic properties. We exhibit the power of our method by constructing three concrete examples of finitary classes. We show that the first and third classes are nonhomogeneous and that the last two are tame, uncountably categorical, and axiomatizable by a single Lω1,ω-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is κ-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph.

The Communication Complexity of Private Simultaneous Messages, Revisited

Private simultaneous message (PSM) protocols were introduced by Feige, Kilian, and Naor (STOC ’94) as a minimal non-interactive model for information theoretic three-party secure computation. While it is known that every function $$f:\{0,1\}^k\times \{0,1\}^k \rightarrow \{0,1\}$$ admits a PSM protocol with exponential communication of $$2^{k/2}$$ (Beimel et al., TCC ’14), the best known (non-explicit) lower-bound is $$3k-O(1)$$ bits. To prove this lower-bound, FKN identified a set of simple requirements, showed that any function that satisfies these requirements is subject to the $$3k-O(1)$$ lower-bound, and proved that a random function is likely to satisfy the requirements. We revisit the FKN lower-bound and prove the following results: (Counterexample) We construct a function that satisfies the FKN requirements but has a PSM protocol with communication of $$2k+O(1)$$ bits, revealing a gap in the FKN proof. (PSM lower-bounds) We show that by imposing additional requirements, the FKN argument can be fixed leading to a $$3k-O(\log k)$$ lower-bound for a random function. We also get a similar lower-bound for a function that can be computed by a polynomial-size circuit (or even polynomial-time Turing machine under standard complexity-theoretic assumptions). This yields the first non-trivial lower-bound for an explicit Boolean function partially resolving an open problem of Data, Prabhakaran, and Prabhakaran (Crypto ’14, IEEE Information Theory ’16). We further extend these results to the setting of imperfect PSM protocols which may have small correctness or privacy error. (CDS lower-bounds) We show that the original FKN argument applies (as is) to some weak form of PSM protocols which are strongly related to the setting of Conditional Disclosure of Secrets (CDS). This connection yields a simple combinatorial criterion for establishing linear $$\varOmega (k)$$-bit CDS lower-bounds. As a corollary, we settle the complexity of the inner-product predicate resolving an open problem of Gay, Kerenidis, and Wee (Crypto ’15).