arXiv subjects

Johannes Schmitt

Publications and source records attributed to Johannes Schmitt.

49 records · Page 3Linked to original sources

arXiv2021

Parameterized (Modular) Counting and Cayley Graph Expanders

We study the problem $\#\mathrm{EdgeSub}(Φ)$ of counting $k$-edge subgraphs satisfying a given graph property $Φ$ in a large host graph $G$. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of $p$-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field $\mathbb{F}_p$ which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of $\#\mathrm{EdgeSub}(Φ)$ for minor-closed properties $Φ$, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21). Additionally, we observe that our methods also apply to modular counting. Among others, we investigate the problems of modular counting of paths, cycles, forests and matroid bases. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime $p$.

cs.CC↗

arXiv2021

Tevelev degrees and Hurwitz moduli spaces

We interpret the degrees which arise in Tevelev's study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for the Tevelev degrees (together with their natural two parameter generalization). We find exact solutions which specialize to Tevelev's formula in his cases and connect to the projective geometry of lines and Castelnuovo's classical count of linear series in other cases. For almost all values, the calculation of the two parameter generalization of the Tevelev degree is new. A related count of refined Dyck paths is solved along the way.

math.AG↗

arXiv2020

Zero cycles on the moduli space of curves

While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed.

math.AG↗

arXiv2020

Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $Φ$. The search for explicit criteria on $Φ$ ensuring that $\#\mathsf{IndSub}(Φ)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and Dörfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#\mathsf{W}[1]$-completeness result.

cs.CC↗

arXiv2020

admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves

The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are available. In this paper we show how to apply the package, including concrete example computations.

math.AG↗

arXiv2019

Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness

We study the problem #IndSub(P) of counting all induced subgraphs of size k in a graph G that satisfy the property P. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties P including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide range of properties P. In this work, we refine this technique for graph properties that are non-trivial on edge-transitive graphs with a prime power number of edges. In particular, we fully classify the case of monotone bipartite graph properties: It is shown that, given any graph property P that is closed under the removal of vertices and edges, and that is non-trivial for bipartite graphs, the problem #IndSub(P) is #W[1]-hard and cannot be solved in time f(k)*n^{o(k)} for any computable function f, unless the Exponential Time Hypothesis fails. This holds true even if the input graph is restricted to be bipartite and counting is done modulo a fixed prime. A similar result is shown for properties that are closed under the removal of edges only.

cs.CC↗

arXiv2018

Intersections of loci of admissible covers with tautological classes

For a finite group $G$, let $\H_{g,G,ξ}$ be the stack of admissible $G$-covers $C\to D$ of stable curves with ramification data $ξ$, $g(C)=g$ and $g(D)=g'$. There are source and target morphisms $ϕ\colon \H_{g,G,ξ}\to \M_{g,r}$ and $δ\colon \H_{g,G,ξ}\to \M_{g',b}$, remembering the curves $C$ and $D$ together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form $ϕ_* [\H_{g,G,ξ}]$. Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves $C$ with marked ramification points. The two main results of this paper are as follows: Firstly, for the gluing morphism $ξ_A\colon \M_A\to \M_{g,r}$ associated to to a stable graph $A$ we give a combinatorial formula for the pullback $ξ^*_A ϕ_*[\H_{g,G,ξ}]$ in terms of spaces of admissible $G$-covers and $ψ$ classes. This allows us to describe the intersection of the cycles $ϕ_*[\H_{g,G,ξ}]$ with tautological classes. Secondly, the pull-push $δ_*ϕ^*$ sends tautological classes to tautological classes and we also give a combinatorial description of this map in terms of standard generators of the tautological rings. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form $ϕ_* [\H_{g,G,ξ}]$. In particular, we compute the classes $[\Hyp_5]$ and $[\Hyp_6]$ of the hyperelliptic loci in $\M_5$ and $\M_6$ and the class $[\B_4]$ of the bielliptic locus in $\M_4$.

math.AG↗

arXiv2018

Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

We investigate the problem $\#\mathsf{IndSub}(Φ)$ of counting all induced subgraphs of size $k$ in a graph $G$ that satisfy a given property $Φ$. This continues the work of Jerrum and Meeks who proved the problem to be $\#\mathrm{W[1]}$-hard for some families of properties which include, among others, (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ is hard for $\#\mathrm{W[1]}$ if the reduced Euler characteristic of the associated simplicial (graph) complex of $Φ$ is non-zero. This observation links $\#\mathsf{IndSub}(Φ)$ to Karp's famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the "topological approach to evasiveness" which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that $\#\mathsf{IndSub}(Φ)$ is $\#\mathrm{W[1]}$-hard for every monotone property $Φ$ that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not $k$-edge-connected for $k > 2$. Moreover, we show that for those properties $\#\mathsf{IndSub}(Φ)$ can not be solved in time $f(k)\cdot n^{o(k)}$ for any computable function $f$ unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that $\#\mathsf{IndSub}(Φ)$ is $\#\mathrm{W[1]}$-hard if $Φ$ is any non-trivial modularity constraint on the number of edges with respect to some prime $q$ or if $Φ$ enforces the presence of a fixed isolated subgraph.

cs.CC↗

arXiv2017

Multiplicativity of the double ramification cycle

The double ramification cycle satisfies a basic multiplicative relation DRC(a).DRC(b) = DRC(a).DRC(a + b) over the locus of compact-type curves, but this relation fails in the Chow ring of the moduli space of stable curves. We restore this relation over the moduli space of stable curves by introducing an extension of the double ramification cycle to the small b-Chow ring (the colimit of the Chow rings of all smooth blowups of the moduli space). We use this to give evidence for the conjectured equality between the (twisted) double ramification cycle and a cycle constructed in previous work by the second author.

math.AG↗

arXiv2016

A correspondence of good G-sets under partial geometric quotients

For a complex variety $\hat X$ with an action of a reductive group $\hat G$ and a geometric quotient $π: \hat X \to X$ by a closed normal subgroup $H \subset \hat G$, we show that open sets of $X$ admitting good quotients by $G=\hat G / H$ correspond bijectively to open sets in $\hat X$ with good $\hat G$-quotients. We use this to compute GIT-chambers and their associated quotients for the diagonal action of $\text{PGL}_2$ on $(\mathbb{P}^1)^n$ in certain subcones of the $\text{PGL}_2$-effective cone via a torus action on affine space. This allows us to represent these quotients as toric varieties with fans determined by convex geometry.

math.AG↗

arXiv2016

Dimension theory of the moduli space of twisted $k$-differentials

In this note we extend the dimension theory for the spaces $\widetilde{\mathcal H}_g^k(μ)$ of twisted $k$-differentials defined by Farkas and Pandharipande in [FP15] to the case $k>1$. In particular, we show that the intersection $\widetilde{\mathcal H}_g^k(μ) \cap \mathcal{M}_{g,n}$ is a union of smooth components of the expected dimensions for all $k\geq 0$. We also extend a conjectural formula from [FP15] for a weighted fundamental class of $\widetilde{\mathcal H}_g^k(μ)$ and provide evidence in low genus.

math.AG↗

arXiv2016

A compactification of the moduli space of self-maps of $\mathbb{CP}^1$ using stable maps

We present a new compactification $M(d,n)$ of the moduli space of self-maps of $\mathbb{CP}^1$ of degree $d$ with $n$ markings. It is constructed via GIT from the stable maps moduli space $\ ar M_{0,n}(\mathbb{CP}^1 \times \mathbb{CP}^1, (1,d))$. We show that it is the coarse moduli space of a smooth Deligne-Mumford stack and we compute its rational Picard group. Using the recursive boundary structure inherited from the stable maps space, we give an explicit algorithm for computing top-intersection numbers of divisors on $M(d,n)$. We also study the $m$-fold iteration map $M(d,n) \dashrightarrow M(d^m,n)$ and we give a geometric way to extend this rational map to parts of the boundary of $M(d,n)$.

math.AG↗

arXiv2015

Semantic Degrees for Industrie 4.0

Under the context of Industrie 4.0 (I4.0), future production systems provide balanced operations between manufacturing flexibility and efficiency, realized in an autonomous, horizontal, and decentralized item-level production control framework. Structured interoperability via precise formulations on an appropriate degree is crucial to achieve engineering efficiency in the system life cycle. However, selecting the degree of formalization can be challenging, as it crucially depends on the desired common understanding (semantic degree) between multiple parties. In this paper, we categorize different semantic degrees and map a set of technologies in industrial automation to their associated degrees. Furthermore, we created guidelines to assist engineers selecting appropriate semantic degrees in their design. We applied these guidelines on publically available scenarios to examine the validity of the approach, and identified semantic elements over internally developed use cases targeting semantically-enabled plug-and-produce.

cs.SE↗