IA Scholar Query: An Approach to the Subgraph Homeomorphism Problem.
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Internet Archive Scholar query results feedeninfo@archive.orgSat, 01 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Descriptive Combinatorics and Distributed Algorithms
https://scholar.archive.org/work/pjgjlnfrkzd5vmd7p7c5yr66pe
In this article we shall explore a fascinating area called descriptive combinatorics and its recently discovered connections to distributed algorithms-a fundamental part of computer science that is becoming increasingly important in the modern era of decentralized computation. The interdisciplinary nature of these connections means that there is very little common background shared by the researchers who are interested in them. With this in mind, this article was written under the assumption that the reader would have close to no background in either descriptive set theory or computer science. The reader will judge to what degree this endeavor was successful. The article comprises two parts. In the first part we give a brief introduction to some of the central notions and problems of descriptive combinatorics. The second part is devoted to a survey of some of the results concerning theAnton Bernshteynwork_pjgjlnfrkzd5vmd7p7c5yr66peSat, 01 Oct 2022 00:00:00 GMTEvolutions of finite graphs
https://scholar.archive.org/work/cpzjxqiz3baxvbitrlapw4ezvi
Every countable graph can be built from finite graphs by a suitable infinite process, either adding new vertices randomly or imposing some rules on the new edges. On the other hand, a profinite topological graph is built as the inverse limit of finite graphs with graph epimorphisms. We propose to look at both constructions simultaneously. We consider countable graphs that can be built from finite ones by using both embeddings and projections, possibly adding a single vertex at each step. We show that the Rado graph can be built this way, while Henson's universal triangle-free graph cannot. We also study the corresponding profinite graphs. Finally, we present a concrete model of the projectively universal profinite graph, the projective Fraisse limit of finite graphs, showing in particular that it has a dense subset of isolated vertices.Stefan Geschke, Szymon Głąb, Wiesław Kubiśwork_cpzjxqiz3baxvbitrlapw4ezviThu, 29 Sep 2022 00:00:00 GMTCovering-based numbers related to the LS-category of finite spaces
https://scholar.archive.org/work/2coxuasw6bdlpkzcbc72wytg4m
In this paper, Lusternik-Schinrelmann and geometric category of finite spaces are considered. We define new numerical invariants of these spaces derived from the geometric category and present an algorithmic approach for its effective computation. The analysis is undertaken by combining homotopic features of the spaces, algorithms and tools from the theory of graphs and hypergraphs. We also provide a number of examples.Manuel Cárdenas, Ramón Flores, Antonio Quintero, Maria Trinidad Villar-Liñánwork_2coxuasw6bdlpkzcbc72wytg4mThu, 29 Sep 2022 00:00:00 GMTThurston's Theorem: Entropy in Dimension One
https://scholar.archive.org/work/2kgwi4eg3bbq3gbcxrylvwk5h4
In his paper, Thurston shows that a positive real number h is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant λ = e^h is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.Ryan Dickmann, George Domat, Thomas Hill, Sanghoon Kwak, Carlos Ospina, Priyam Patel, Rebecca Rechkinwork_2kgwi4eg3bbq3gbcxrylvwk5h4Thu, 29 Sep 2022 00:00:00 GMTOptimal transport methods for combinatorial optimization over two random point sets
https://scholar.archive.org/work/lzgi6js5gvfatmtszdhxg2myre
We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in ℝ^d where the edge cost between two points is given by a p-th power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and d ≥ 3, 1≤ p<d. Previous results were limited to the range p<d/2. Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.Michael Goldman, Dario Trevisanwork_lzgi6js5gvfatmtszdhxg2myreThu, 29 Sep 2022 00:00:00 GMTTopology and adjunction in promise constraint satisfaction
https://scholar.archive.org/work/tuzqedfdtfd7jd7nvikpmdqpie
The approximate graph colouring problem, whose complexity is unresolved in most cases, concerns finding a c-colouring of a graph that is promised to be k-colourable, where c≥ k. This problem naturally generalises to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.Andrei Krokhin, Jakub Opršal, Marcin Wrochna, Stanislav Živnýwork_tuzqedfdtfd7jd7nvikpmdqpieThu, 29 Sep 2022 00:00:00 GMTDefining and Characterizing Reward Hacking
https://scholar.archive.org/work/hmhy7zpfjfddjes2f3zemt4ily
We provide the first formal definition of reward hacking, a phenomenon where optimizing an imperfect proxy reward function, ℛ̃, leads to poor performance according to the true reward function, ℛ. We say that a proxy is unhackable if increasing the expected proxy return can never decrease the expected true return. Intuitively, it might be possible to create an unhackable proxy by leaving some terms out of the reward function (making it "narrower") or overlooking fine-grained distinctions between roughly equivalent outcomes, but we show this is usually not the case. A key insight is that the linearity of reward (in state-action visit counts) makes unhackability a very strong condition. In particular, for the set of all stochastic policies, two reward functions can only be unhackable if one of them is constant. We thus turn our attention to deterministic policies and finite sets of stochastic policies, where non-trivial unhackable pairs always exist, and establish necessary and sufficient conditions for the existence of simplifications, an important special case of unhackability. Our results reveal a tension between using reward functions to specify narrow tasks and aligning AI systems with human values.Joar Skalse, Nikolaus H. R. Howe, Dmitrii Krasheninnikov, David Kruegerwork_hmhy7zpfjfddjes2f3zemt4ilyTue, 27 Sep 2022 00:00:00 GMTThe fuzzy Potts model in the plane: Scaling limits and arm exponents
https://scholar.archive.org/work/raboh4pju5cwzkk2fsmfgb5zqy
We study the fuzzy Potts model on a critical FK percolation in the plane, which is obtained by coloring the clusters of the percolation model independently at random. We show that under the assumption that this critical FK percolation model converges to a conformally invariant scaling limit (which is known to hold for the FK-Ising model), the obtained coloring converges to variants of Conformal Loop Ensembles constructed, described and studied by Miller, Sheffield and Werner. We also show, using discrete considerations that the arm exponents for this coloring in the discrete model are identical to the ones of the continuum model. Using the values of these arm exponents in the continuum, we determine the arm exponents for the fuzzy Potts model.Laurin Köhler-Schindler, Matthis Lehmkuehlerwork_raboh4pju5cwzkk2fsmfgb5zqyMon, 26 Sep 2022 00:00:00 GMTInvariance of immersed Floer cohomology under Lagrangian surgery
https://scholar.archive.org/work/mmtjxyj6vnck3g4dh5x3g73q2e
We show that cellular Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The chain-level map replaces the two orderings of the self-intersection point with meridianal and longitudinal cells on the handle created by the surgery, and uses a bijection between holomorphic disks developed by Fukaya-Oh-Ohta-Ono. Our result generalizes invariance of potentials for certain Lagrangian surfaces in Dimitroglou-Rizell--Ekholm--Tonkonog, and implies the invariance of Floer cohomology under mean curvature flow with this type of surgery, as conjectured by Joyce.Joseph Palmer, Chris Woodwardwork_mmtjxyj6vnck3g4dh5x3g73q2eSun, 25 Sep 2022 00:00:00 GMTThe topology of Gelfand-Zeitlin fibers
https://scholar.archive.org/work/vymrnidg7rcb5gzgrr2tjikduq
We prove several new results about the topology of fibers of Gelfand--Zeitlin systems on unitary and orthogonal coadjoint orbits, at the same time finding a unifying framework recovering and shedding light on essentially all known results. We find completely explicit descriptions of the diffeomorphism type of the fiber in many instances a direct factor decomposition of the fiber, and a torus factor corresponding to the action given by the Thimm trick. The new description also gives us a weak local normal form for a coadjoint orbit, which we use to define a topological toric degeneration, new in the orthogonal case. We also compute the first three homotopy groups (new in the orthogonal case) and cohomology rings of a fiber (new in both cases). All these descriptions can be read in a straightforward manner from the combinatorics of the associated Gelfand--Zeitlin pattern.Jeffrey D. Carlson, Jeremy Lanework_vymrnidg7rcb5gzgrr2tjikduqSat, 24 Sep 2022 00:00:00 GMTA topological zero-one law and elementary equivalence of finitely generated groups
https://scholar.archive.org/work/ljrgwkhtazbnlem7ly27rkzkq4
Let 𝒢 denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces 𝒮⊆𝒢 satisfying the following zero-one law: for any sentence σ in the infinitary logic ℒ_ω_1, ω, the set of all models of σ in 𝒮 is either meager or comeager. In particular, we show that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we obtain that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same claim holds for lacunary hyperbolic groups without non-trivial finite normal subgroups. Our paper has a substantial expository component. We give streamlined proofs of some known results and survey ideas from topology, logic, and geometric group theory relevant to our work. We also discuss some open problems.D. Osinwork_ljrgwkhtazbnlem7ly27rkzkq4Sat, 24 Sep 2022 00:00:00 GMTClique Homology is QMA1-hard
https://scholar.archive.org/work/xebfmtmnw5hvtfvtnxro7jjnay
We tackle the long-standing question of the computational complexity of determining homology groups of simplicial complexes, a fundamental task in computational topology, posed by Kaibel and Pfetsch 20 years ago. We show that this decision problem is QMA1-hard. Moreover, we show that a version of the problem satisfying a suitable promise and certain constraints is contained in QMA. This suggests that the seemingly classical problem may in fact be quantum mechanical. In fact, we are able to significantly strengthen this by showing that the problem remains QMA1-hard in the case of clique complexes, a family of simplicial complexes specified by a graph which is relevant to the problem of topological data analysis. The proof combines a number of techniques from Hamiltonian complexity and homological algebra. We discuss potential implications for the problem of quantum advantage in topological data analysis.Marcos Crichigno, Tamara Kohlerwork_xebfmtmnw5hvtfvtnxro7jjnayFri, 23 Sep 2022 00:00:00 GMTThickness and relative hyperbolicity for graphs of multicurves
https://scholar.archive.org/work/uecfpceqevh2jdwnx7zex4xm5q
We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex of the graph. This extends previously established results for the pants graph and the separating curve graph to a broad family of graphs associated to surfaces.Jacob Russell, Kate M. Vokeswork_uecfpceqevh2jdwnx7zex4xm5qWed, 21 Sep 2022 00:00:00 GMTOn cubulated relatively hyperbolic groups
https://scholar.archive.org/work/lny4oojzdbcg5kv6ugawlo6bn4
We show that properly and cocompactly cubulated relatively hyperbolic groups are virtually special, provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. This extends Agol's result for cubulated hyperbolic groups, and applies to a wide range of peripheral subgroups. In particular, we deduce virtual specialness for properly and cocompactly cubulated groups that are hyperbolic relative to virtually abelian groups. As another consequence, by using a theorem of Martin and Steenbock we obtain virtual specialness for groups obtained as a quotient of a free product of finitely many virtually compact special groups by a finite set of relators satisfying the classical C'(1/6)-small cancellation condition.Eduardo Oregón-Reyeswork_lny4oojzdbcg5kv6ugawlo6bn4Tue, 20 Sep 2022 00:00:00 GMTRoot polytopes, tropical types, and toric edge ideals
https://scholar.archive.org/work/cvbnyr3xvba55jghra3fuqtgae
We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its 'type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincón, these 'tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal, we show how a natural monomial labeling of these complexes describes polynomial relations (syzygies) among 'type ideals' which arise naturally from the combinatorial data of the arrangement. In particular, we show that the cotype ideal is Alexander dual to a corresponding initial ideal of the lattice ideal of the underlying root polytope. This leads to novel ways of studying algebraic properties of various monomial and toric ideals, as well as relating them to combinatorial and geometric properties. In particular, our methods of studying the dimension of the tropical complex leads to new formulas for homological invariants of toric edge ideals of bipartite graphs, which have been extensively studied in the commutative algebra community.Ayah Almousa, Anton Dochtermann, Ben Smithwork_cvbnyr3xvba55jghra3fuqtgaeTue, 20 Sep 2022 00:00:00 GMTNodal domain theorems for p-Laplacians on signed graphs
https://scholar.archive.org/work/anjetek4xvclxjojnjnrouetvu
We establish various nodal domain theorems for generalized p-Laplacians on signed graphs, which unify most of the results on nodal domains of graph p-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we also obtain a higher order Cheeger inequality that relates the variational eigenvalues of p-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs. Moreover, we show new results for 1-Laplacian on signed graphs, including a sharp upper bound of the number of nonzeros of certain eigenfunction corresponding to the k-th variational eigenvalues, and some identities relating variational eigenvalues and k-way Cheeger constants.Chuanyuan Ge, Shiping Liu, Dong Zhangwork_anjetek4xvclxjojnjnrouetvuMon, 19 Sep 2022 00:00:00 GMTIsing model on random triangulations of the disk: phase transition
https://scholar.archive.org/work/pavuypmzzzfbznsk43tk4vcd3m
In [arXiv:1806.06668], we have studied the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the infinite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform infinite half-planar triangulation (UIHPT) decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of finite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in [arXiv:1806.06668] to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a.\ Liouville quantum gravity).Linxiao Chen, Joonas Turunenwork_pavuypmzzzfbznsk43tk4vcd3mThu, 15 Sep 2022 00:00:00 GMTFinite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups
https://scholar.archive.org/work/eynsidhrirddjczdxo67ikxr7q
For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.Yi Liuwork_eynsidhrirddjczdxo67ikxr7qTue, 13 Sep 2022 00:00:00 GMTHitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable
https://scholar.archive.org/work/hwtsnv2fqfcezcbl2bjtqxls7q
For a finite collection of graphs ℱ, the ℱ-TM-Deletion problem has as input an n-vertex graph G and an integer k and asks whether there exists a set S ⊆ V(G) with |S| ≤ k such that G ∖ S does not contain any of the graphs in ℱ as a topological minor. We prove that for every such ℱ, ℱ-TM-Deletion is fixed parameter tractable on planar graphs. Our algorithm runs in a 2^𝒪(k^2)· n^2 time or, alternatively in 2^𝒪(k)· n^4 time. Our techniques can easily be extended to graphs that are embeddable to any fixed surface.Petr A. Golovach, Giannos Stamoulis, Dimitrios M. Thilikoswork_hwtsnv2fqfcezcbl2bjtqxls7qMon, 12 Sep 2022 00:00:00 GMTConvergence of the Probabilistic Interpretation of Modulus
https://scholar.archive.org/work/g5bpd43izrdffgkmzskttdug2y
Given a Jordan domain Ω⊂ℂ and two disjoint arcs A, B on ∂Ω, the modulus m of the curve family connecting A and B in Ω is equal to the corresponding modulus in the rectangle R=[0,1]×[0,m], and m>0 has the property that there is a conformal map ϕ mapping Ω to int(R) so that ϕ extends continuously to a homeomorphism of ∂Ω onto ∂ R and the arcs A and B are sent to the vertical sides of R. Moreover, in the case of the rectangle the family of horizontal segments connecting the two sides has the same modulus as the entire connecting family. Pulling these segments back to Ω via ϕ yields a family of extremal curves (also known as horizontal trajectories) connecting A to B in Ω. In this paper, we show that these extremal curves can be approximated by some discrete paths arising from an orthodiagonal approximation of Ω. Moreover, we show that there is a natural probability mass function (pmf) on these paths, deriving from the theory of discrete modulus, which converges to the transverse measure on the set of extremal curves. The key ingredient is an algorithm that, for an embedded planar graph, takes the current flow between two sets of nodes A and B, and produces a unique path decomposition with non-crossing paths. Moreover, some care was taken to adapt recent results for harmonic convergence on orthodiagonal maps, due to Gurel-Gurevich, Jerison, and Nachmias, to our context. Finally, we generalize a result of N. Alrayes from the square grid setting to the orthodiagonal setting, and prove that the discrete modulus of the approximating non-overlapping paths converges to the continuous modulus.Nathan Albin, Joan Lind, Pietro Poggi-Corradiniwork_g5bpd43izrdffgkmzskttdug2yFri, 09 Sep 2022 00:00:00 GMT