arXiv subjects

Simon Weber

Publications and source records attributed to Simon Weber.

At least 37 records · Page 2Linked to original sources

arXiv2023

On Phases of Unique Sink Orientations

A unique sink orientation (USO) is an orientation of the $n$-dimensional hypercube graph such that every non-empty face contains a unique sink. Schurr showed that given any $n$-dimensional USO and any dimension $i$, the set of edges $E_i$ in that dimension can be decomposed into equivalence classes (so-called phases), such that flipping the orientation of a subset $S$ of $E_i$ yields another USO if and only if $S$ is a union of a set of these phases. In this paper we prove various results on the structure of phases. Using these results, we show that all phases can be computed in $O(3^n)$ time, significantly improving upon the previously known $O(4^n)$ trivial algorithm. Furthermore, we show that given a boolean circuit of size $poly(n)$ succinctly encoding an $n$-dimensional (acyclic) USO, it is PSPACE-complete to determine whether two given edges are in the same phase. The problem is thus equally difficult as determining whether the hypercube orientation encoded by a given circuit is an acyclic USO [Gärtner and Thomas, STACS'15].

math.CO↗

arXiv2023

Matching Function Equilibria with Partial Assignment: Existence, Uniqueness and Estimation

We argue that models coming from a variety of fields, such as matching models and discrete choice models among others, share a common structure that we call matching function equilibria with partial assignment. This structure includes an aggregate matching function and a system of nonlinear equations. We provide a proof of existence and uniqueness of an equilibrium and propose an efficient algorithm to compute it. For a subclass of matching models, we also develop a new parameter-free approach for constructing the counterfactual matching equilibrium. It has the advantage of not requiring parametric estimation when computing counterfactuals. We use our procedure to analyze the impact of the elimination of the Social Security Student Benefit Program in 1982 on the marriage market in the United States. We estimate several candidate models from our general class of matching functions and select the best fitting model using information based criterion.

econ.GN↗

arXiv2023

Existence of a Competitive Equilibrium with Substitutes, with Applications to Matching and Discrete Choice Models

We propose new results for the existence and uniqueness of a general nonparametric and nonseparable competitive equilibrium with substitutes. These results ensure the invertibility of a general competitive system. The existing literature has focused on the uniqueness of a competitive equilibrium assuming that existence holds. We introduce three properties that our supply system must satisfy: weak substitutes, pivotal substitutes, and responsiveness. These properties are sufficient to ensure the existence of an equilibrium, thus providing the existence counterpart to Berry, Gandhi, and Haile (2013)'s uniqueness results. For two important classes of models, bipartite matching models with full assignment and discrete choice models, we show that both models can be reformulated as a competitive system such that our existence and uniqueness results can be readily applied. We also provide an algorithm to compute the unique competitive equilibrium. Furthermore, we argue that our results are particularly useful for studying imperfectly transferable utility matching models with full assignment and non-additive random utility models.

econ.GN↗

arXiv2023

The Complexity of Recognizing Geometric Hypergraphs

As set systems, hypergraphs are omnipresent and have various representations ranging from Euler and Venn diagrams to contact representations. In a geometric representation of a hypergraph $H=(V,E)$, each vertex $v\in V$ is associated with a point $p_v\in \mathbb{R}^d$ and each hyperedge $e\in E$ is associated with a connected set $s_e\subset \mathbb{R}^d$ such that $\{p_v\mid v\in V\}\cap s_e=\{p_v\mid v\in e\}$ for all $e\in E$. We say that a given hypergraph $H$ is representable by some (infinite) family $F$ of sets in $\mathbb{R}^d$, if there exist $P\subset \mathbb{R}^d$ and $S \subseteq F$ such that $(P,S)$ is a geometric representation of $H$. For a family F, we define RECOGNITION(F) as the problem to determine if a given hypergraph is representable by F. It is known that the RECOGNITION problem is $\exists\mathbb{R}$-hard for halfspaces in $\mathbb{R}^d$. We study the families of translates of balls and ellipsoids in $\mathbb{R}^d$, as well as of other convex sets, and show that their RECOGNITION problems are also $\exists\mathbb{R}$-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution.

cs.CG↗

arXiv2023

On Connectivity in Random Graph Models with Limited Dependencies

For any positive edge density $p$, a random graph in the Erdős-Renyi $G_{n,p}$ model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability $ρ(n)$, such that for any distribution $\mathcal{G}$ (in this model) on graphs with $n$ vertices in which each potential edge has a marginal probability of being present at least $ρ(n)$, a graph drawn from $\mathcal{G}$ is connected with non-zero probability? As it turns out, the condition ``edges that do not share endpoints are independent'' needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold $ρ(n)$. For each condition, we provide upper and lower bounds for $ρ(n)$. In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that $ρ(n)\rightarrow 2-ϕ\approx 0.38$ for $n\rightarrow\infty$, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that $ρ(n)>0.5-o(n)$. In stark contrast to the coloring model, for our weakest independence condition -- pairwise independence of non-adjacent edges -- we show that $ρ(n)$ lies within $O(1/n^2)$ of the threshold $1-2/n$ for completely arbitrary distributions.

math.CO↗

arXiv2023

A Topological Version of Schaefer's Dichotomy Theorem

Schaefer's dichotomy theorem [Schaefer, STOC'78] states that a boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of six given conditions holds for every type of constraint allowed in its instances. Otherwise, it is NP-complete. In this paper, we analyze boolean CSPs in terms of their topological complexity, instead of their computational complexity. We attach a natural topological space to the set of solutions of a boolean CSP and introduce the notion of projection-universality. We prove that a boolean CSP is projection-universal if and only if it is categorized as NP-complete by Schaefer's dichotomy theorem, showing that the dichotomy translates exactly from computational to topological complexity. We show a similar dichotomy for SAT variants and homotopy-universality.

cs.CC↗

arXiv2023

An FPT Algorithm for Splitting a Necklace Among Two Thieves

It is well-known that the 2-Thief-Necklace-Splitting problem reduces to the discrete Ham Sandwich problem. In fact, this reduction was crucial in the proof of the PPA-completeness of the Ham Sandwich problem [Filos-Ratsikas and Goldberg, STOC'19]. Recently, a variant of the Ham Sandwich problem called $α$-Ham Sandwich has been studied, in which the point sets are guaranteed to be well-separated [Steiger and Zhao, DCG'10]. The complexity of this search problem remains unknown, but it is known to lie in the complexity class UEOPL [Chiu, Choudhary and Mulzer, ICALP'20]. We define the analogue of this well-separability condition in the necklace splitting problem -- a necklace is $n$-separable, if every subset $A$ of the $n$ types of jewels can be separated from the types $[n]\setminus A$ by at most $n$ separator points. By the reduction to the Ham Sandwich problem it follows that this version of necklace splitting has a unique solution. We furthermore provide two FPT algorithms: The first FPT algorithm solves 2-Thief-Necklace-Splitting on $(n-1+\ell)$-separable necklaces with $n$ types of jewels and $m$ total jewels in time $2^{O(\ell\log\ell)}+m^2$. In particular, this shows that 2-Thief-Necklace-Splitting is polynomial-time solvable on $n$-separable necklaces. Thus, attempts to show hardness of $α$-Ham Sandwich through reduction from the 2-Thief-Necklace-Splitting problem cannot work. The second FPT algorithm tests $(n-1+\ell)$-separability of a given necklace with $n$ types of jewels in time $2^{O(\ell^2)}\cdot n^4$. In particular, $n$-separability can thus be tested in polynomial time, even though testing well-separation of point sets is coNP-complete [Bergold et al., SWAT'22].

math.CO↗

arXiv2023

Topological Art in Simple Galleries

Let $P$ be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in $P$. We say two points $a,b\in P$ can see each other if the line segment $seg(a,b)$ is contained in $P$. We denote by $V(P)$ the family of all minimum guard placements. The Hausdorff distance makes $V(P)$ a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set $S$ there is a polygon $P$ such that $V(P)$ is homotopy equivalent to $S$. Furthermore, for various concrete topological spaces $T$, we describe instances $I$ of the art gallery problem such that $V(I)$ is homeomorphic to $T$.

cs.CG↗

arXiv2023

Power Bundle Adjustment for Large-Scale 3D Reconstruction

We introduce Power Bundle Adjustment as an expansion type algorithm for solving large-scale bundle adjustment problems. It is based on the power series expansion of the inverse Schur complement and constitutes a new family of solvers that we call inverse expansion methods. We theoretically justify the use of power series and we prove the convergence of our approach. Using the real-world BAL dataset we show that the proposed solver challenges the state-of-the-art iterative methods and significantly accelerates the solution of the normal equation, even for reaching a very high accuracy. This easy-to-implement solver can also complement a recently presented distributed bundle adjustment framework. We demonstrate that employing the proposed Power Bundle Adjustment as a sub-problem solver significantly improves speed and accuracy of the distributed optimization.

cs.CV↗

arXiv2023

On the Complexity of Recognizing Nerves of Convex Sets

We study the problem of recognizing whether a given abstract simplicial complex $K$ is the $k$-skeleton of the nerve of $j$-dimensional convex sets in $\mathbb{R}^d$. We denote this problem by $R(k,j,d)$. As a main contribution, we unify the results of many previous works under this framework and show that many of these works in fact imply stronger results than explicitly stated. This allows us to settle the complexity status of $R(1,j,d)$, which is equivalent to the problem of recognizing intersection graphs of $j$-dimensional convex sets in $\mathbb{R}^d$, for any $j$ and $d$. Furthermore, we point out some trivial cases of $R(k,j,d)$, and demonstrate that $R(k,j,d)$ is ER-complete for $j\in\{d-1,d\}$ and $k\geq d$.

cs.CG↗

arXiv2023

Reducing Nearest Neighbor Training Sets Optimally and Exactly

In nearest-neighbor classification, a training set $P$ of points in $\mathbb{R}^d$ with given classification is used to classify every point in $\mathbb{R}^d$: Every point gets the same classification as its nearest neighbor in $P$. Recently, Eppstein [SOSA'22] developed an algorithm to detect the relevant training points, those points $p\in P$, such that $P$ and $P\setminus\{p\}$ induce different classifications. We investigate the problem of finding the minimum cardinality reduced training set $P'\subseteq P$ such that $P$ and $P'$ induce the same classification. We show that the set of relevant points is such a minimum cardinality reduced training set if $P$ is in general position. Furthermore, we show that finding a minimum cardinality reduced training set for possibly degenerate $P$ is in P for $d=1$, and NP-complete for $d\geq 2$.

cs.CG↗

arXiv2022

Collective models and the marriage market

In this paper, I develop an integrated approach to collective models and matching models of the marriage market. In the collective framework, both household formation and the intra-household allocation of bargaining power are taken as given. This is no longer the case in the present contribution, where both are endogenous to the determination of equilibrium on the marriage market. I characterize a class of "proper" collective models which can be embedded into a general matching framework with imperfectly transferable utility. In such models, the bargaining sets are parametrized by an analytical device called distance function, which plays a key role both for writing down the usual stability conditions and for estimation. In general, however, distance functions are not known in closed-form. I provide an efficient method for computing distance functions, that works even with the most complex collective models. Finally, I provide a fully-fledged application using PSID data. I identify the sharing rule and its distribution and study the evolution of the sharing rule and housework time sharing in the United States since 1969. In a counterfactual experiment, I simulate the impact of closing the gender wage gap.

econ.GN↗

arXiv2022

A Universal Construction for Unique Sink Orientations

Unique Sink Orientations (USOs) of cubes can be used to capture the combinatorial structure of many essential algebraic and geometric problems. For various structural and algorithmic questions, including enumeration of USOs and algorithm analysis, it is crucial to have systematic constructions of USOs. While some construction methods for USOs already exist, each one of them has some significant downside. Most of the construction methods have limited expressivity -- USOs with some desired properties cannot be constructed. In contrast, the phase flips of Schurr can construct all USOs, but the operation is not well understood. We were inspired by techniques from cube tilings of space; we expand upon existing techniques in the area to develop generalized rewriting rules for USOs. These rewriting rules are a new construction framework which can be applied to all USOs. The rewriting rules can generate every USO using only USOs of lower dimension. The effect of any specific rewriting rule on an USO is simple to understand. A special case of our construction produces a new elementary transformation of USOs, which we call a partial swap. We further investigate the relationship between partial swaps and phase flips and generalize partial swaps to phase swaps.

math.CO↗

arXiv2022

Realizability Makes a Difference: A Complexity Gap for Sink-Finding in USOs

Algorithms for finding the sink in Unique Sink Orientations (USOs) of the hypercube can be used to solve many algebraic and geometric problems, most importantly including the P-Matrix Linear Complementarity Problem and Linear Programming. The realizable USOs are those that arise from the reductions of these problems to the USO sink-finding problem. Finding the sink of realizable USOs is thus highly practically relevant, yet it is unknown whether realizability can be exploited algorithmically to find the sink more quickly. However, all (non-trivial) known unconditional lower bounds for sink-finding make use of USOs that are provably not realizable. This indicates that the sink-finding problem might indeed be strictly easier on realizable USOs. In this paper we show that this is true for a subclass of all USOs. We consider the class of Matoušek-type USOs, which are a translation of Matoušek's LP-type problems into the language of USOs. We show a query complexity gap between sink-finding in all, and sink-finding in only the realizable $n$-dimensional Matoušek-type USOs. We provide concrete deterministic algorithms and lower bounds for both cases, and show that in the realizable case $O(log^2 n)$ vertex evaluation queries suffice, while in general exactly $n$ queries are needed. The Matoušek-type USOs are the first USO class found to admit such a gap.

cs.DS↗

arXiv2021

Multidirectional Conjugate Gradients for Scalable Bundle Adjustment

We revisit the problem of large-scale bundle adjustment and propose a technique called Multidirectional Conjugate Gradients that accelerates the solution of the normal equation by up to 61%. The key idea is that we enlarge the search space of classical preconditioned conjugate gradients to include multiple search directions. As a consequence, the resulting algorithm requires fewer iterations, leading to a significant speedup of large-scale reconstruction, in particular for denser problems where traditional approaches notoriously struggle. We provide a number of experimental ablation studies revealing the robustness to variations in the hyper-parameters and the speedup as a function of problem density.

cs.CV↗

arXiv2021

A Characterization of the Realizable Matoušek Unique Sink Orientations

The Matoušek LP-type problems were used by Matoušek to show that the Sharir-Welzl algorithm may require at least subexponential time. Later, Gärtner translated this result into the language of Unique Sink Orientations (USOs) and introduced the Matoušek USOs, the USOs equivalent to Matoušek's LP-type problems. He further showed that the Random Facet algorithm only requires quadratic time on the realizable subset of the Matoušek USOs, but without characterizing this subset. In this paper, we deliver this missing characterization and also provide concrete realizations for all realizable Matoušek USOs. Furthermore, we show that the realizable Matoušek USOs are exactly the orientations arising from simple extensions of cyclic-P-matroids.

math.CO↗

arXiv2021

Slim Graph: Practical Lossy Graph Compression for Approximate Graph Processing, Storage, and Analytics

We propose Slim Graph: the first programming model and framework for practical lossy graph compression that facilitates high-performance approximate graph processing, storage, and analytics. Slim Graph enables the developer to express numerous compression schemes using small and programmable compression kernels that can access and modify local parts of input graphs. Such kernels are executed in parallel by the underlying engine, isolating developers from complexities of parallel programming. Our kernels implement novel graph compression schemes that preserve numerous graph properties, for example connected components, minimum spanning trees, or graph spectra. Finally, Slim Graph uses statistical divergences and other metrics to analyze the accuracy of lossy graph compression. We illustrate both theoretically and empirically that Slim Graph accelerates numerous graph algorithms, reduces storage used by graph datasets, and ensures high accuracy of results. Slim Graph may become the common ground for developing, executing, and analyzing emerging lossy graph compression schemes.

cs.DS↗

arXiv2015

The nonlinear Bernstein-Schrödinger equation in Economics

In this paper we relate the Equilibrium Assignment Problem (EAP), which is underlying in several economics models, to a system of nonlinear equations that we call the "nonlinear Bernstein-Schrödinger system", which is well-known in the linear case, but whose nonlinear extension does not seem to have been studied. We apply this connection to derive an existence result for the EAP, and an efficient computational method.

econ.GN↗