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The Necklace Splitting problem is a classical problem in combinatorics that has been intensively studied both from a combinatorial and a computational point of view. It is well-known that the Necklace Splitting problem reduces to the discrete Ham Sandwich problem. This reduction was crucial in the proof of PPA-completeness of the Ham Sandwich problem. Recently, Borzechowski, Schnider and Weber [ISAAC'23] introduced a variant of Necklace Splitting that similarly reduces to the $\alpha$-Ham Sandwich problem, which lies in the complexity class UEOPL but is not known to be complete. To make this reduction work, the input necklace is guaranteed to be n-separable. They showed that these necklaces can be fairly split in polynomial time and thus this subproblem cannot be used to prove UEOPL-hardness for $\alpha$-Ham Sandwich. We consider the more general unfair necklace splitting problem on n-separable necklaces, i.e., the problem of splitting these necklaces such that each thief gets a desired fraction of each type of jewels. This more general problem is the natural necklace-splitting-type version of $\alpha$-Ham Sandwich, and its complexity status is one of the main open questions posed by Borzechowski, Schnider and Weber. We show that the unfair splitting problem is also polynomial-time solvable, and can thus also not be used to show UEOPL-hardness for $\alpha$-Ham Sandwich., Comment: 34 pages, 14 figures

MaxCut is a classical NP-complete problem and a crucial building block in many combinatorial algorithms. The famous Edwards-Erd\H{o}s bound states that any connected graph on n vertices with m edges contains a cut of size at least $m/2 + (n-1)/4$. Crowston, Jones and Mnich [Algorithmica, 2015] showed that the MaxCut problem on simple connected graphs admits an FPT algorithm, where the parameter k is the difference between the desired cut size c and the lower bound given by the Edwards-Erd\H{o}s bound. This was later improved by Etscheid and Mnich [Algorithmica, 2017] to run in parameterized linear time, i.e., $f(k)\cdot O(m)$. We improve upon this result in two ways: Firstly, we extend the algorithm to work also for multigraphs (alternatively, graphs with positive integer weights). Secondly, we change the parameter; instead of the difference to the Edwards-Erd\H{o}s bound, we use the difference to the Poljak-Turz\'ik bound. The Poljak-Turz\'ik bound states that any weighted graph G has a cut of size at least $w(G)/2 + w_{MSF}(G)/4$, where w(G) denotes the total weight of G, and $w_{MSF}(G)$ denotes the weight of its minimum spanning forest. In connected simple graphs the two bounds are equivalent, but for multigraphs the Poljak-Turz\'ik bound can be larger and thus yield a smaller parameter k. Our algorithm also runs in parameterized linear time, i.e., $f(k)\cdot O(m+n)$., Comment: 20 pages

Most Bundle Adjustment (BA) solvers like the Levenberg-Marquardt algorithm require a good initialization. Instead, initialization-free BA remains a largely uncharted territory. The under-explored Variable Projection algorithm (VarPro) exhibits a wide convergence basin even without initialization. Coupled with object space error formulation, recent works have shown its ability to solve small-scale initialization-free bundle adjustment problem. To make such initialization-free BA approaches scalable, we introduce Power Variable Projection (PoVar), extending a recent inverse expansion method based on power series. Importantly, we link the power series expansion to Riemannian manifold optimization. This projective framework is crucial to solve large-scale bundle adjustment problems without initialization. Using the real-world BAL dataset, we experimentally demonstrate that our solver achieves state-of-the-art results in terms of speed and accuracy. To our knowledge, this work is the first to address the scalability of BA without initialization opening new venues for initialization-free structure-from-motion.

The Laplace-Beltrami operator (LBO) emerges from studying manifolds equipped with a Riemannian metric. It is often called the Swiss army knife of geometry processing as it allows to capture intrinsic shape information and gives rise to heat diffusion, geodesic distances, and a multitude of shape descriptors. It also plays a central role in geometric deep learning. In this work, we explore Finsler manifolds as a generalization of Riemannian manifolds. We revisit the Finsler heat equation and derive a Finsler heat kernel and a Finsler-Laplace-Beltrami Operator (FLBO): a novel theoretically justified anisotropic Laplace-Beltrami operator (ALBO). In experimental evaluations we demonstrate that the proposed FLBO is a valuable alternative to the traditional Riemannian-based LBO and ALBOs for spatial filtering and shape correspondence estimation. We hope that the proposed Finsler heat kernel and the FLBO will inspire further exploration of Finsler geometry in the computer vision community.

Hierarchy is a natural representation of semantic taxonomies, including the ones routinely used in image segmentation. Indeed, recent work on semantic segmentation reports improved accuracy from supervised training leveraging hierarchical label structures. Encouraged by these results, we revisit the fundamental assumptions behind that work. We postulate and then empirically verify that the reasons for the observed improvement in segmentation accuracy may be entirely unrelated to the use of the semantic hierarchy. To demonstrate this, we design a range of cross-domain experiments with a representative hierarchical approach. We find that on the new testing domains, a flat (non-hierarchical) segmentation network, in which the parents are inferred from the children, has superior segmentation accuracy to the hierarchical approach across the board. Complementing these findings and inspired by the intrinsic properties of hyperbolic spaces, we study a more principled approach to hierarchical segmentation using the Poincar\'e ball model. The hyperbolic representation largely outperforms the previous (Euclidean) hierarchical approach as well and is on par with our flat Euclidean baseline in terms of segmentation accuracy. However, it additionally exhibits surprisingly strong calibration quality of the parent nodes in the semantic hierarchy, especially on the more challenging domains. Our combined analysis suggests that the established practice of hierarchical segmentation may be limited to in-domain settings, whereas flat classifiers generalize substantially better, especially if they are modeled in the hyperbolic space.

In this paper, we examine matching models with imperfectly transferable utility (ITU). We provide motivating examples, discuss the theoretical foundations of ITU matching models and present methods for estimating them. We also explore connected topics and provide an overview of the related literature. This paper has been submitted as a draft chapter for the Handbook of the Economics of Matching, edited by Che, Chiappori and Salani\'e.

We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the $\alpha$-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our $\alpha$-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs., Comment: 29 pages, 9 figures

Given a set of objects O in the plane, the corresponding intersection graph is defined as follows. A vertex is created for each object and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly k bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as the intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are $\exists\mathbb{R}$-complete, leaving unit segments and polylines as few remaining natural cases. We show that recognition for both families of objects is $\exists\mathbb{R}$-complete., Comment: 17 pages, 15 figures

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 [G\"artner and Thomas, STACS'15]., Comment: 21 pages, 7 figures

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.

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., Comment: 18 pages, 1 figure

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 $\alpha$-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 $\alpha$-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]., Comment: 16 pages, 7 figures

For any positive edge density $p$, a random graph in the Erd\H{o}s-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 $\rho(n)$, 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 $\rho(n)$, 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 $\rho(n)$. For each condition, we provide upper and lower bounds for $\rho(n)$. In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that $\rho(n)\rightarrow 2-\phi\approx 0.38$ 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 $\rho(n)>0.5-o(n)$. In stark contrast to the coloring model, for our weakest independence condition -- pairwise independence of non-adjacent edges -- we show that $\rho(n)$ lies within $O(1/n^2)$ of the threshold $1-2/n$ for completely arbitrary distributions., Comment: 35 pages, 6 figures. [v2] adds related work and is intended as a full version accompanying the version to appear at RANDOM'23

Klaus showed that the Oriented Matroid Complementarity Problem (OMCP) can be solved by a reduction to the problem of sink-finding in a unique sink orientation (USO) if the input is promised to be given by a non-degenerate extension of a P-matroid. In this paper, we investigate the effect of degeneracy on this reduction. On the one hand, this understanding of degeneracies allows us to prove a linear lower bound on the number of vertex evaluations required for sink-finding in P-matroid USOs, the set of USOs obtainable through Klaus' reduction. On the other hand, it allows us to adjust Klaus' reduction to also work with degenerate instances. Furthermore, we introduce a total search version of the P-Matroid Oriented Matroid Complementarity Problem (P-OMCP). Given any extension of any oriented matroid M, by reduction to a total search version of USO sink-finding we can either solve the OMCP, or provide a polynomial-time verifiable certificate that M is not a P-matroid. This places the total search version of the P-OMCP in the complexity class Unique End of Potential Line (UEOPL)., Comment: 20 pages, 6 figures

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., Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figures

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 $k\geq d$., Comment: 6 pages, 1 figure, presented at EuroCG'23

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 $d\geq 2$., Comment: 10 pages, 9 figures

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.

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., Comment: 18 pages

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 Matou\v{s}ek-type USOs, which are a translation of Matou\v{s}ek'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 Matou\v{s}ek-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 Matou\v{s}ek-type USOs are the first USO class found to admit such a gap., Comment: 18 pages, 5 figures

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.

We consider the problem of finding weights and biases for a two-layer fully connected neural network to fit a given set of data points as well as possible, also known as EmpiricalRiskMinimization. Our main result is that the associated decision problem is $\exists\mathbb{R}$-complete, that is, polynomial-time equivalent to determining whether a multivariate polynomial with integer coefficients has any real roots. Furthermore, we prove that algebraic numbers of arbitrarily large degree are required as weights to be able to train some instances to optimality, even if all data points are rational. Our result already applies to fully connected instances with two inputs, two outputs, and one hidden layer of ReLU neurons. Thereby, we strengthen a result by Abrahamsen, Kleist and Miltzow [NeurIPS 2021]. A consequence of this is that a combinatorial search algorithm like the one by Arora, Basu, Mianjy and Mukherjee [ICLR 2018] is impossible for networks with more than one output dimension, unless $\mathsf{NP}=\exists\mathbb{R}$., Comment: 39 pages, 17 figures. Changes in version 2: Added algebraic universality result, improved interpretation of results Changes in version 3: Improved exposition by formalizing properties of gadgets

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.

The Matou\v{s}ek LP-type problems were used by Matou\v{s}ek to show that the Sharir-Welzl algorithm may require at least subexponential time. Later, G\"artner translated this result into the language of Unique Sink Orientations (USOs) and introduced the Matou\v{s}ek USOs, the USOs equivalent to Matou\v{s}ek's LP-type problems. He further showed that the Random Facet algorithm only requires quadratic time on the realizable subset of the Matou\v{s}ek USOs, but without characterizing this subset. In this paper, we deliver this missing characterization and also provide concrete realizations for all realizable Matou\v{s}ek USOs. Furthermore, we show that the realizable Matou\v{s}ek USOs are exactly the orientations arising from simple extensions of cyclic-P-matroids., Comment: 17 pages, 5 figures

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 $T$., Comment: 32 pages, 36 figures. For associated GeoGebra files, see source files. For associated video, see http://youtube.com/playlist?list=PLh3Niobwkd8pZcSF_Al7e2eeZ-8vqNm-b . Version v2 adds some additional details and references to publications that appeared after v1

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.

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.