arXiv subjects

Nicole Wein

Publications and source records attributed to Nicole Wein.

35 records · Page 2Linked to original sources

arXiv2023

A Local-to-Global Theorem for Congested Shortest Paths

Amiri and Wargalla (2020) proved the following local-to-global theorem in directed acyclic graphs (DAGs): if $G$ is a weighted DAG such that for each subset $S$ of 3 nodes there is a shortest path containing every node in $S$, then there exists a pair $(s,t)$ of nodes such that there is a shortest $st$-path containing every node in $G$. We extend this theorem to general graphs. For undirected graphs, we prove that the same theorem holds (up to a difference in the constant 3). For directed graphs, we provide a counterexample to the theorem (for any constant), and prove a roundtrip analogue of the theorem which shows there exists a pair $(s,t)$ of nodes such that every node in $G$ is contained in the union of a shortest $st$-path and a shortest $ts$-path. The original theorem for DAGs has an application to the $k$-Shortest Paths with Congestion $c$ (($k,c$)-SPC) problem. In this problem, we are given a weighted graph $G$, together with $k$ node pairs $(s_1,t_1),\dots,(s_k,t_k)$, and a positive integer $c\leq k$. We are tasked with finding paths $P_1,\dots, P_k$ such that each $P_i$ is a shortest path from $s_i$ to $t_i$, and every node in the graph is on at most $c$ paths $P_i$, or reporting that no such collection of paths exists. When $c=k$ the problem is easily solved by finding shortest paths for each pair $(s_i,t_i)$ independently. When $c=1$, the $(k,c)$-SPC problem recovers the $k$-Disjoint Shortest Paths ($k$-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed $k$, $k$-DSP can be solved in polynomial time on DAGs and undirected graphs. Previous work shows that the local-to-global theorem for DAGs implies that $(k,c)$-SPC on DAGs whenever $k-c$ is constant. In the same way, our work implies that $(k,c)$-SPC can be solved in polynomial time on undirected graphs whenever $k-c$ is constant.

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arXiv2022

Approximation Algorithms and Hardness for $n$-Pairs Shortest Paths and All-Nodes Shortest Cycles

We study the approximability of two related problems on graphs with $n$ nodes and $m$ edges: $n$-Pairs Shortest Paths ($n$-PSP), where the goal is to find a shortest path between $O(n)$ prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate $n$-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate $n$-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of $n$-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio. A highlight of our conditional lower bounds results is that for any integer $k\ge 1$, under the combinatorial $4k$-clique hypothesis, there is no combinatorial algorithm for unweighted undirected $n$-PSP with approximation ratio better than $1+1/k$ that runs in $O(m^{2-2/(k+1)}n^{1/(k+1)-ε})$ time. This nearly matches an upper bound implied by the result of Agarwal (2014). A highlight of our algorithmic results is that one can solve both $n$-PSP and ANSC in $\tilde O(m+ n^{3/2+ε})$ time with approximation factor $2+ε$ (and additive error that is function of $ε$), for any constant $ε>0$. For $n$-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for $n$-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms.

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arXiv2022

Online List Labeling: Breaking the $\log^2n$ Barrier

The online list labeling problem is an algorithmic primitive with a large literature of upper bounds, lower bounds, and applications. The goal is to store a dynamically-changing set of $n$ items in an array of $m$ slots, while maintaining the invariant that the items appear in sorted order, and while minimizing the relabeling cost, defined to be the number of items that are moved per insertion/deletion. For the linear regime, where $m = (1 + Θ(1)) n$, an upper bound of $O(\log^2 n)$ on the relabeling cost has been known since 1981. A lower bound of $Ω(\log^2 n)$ is known for deterministic algorithms and for so-called smooth algorithms, but the best general lower bound remains $Ω(\log n)$. The central open question in the field is whether $O(\log^2 n)$ is optimal for all algorithms. In this paper, we give a randomized data structure that achieves an expected relabeling cost of $O(\log^{3/2} n)$ per operation. More generally, if $m = (1 + \varepsilon) n$ for $\varepsilon = O(1)$, the expected relabeling cost becomes $O(\varepsilon^{-1} \log^{3/2} n)$. Our solution is history independent, meaning that the state of the data structure is independent of the order in which items are inserted/deleted. For history-independent data structures, we also prove a matching lower bound: for all $ε$ between $1 / n^{1/3}$ and some sufficiently small positive constant, the optimal expected cost for history-independent list-labeling solutions is $Θ(\varepsilon^{-1}\log^{3/2} n)$.

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arXiv2022

Hardness of Token Swapping on Trees

Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree. These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems); polynomial-time algorithms for simple graph classes such as cliques, stars, paths, and cycles; and constant-factor approximation algorithms in some cases. The two natural cases of sequential and parallel token swapping in trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown. We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is $2$) and show that no such algorithm can achieve an approximation factor less than $2$.

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arXiv2022

Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost

We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of $w$ workers and a multiset $T \subseteq[t]$ of $|T|=w$ tasks, a memoryless worker-task assignment function is any function $ϕ$ that assigns the workers $[w]$ to the tasks $T$ based only on the current value of $T$. The assignment function $ϕ$ is said to have switching cost at most $k$ if, for every task multiset $T$, changing the contents of $T$ by one task changes $ϕ(T)$ by at most $k$ worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost. In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sub-linear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020). We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost $O(\log w \log (wt))$ and an explicit construction that achieves switching cost $\operatorname{polylog} (wt)$. We also prove a super-constant lower bound on switching cost: we show that for any value of $w$, there exists a value of $t$ for which the optimal switching cost is $w$. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of $w$. Finally, we present an application of the worker-task assignment problem to a metric embeddings problem. In particular, we use our results to give the first low-distortion embedding from sparse binary vectors into low-dimensional Hamming space.

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arXiv2021

New Techniques and Fine-Grained Hardness for Dynamic Near-Additive Spanners

Maintaining and updating shortest paths information in a graph is a fundamental problem with many applications. As computations on dense graphs can be prohibitively expensive, and it is preferable to perform the computations on a sparse skeleton of the given graph that roughly preserves the shortest paths information. Spanners and emulators serve this purpose. This paper develops fast dynamic algorithms for sparse spanner and emulator maintenance and provides evidence from fine-grained complexity that these algorithms are tight. Under the popular OMv conjecture, we show that there can be no decremental or incremental algorithm that maintains an $n^{1+o(1)}$ edge (purely additive) $+n^δ$-emulator for any $δ<1/2$ with arbitrary polynomial preprocessing time and total update time $m^{1+o(1)}$. Also, under the Combinatorial $k$-Clique hypothesis, any fully dynamic combinatorial algorithm that maintains an $n^{1+o(1)}$ edge $(1+ε,n^{o(1)})$-spanner or emulator must either have preprocessing time $mn^{1-o(1)}$ or amortized update time $m^{1-o(1)}$. Both of our conditional lower bounds are tight. As the above fully dynamic lower bound only applies to combinatorial algorithms, we also develop an algebraic spanner algorithm that improves over the $m^{1-o(1)}$ update time for dense graphs. For any constant $ε\in (0,1]$, there is a fully dynamic algorithm with worst-case update time $O(n^{1.529})$ that whp maintains an $n^{1+o(1)}$ edge $(1+ε,n^{o(1)})$-spanner. Our new algebraic techniques and spanner algorithms allow us to also obtain (1) a new fully dynamic algorithm for All-Pairs Shortest Paths (APSP) with update and path query time $O(n^{1.9})$; (2) a fully dynamic $(1+ε)$-approximate APSP algorithm with update time $O(n^{1.529})$; (3) a fully dynamic algorithm for near-$2$-approximate Steiner tree maintenance.

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arXiv2021

Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial

We revisit the minimum dominating set problem on graphs with arboricity bounded by $α$. Bansal and Umboh [BU17] gave an $O(α)$-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [KY14, You14, AZO19, Qua10]. Moreover, [BU17] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [LW10], and Jones et al. [JLR+13], achieve an approximation factor of $O(α^2)$ in linear time. There is a similar situation in the distributed setting: While there is an $O(\log^2 n)$-round LP-based $O(α)$-approximation algorithm implied in [KMW06], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an $O(α^2)$-approximation within $O(\log n)$ rounds. We address the questions of whether one can achieve an $O(α)$-approximation algorithm that is not LP-based, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. More specifically, our contribution is two-fold: 1. In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an $O(α)$-approximation in linear time. 2. Based on our centralized algorithm, we design a distributed combinatorial $O(α)$-approximation algorithm in the CONGEST model that runs in $O(α\log n )$ rounds with high probability.

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arXiv2021

Towards Tight Approximation Bounds for Graph Diameter and Eccentricities

Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this parameter can be approximated. Chechik et al. showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $\tilde{O}(m^{3/2})$ time. Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-ε})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs. Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$. It was, however, completely plausible that a $3/2$-approximation is possible in linear time. In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-ε})$ time algorithm can achieve an approximation factor better than $5/3$. Another fundamental set of graph parameters are the Eccentricities. The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$. Chechik et al. showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $\tilde{O}(m^{3/2})$ time and Abboud et al. showed that no $O(n^{2-ε})$ algorithm can achieve better than $5/3$ approximation in sparse graphs. We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH. We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+δ$ factor in $\tilde{O}(m/δ)$ time for any $0<δ<1$. This beats all Eccentricity algorithms in Cairo et al.

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arXiv2020

Tight Conditional Lower Bounds for Approximating Diameter in Directed Graphs

Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices. Computing the Diameter of a graph with $m$ edges requires $m^{2-o(1)}$ time under the Strong Exponential Time Hypothesis (SETH), which can be prohibitive for very large graphs, so efficient approximation algorithms for Diameter are desired. There is a folklore algorithm that gives a $2$-approximation for Diameter in $\tilde{O}(m)$ time. Additionally, a line of work concludes with a $3/2$-approximation algorithm for Diameter in weighted directed graphs that runs in $\tilde{O}(m^{3/2})$ time. The $3/2$-approximation algorithm is known to be tight under SETH: Roditty and Vassilevska W. proved that under SETH any $3/2-ε$ approximation algorithm for Diameter in undirected unweighted graphs requires $m^{2-o(1)}$ time, and then Backurs, Roditty, Segal, Vassilevska W., and Wein and the follow-up work of Li proved that under SETH any $5/3-ε$ approximation algorithm for Diameter in undirected unweighted graphs requires $m^{3/2-o(1)}$ time. Whether or not the folklore 2-approximation algorithm is tight, however, is unknown, and has been explicitly posed as an open problem in numerous papers. Towards this question, Bonnet recently proved that under SETH, any $7/4-ε$ approximation requires $m^{4/3-o(1)}$, only for directed weighted graphs. We completely resolve this question for directed graphs by proving that the folklore 2-approximation algorithm is conditionally optimal. In doing so, we obtain a series of conditional lower bounds that together with prior work, give a complete time-accuracy trade-off that is tight with all known algorithms for directed graphs. Specifically, we prove that under SETH for any $δ>0$, a $(\frac{2k-1}{k}-δ)$-approximation algorithm for Diameter on directed unweighted graphs requires $m^{\frac{k}{k-1}-o(1)}$ time.

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arXiv2020

Lower Bounds for Dynamic Distributed Task Allocation

We study the problem of distributed task allocation in multi-agent systems. Suppose there is a collection of agents, a collection of tasks, and a demand vector, which specifies the number of agents required to perform each task. The goal of the agents is to cooperatively allocate themselves to the tasks to satisfy the demand vector. We study the dynamic version of the problem where the demand vector changes over time. Here, the goal is to minimize the switching cost, which is the number of agents that change tasks in response to a change in the demand vector. The switching cost is an important metric since changing tasks may incur significant overhead. We study a mathematical formalization of the above problem introduced by Su, Su, Dornhaus, and Lynch, which can be reformulated as a question of finding a low distortion embedding from symmetric difference to Hamming distance. In this model it is trivial to prove that the switching cost is at least 2. We present the first non-trivial lower bounds for the switching cost, by giving lower bounds of 3 and 4 for different ranges of the parameters.

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arXiv2020

Improved Dynamic Graph Coloring

This paper studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within $n^{1-ε}$ for any $ε> 0$, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring, or alternatively, study restricted families of graphs. Towards understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for $C$-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. In WADS'17, Barba et al. devised two complementary algorithms: For any $β> 0$ the first (respectively, second) maintains an $O(C βn^{1/β})$ (resp., $O(C β)$)-coloring while recoloring $O(β)$ (resp., $O(βn^{1/β})$) vertices per update. Our contribution is two-fold: - We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: For any $β> 0$, we get a $\tilde{O}(\frac{C}β\log^2 n)$-coloring with $O(β)$ recolorings per update, where the $\tilde{O}$ notation supresses polyloglog$(n)$ factors. In particular, for $β=O(1)$ we get constant recolorings with polylog$(n)$ colors; this is an exponential improvement over the previous bound. - For uniformly sparse graphs, we use low out-degree orientations to strengthen the above result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest.

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arXiv2020

New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs

In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a graph $G=(V,E)$ subject to edge insertions and deletions and a source vertex $s\in V$, and the goal is to maintain the distance $d(s,t)$ for all $t\in V$. Fine-grained complexity has provided strong lower bounds for exact partially dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15]. Thus much focus has been directed towards finding efficient partially dynamic $(1+ε)$-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14, FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20]. Despite this rich literature, for directed graphs there are no known deterministic algorithms for $(1+ε)$-approximate dynamic SSSP that perform better than the classic ES-tree [JACM'81]. We present the first such algorithm. We present a \emph{deterministic} data structure for incremental SSSP in weighted digraphs with total update time $\tilde{O}(n^2 \log W)$ which is near-optimal for very dense graphs; here $W$ is the ratio of the largest weight in the graph to the smallest. Our algorithm also improves over the best known partially dynamic \emph{randomized} algorithm for directed SSSP by Henzinger et al. [STOC'14, ICALP'15] if $m=ω(n^{1.1})$. We also provide improved conditional lower bounds. Henzinger et al. [STOC'15] showed that under the OMv Hypothesis, the partially dynamic exact $s$-$t$ Shortest Path problem in undirected graphs requires amortized update or query time $m^{1/2-o(1)}$, given polynomial preprocessing time. Under a hypothesis about finding Cliques, we improve the update and query lower bound for algorithms with polynomial preprocessing time to $m^{0.626-o(1)}$. Further, under the $k$-Cycle hypothesis, we show that any partially dynamic SSSP algorithm with $O(m^{2-ε})$ preprocessing time requires amortized update or query time $m^{1-o(1)}$.

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arXiv2019

Algorithms and Hardness for Diameter in Dynamic Graphs

The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: - Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. - Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly $(3/2+ε)$-approximation to Diameter in directed or undirected graphs can be maintained decrementally in total time $m^{1+o(1)}\sqrt{n}/ε^2$. This nearly matches the static $3/2$-approximation algorithm for the problem that is known to be conditionally optimal.

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arXiv2019

Approximation Algorithms for Min-Distance Problems

We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between $u$ and $v$ is the minimum of the shortest path distances from $u$ to $v$ and from $v$ to $u$. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help. By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we study can be computed exactly in $\tilde{O}(mn)$ time for directed graphs on $n$ vertices, $m$ edges and nonnegative edge weights. Furthermore, this time bound is tight under the Strong Exponential Time Hypothesis [Roditty-Vassilevska W. STOC 2013] so it is natural to study how well these parameters can be approximated in $O(mn^{1-ε})$ time for constant $ε>0$. Abboud, Vassilevska Williams, and Wang [SODA 2016] gave a polynomial factor approximation for Diameter and Radius, as well as a constant factor approximation for both problems in the special case where the graph is a DAG. We greatly improve upon these bounds by providing the first constant factor approximations for Diameter, Radius and the related Eccentricities problem in general graphs. Additionally, we provide a hierarchy of algorithms for Diameter that gives a time/accuracy trade-off.

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arXiv2019

Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems

Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important $ST$-variant considers two subsets $S$ and $T$ of the vertex set and lets the $ST$-diameter be the maximum distance between a node in $S$ and a node in $T$, and the $ST$-radius be the minimum distance for a node of $S$ to reach all nodes of $T$. The bichromatic variant is the special case in which $S$ and $T$ partition the vertex set. In this paper we present a comprehensive study of the approximability of $ST$ and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis. For instance, for Bichromatic Diameter in undirected weighted graphs with $m$ edges, we present an $\tilde{O}(m^{3/2})$ time $5/3$-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged.

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arXiv2018

Fully Dynamic MIS in Uniformly Sparse Graphs

We consider the problem of maintaining a maximal independent set (MIS) in a dynamic graph subject to edge insertions and deletions. Recently, Assadi, Onak, Schieber and Solomon (STOC 2018) showed that an MIS can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this paper we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity $α$, the amortized update time of our algorithm is $O(α^2 \cdot \log^2 n)$, where $n$ is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs as well as some classes of `real world' graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by $m^{3/8 - ε}$, for any constant $ε> 0$. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed $m^{1/2}$.

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arXiv2018

Finding Cliques in Social Networks: A New Distribution-Free Model

We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a "$c$-closed" graph, where for every pair of vertices $u,v$ with at least $c$ common neighbors, $u$ and $v$ are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to $c$ on $c$-closed graphs. Our results carry over to "weakly $c$-closed graphs", which only require a vertex deletion ordering that avoids pairs of non-adjacent vertices with $c$ common neighbors. Numerical experiments show that well-studied social networks tend to be weakly $c$-closed for modest values of $c$.

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