arXiv subjects

Glencora Borradaile

Publications and source records attributed to Glencora Borradaile.

At least 19 recordsLinked to original sources

arXiv2026

Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices, assuming that the connected components of the inner layer are two-connected.

math.CO↗

arXiv2021

The Motivated Can Encrypt (Even with PGP)

Existing end-to-end-encrypted (E2EE) email systems, mainly PGP, have long been evaluated in controlled lab settings. While these studies have exposed usability obstacles for the average user and offer design improvements, there exist users with an immediate need for private communication, who must cope with existing software and its limitations. We seek to understand whether individuals motivated by concrete privacy threats, such as those vulnerable to state surveillance, can overcome usability issues to adopt complex E2EE tools for long-term use. We surveyed regional activists, as surveillance of social movements is well-documented. Our study group includes individuals from 9 social movement groups in the US who had elected to participate in a workshop on using Thunderbird+Enigmail for email encryption. These workshops tool place prior to mid-2017, via a partnership with a non-profit which supports social movement groups. Six to 40 months after their PGP email encryption training, more than half of the study participants were continuing to use PGP email encryption despite intervening widespread deployment of simple E2EE messaging apps such as Signal. We study the interplay of usability with social factors such as motivation and the risks that individuals undertake through their activism. We find that while usability is an important factor, it is not enough to explain long term use. For example, we find that riskiness of one's activism is negatively correlated with long-term PGP use. This study represents the first long-term study, and the first in-the-wild study, of PGP email encryption adoption.

cs.CR↗

arXiv2020

Low-stretch spanning trees of graphs with bounded width

We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph $G$ with a linear arrangement of bandwidth $b$ can be embedded into a distribution $\mathcal T$ of spanning trees such that the expected stretch of each edge of $G$ is $O(b^2)$. Our proof implies a linear time algorithm for sampling from $\mathcal T$. Therefore, we have a linear time algorithm that finds a spanning tree of $G$ with average stretch $O(b^2)$ with high probability. We also describe a deterministic linear-time algorithm for computing a spanning tree of $G$ with average stretch $O(b^3)$. For graphs of cutwidth $c$, we construct a spanning tree with stretch $O(c^2)$ in linear time. Finally, when $G$ has treewidth $k$ we provide a dynamic programming algorithm computing a minimum stretch spanning tree of $G$ that runs in polynomial time with respect to the number of vertices of $G$.

cs.DS↗

arXiv2020

Minimum bounded chains and minimum homologous chains in embedded simplicial complexes

We study two optimization problems on simplicial complexes with homology over $\mathbb{Z}_2$, the minimum bounded chain problem: given a $d$-dimensional complex $\mathcal{K}$ embedded in $\mathbb{R}^{d+1}$ and a null-homologous $(d-1)$-cycle $C$ in $\mathcal{K}$, find the minimum $d$-chain with boundary $C$, and the minimum homologous chain problem: given a $(d+1)$-manifold $\mathcal{M}$ and a $d$-chain $D$ in $\mathcal{M}$, find the minimum $d$-chain homologous to $D$. We show strong hardness results for both problems even for small values of $d$; $d = 2$ for the former problem, and $d=1$ for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed parameter tractable with respect to the size of the optimal solution. Moreover, we provide an $O(\sqrt{\log β_d})$-approximation algorithm for the minimum bounded chain problem where $β_d$ is the $d$th Betti number of $\mathcal{K}$. Finally, we provide an $O(\sqrt{\log n_{d+1}})$-approximation algorithm for the minimum homologous chain problem where $n_{d+1}$ is the number of $d$-simplices in $\mathcal{M}$.

cs.CG↗

arXiv2020

Whose Tweets are Surveilled for the Police: An Audit of Social-Media Monitoring Tool via Log Files

Social media monitoring by law enforcement is becoming commonplace, but little is known about what software packages for it do. Through public records requests, we obtained log files from the Corvallis (Oregon) Police Department's use of social media monitoring software called DigitalStakeout. These log files include the results of proprietary searches by DigitalStakeout that were running over a period of 13 months and include 7240 social media posts. In this paper, we focus on the Tweets logged in this data and consider the racial and ethnic identity (through manual coding) of the users that are therein flagged by DigitalStakeout. We observe differences in the demographics of the users whose Tweets are flagged by DigitalStakeout compared to the demographics of the Twitter users in the region, however, our sample size is too small to determine significance. Further, the demographics of the Twitter users in the region do not seem to reflect that of the residents of the region, with an apparent higher representation of Black and Hispanic people. We also reconstruct the keywords related to a Narcotics report set up by DigitalStakeout for the Corvallis Police Department and find that these keywords flag Tweets unrelated to narcotics or flag Tweets related to marijuana, a drug that is legal for recreational use in Oregon. Almost all of the keywords have a common meaning unrelated to narcotics (e.g.\ broken, snow, hop, high) that call into question the utility that such a keyword based search could have to law enforcement.

cs.CY↗

arXiv2018

Designing Practical PTASes for Minimum Feedback Vertex Set in Planar Graphs

We present two algorithms for the minimum feedback vertex set problem in planar graphs: an $O(n \log n)$ PTAS using a linear kernel and balanced separator, and a heuristic algorithm using kernelization and local search. We implemented these algorithms and compared their performance with Becker and Geiger's 2-approximation algorithm. We observe that while our PTAS is competitive with the 2-approximation algorithm on large planar graphs, its running time is much longer. And our heuristic algorithm can outperform the 2-approximation algorithm on most large planar graphs and provide a trade-off between running time and solution quality, i.e. a "PTAS behavior".

cs.DS↗

arXiv2017

Density decompositions of networks

We introduce a new topological descriptor of a network called the density decomposition which is a partition of the nodes of a network into regions of uniform density. The decomposition we define is unique in the sense that a given network has exactly one density decomposition. The number of nodes in each partition defines a density distribution which we find is measurably similar to the degree distribution of given real networks (social, internet, etc.) and measurably dissimilar in synthetic networks (preferential attachment, small world, etc.).

cs.SI↗

arXiv2017

Greedy spanners are optimal in doubling metrics

We show that the greedy spanner algorithm constructs a $(1+ε)$-spanner of weight $ε^{-O(d)}w(\mathrm{MST})$ for a point set in metrics of doubling dimension $d$, resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in $d$-dimension Euclidean space has a $(1+ε)$-spanner of weight at most $ε^{-O(d)}w(\mathrm{MST})$. Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space.

cs.CG↗

arXiv2017

Egalitarian Graph Orientations

Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order of the indegrees is NP-hard when the resulting orientation is required to be acyclic.

cs.DM↗

arXiv2017

Minor-free graphs have light spanners

We show that every $H$-minor-free graph has a light $(1+ε)$-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung. Our lightness bound is \[O\left(\frac{σ_H}{ε^3}\log \frac{1}ε\right)\] where $σ_H = |V(H)|\sqrt{\log |V(H)|}$ is the sparsity coefficient of $H$-minor-free graphs. That is, it has a practical dependency on the size of the minor $H$. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in $H$-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is $2^{O_H\left(\frac{1}{ε^4}\log \frac{1}ε\right)}n^{O(1)}$ where $O_H$ ignores dependencies on the size of $H$. Our techniques significantly deviate from existing lines of research on spanners for $H$-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs.

cs.DS↗

arXiv2017

Time-dependent shortest paths in bounded treewidth graphs

We present a proof that the number of breakpoints in the arrival function between two terminals in graphs of treewidth $w$ is $n^{O(\log^2 w)}$ when the edge arrival functions are piecewise linear. This is an improvement on the bound of $n^{Θ(\log n)}$ by Foschini, Hershberger, and Suri for graphs without any bound on treewidth. We provide an algorithm for calculating this arrival function using star-mesh transformations, a generalization of the wye-delta-wye transformations.

cs.DS↗

arXiv2017

Light spanners for bounded treewidth graphs imply light spanners for $H$-minor-free graphs

Grigni and Hung~\cite{GH12} conjectured that H-minor-free graphs have $(1+ε)$-spanners that are light, that is, of weight $g(|H|,ε)$ times the weight of the minimum spanning tree for some function $g$. This conjecture implies the {\em efficient} polynomial-time approximation scheme (PTAS) of the traveling salesperson problem in $H$-minor free graphs; that is, a PTAS whose running time is of the form $2^{f(ε)}n^{O(1)}$ for some function $f$. The state of the art PTAS for TSP in H-minor-free-graphs has running time $n^{1/ε^c}$. We take a further step toward proving this conjecture by showing that if the bounded treewidth graphs have light greedy spanners, then the conjecture is true. We also prove that the greedy spanner of a bounded pathwidth graph is light and discuss the possibility of extending our proof to bounded treewidth graphs.

cs.DS↗

arXiv2017

Embedded-width: A variant of treewidth for plane graphs

We define a special case of tree decompositions for planar graphs that respect a given embedding of the graph. We study the analogous width of the resulting decomposition we call the embedded-width of a plane graph. We show both upper bounds and lower bounds for the embedded-width of a graph in terms of its treewidth and describe a fixed parameter tractable algorithm to calculate the embedded-width of a plane graph. To do so, we give novel bounds on the size of matchings in planar graphs.

cs.DM↗

arXiv2016

A PTAS for Three-Edge Connectivity in Planar Graphs

We consider the problem of finding the minimum-weight subgraph that satisfies given connectivity requirements. Specifically, given a requirement $r \in \{0,1,2,3\}$ for every vertex, we seek the minimum-weight subgraph that contains, for every pair of vertices $u$ and $v$, at least $\min\{ r(v), r(u)\}$ edge-disjoint $u$-to-$v$ paths. We give a polynomial-time approximation scheme (PTAS) for this problem when the input graph is planar and the subgraph may use multiple copies of any given edge. This generalizes an earlier result for $r \in \{0,1,2\}$. In order to achieve this PTAS, we prove some properties of triconnected planar graphs that may be of independent interest.

cs.DS↗

arXiv2016

Minimum cycle and homology bases of surface embedded graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^ω+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^ω)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.

cs.DS↗

arXiv2015

All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is $2^{O(g^2)}$.

cs.CG↗

arXiv2015

The two-edge connectivity survivable-network design problem in planar graphs

Consider the following problem: given a graph with edge costs and a subset Q of vertices, find a minimum-cost subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient analog of the Steiner tree problem arising, for example, in telecommunications applications. We study a more general mixed-connectivity formulation, also employed in telecommunications optimization. Given a number (or requirement) r(v) in {0, 1, 2} for each vertex v in the graph, find a minimum-cost subgraph in which there are min{r(u), r(v)} edge-disjoint u-to-v paths for every pair u, v of vertices. We address the problem in planar graphs, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy). The problem is max SNP-hard in general graphs and strongly NP-hard in planar graphs. We give the first polynomial-time approximation scheme in planar graphs. The running time is O(n log n). Under the additional restriction that the requirements are only non-zero for vertices on the boundary of a single face of a planar graph, we give a polynomial-time algorithm to find the optimal solution.

cs.DS↗

arXiv2015

Towards single face shortest vertex-disjoint paths in undirected planar graphs

Given $k$ pairs of terminals $\{(s_{1}, t_{1}), \ldots, (s_{k}, t_{k})\}$ in a graph $G$, the min-sum $k$ vertex-disjoint paths problem is to find a collection $\{Q_{1}, Q_{2}, \ldots, Q_{k}\}$ of vertex-disjoint paths with minimum total length, where $Q_{i}$ is an $s_i$-to-$t_i$ path between $s_i$ and $t_i$. We consider the problem in planar graphs, where little is known about computational tractability, even in restricted cases. Kobayashi and Sommer propose a polynomial-time algorithm for $k \le 3$ in undirected planar graphs assuming all terminals are adjacent to at most two faces. Colin de Verdiere and Schrijver give a polynomial-time algorithm when all the sources are on the boundary of one face and all the sinks are on the boundary of another face and ask about the existence of a polynomial-time algorithm provided all terminals are on a common face. We make progress toward Colin de Verdiere and Schrijver's open question by giving an $O(kn^5)$ time algorithm for undirected planar graphs when $\{(s_{1}, t_{1}), \ldots, (s_{k}, t_{k})\}$ are in counter-clockwise order on a common face.

cs.DS↗