On the Obfuscation Complexity of Planar Graphs

Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity of $G$, to be the maximum number of edge crossings in a drawing of $G$. Relating $obf(G)$ to the distribution of vertex degrees in $G$, we show an efficient way of constructing a drawing of $G$ with at least $obf(G)/3$ edge crossings. We prove bounds (\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2 for an $n$-vertex planar graph $G$ with minimum vertex degree $\delta(G)\ge 2$. The shift complexity of $G$, denoted by $shift(G)$, is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of $G$ (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If $\delta(G)\ge 3$, then $shift(G)$ is linear in the number of vertices due to the known fact that the matching number of $G$ is linear. However, in the case $\delta(G)\ge2$ we notice that $shift(G)$ can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing $D$ of a planar graph, it is NP-hard to find an optimum sequence of shifts making $D$ crossing-free.Comment: 12 pages, 1 figure. The proof of Theorem 3 is simplified. An overview of a related work is adde

On the Anonymization of Differentially Private Location Obfuscation

Obfuscation techniques in location-based services (LBSs) have been shown useful to hide the concrete locations of service users, whereas they do not necessarily provide the anonymity. We quantify the anonymity of the location data obfuscated by the planar Laplacian mechanism and that by the optimal geo-indistinguishable mechanism of Bordenabe et al. We empirically show that the latter provides stronger anonymity than the former in the sense that more users in the database satisfy k-anonymity. To formalize and analyze such approximate anonymity we introduce the notion of asymptotic anonymity. Then we show that the location data obfuscated by the optimal geo-indistinguishable mechanism can be anonymized by removing a smaller number of users from the database. Furthermore, we demonstrate that the optimal geo-indistinguishable mechanism has better utility both for users and for data analysts.Comment: ISITA'18 conference pape

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure

Optimal Geo-Indistinguishable Mechanisms for Location Privacy

We consider the geo-indistinguishability approach to location privacy, and the trade-off with respect to utility. We show that, given a desired degree of geo-indistinguishability, it is possible to construct a mechanism that minimizes the service quality loss, using linear programming techniques. In addition we show that, under certain conditions, such mechanism also provides optimal privacy in the sense of Shokri et al. Furthermore, we propose a method to reduce the number of constraints of the linear program from cubic to quadratic, maintaining the privacy guarantees and without affecting significantly the utility of the generated mechanism. This reduces considerably the time required to solve the linear program, thus enlarging significantly the location sets for which the optimal mechanisms can be computed.Comment: 13 page

Privacy-Preserving Vehicle Assignment for Mobility-on-Demand Systems

Urban transportation is being transformed by mobility-on-demand (MoD) systems. One of the goals of MoD systems is to provide personalized transportation services to passengers. This process is facilitated by a centralized operator that coordinates the assignment of vehicles to individual passengers, based on location data. However, current approaches assume that accurate positioning information for passengers and vehicles is readily available. This assumption raises privacy concerns. In this work, we address this issue by proposing a method that protects passengers' drop-off locations (i.e., their travel destinations). Formally, we solve a batch assignment problem that routes vehicles at obfuscated origin locations to passenger locations (since origin locations correspond to previous drop-off locations), such that the mean waiting time is minimized. Our main contributions are two-fold. First, we formalize the notion of privacy for continuous vehicle-to-passenger assignment in MoD systems, and integrate a privacy mechanism that provides formal guarantees. Second, we present a scalable algorithm that takes advantage of superfluous (idle) vehicles in the system, combining multiple iterations of the Hungarian algorithm to allocate a redundant number of vehicles to a single passenger. As a result, we are able to reduce the performance deterioration induced by the privacy mechanism. We evaluate our methods on a real, large-scale data set consisting of over 11 million taxi rides (specifying vehicle availability and passenger requests), recorded over a month's duration, in the area of Manhattan, New York. Our work demonstrates that privacy can be integrated into MoD systems without incurring a significant loss of performance, and moreover, that this loss can be further minimized at the cost of deploying additional (redundant) vehicles into the fleet.Comment: 8 pages; Submitted to IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 201

A polynomial bound for untangling geometric planar graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure