Abstract. The degeneracy of an n-vertex graph G is the smallest number d such that every subgraph of G contains a vertex of degree at most d. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron-Kerbosch algorithm and show that it runs in time O(dn3 d/3 ). We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an n-vertex graph with degeneracy d (when d is a multiple of 3 and n ≥ d + 3) is (n − d)3 d/3 . Therefore, our algorithm matches the) worst-case output size of the problem whenever n − d = Ω (n).

In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the 'longest' subtrajectory cluster is as hard as MaxClique to compute and approximate.

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n 13 ), and prove NP-hardness for some other variants.

Route memory is frequently assessed in virtual environments. These environments can be presented in a fully controlled manner and are easy to use. Yet they lack the physical involvement that participants have when navigating real environments. For some aspects of route memory this may result in reduced performance in virtual environments. We assessed route memory performance in four different environments: real, virtual, virtual with directional information (compass), and hybrid. In the hybrid environment, participants walked the route outside on an open field, while all route information (i.e., path, landmarks) was shown simultaneously on a handheld tablet computer. Results indicate that performance in the real life environment was better than in the virtual conditions for tasks relying on survey knowledge, like pointing to start and end point, and map drawing. Performance in the hybrid condition however, hardly differed from real life performance. Performance in the virtual environment did not benefit from directional information. Given these findings, the hybrid condition may offer the best of both worlds: the performance level is comparable to that of real life for route memory, yet it offers full control of visual input during route learning.

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We investigate the concept of a median among a set of trajectories. We establish criteria that a "median trajectory" should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and evaluate whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better.

The Euclidean k-center problem is a classical problem that has been extensively studied in computer science. Given a set G of n points in Euclidean space, the problem is to determine a set C of k centers (not necessarily part of G) such that the maximum distance between a point in G and its nearest neighbor in C is minimized. In this paper we study the corresponding (k, )-center problem for polygonal curves under the Fréchet distance, that is, given a set G of n polygonal curves in R d , each of complexity m, determine a set C of k polygonal curves in R d , each of complexity , such that the maximum Fréchet distance of a curve in G to its closest curve in C is minimized. In their 2016 paper, Driemel, Krivošija, and Sohler give a near-linear time (1 + ε)approximation algorithm for one-dimensional curves, assuming that k and are constants. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension 2 and higher. Our analysis thus extends to application-relevant input data such as GPS-trajectories and protein backbones. We show that, if is part of the input, then there is no polynomial-time approximation scheme unless P = NP. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to 2.598. This result also holds when k = 1, and the NP-hardness extends to the case that = ∞, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a 3-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.

In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the 'longest' subtrajectory cluster is as hard as MaxClique to compute and approximate.