We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains open. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open.

Generating All Vertices of a Polyhedron is Hard

This article defines and studies a depth for multivariate functional data. By the multivariate nature and by including a weight function, it acknowledges important characteristics of functional data, namely differences in the amount of local amplitude, shape, and phase variation. We study both population and finite sample versions. The multivariate sample of curves may include warping functions, derivatives, and integrals of the original curves for a better overall representation of the functional data via the depth. We present a simulation study and data examples that confirm the good performance of this depth function. Supplementary materials for this article are available online.

/pdf/multivariate-functional-halfspace-depth-16a2fu9fbr.pdf

Multivariate Functional Halfspace Depth

The linear least trimmed squares (LTS) estimator is a statistical technique for fitting a linear model to a set of points. Given a set of n points in ℝ
 d
 and given an integer trimming parameter h≤n, LTS involves computing the (d−1)-dimensional hyperplane that minimizes the sum of the smallest h squared residuals. LTS is a robust estimator with a 50 %-breakdown point, which means that the estimator is insensitive to corruption due to outliers, provided that the outliers constitute less than 50 % of the set. LTS is closely related to the well known LMS estimator, in which the objective is to minimize the median squared residual, and LTA, in which the objective is to minimize the sum of the smallest 50 % absolute residuals. LTS has the advantage of being statistically more efficient than LMS. Unfortunately, the computational complexity of LTS is less understood than LMS. In this paper we present new algorithms, both exact and approximate, for computing the LTS estimator. We also present hardness results for exact and approximate LTS. A number of our results apply to the LTA estimator as well.

/pdf/on-the-least-trimmed-squares-estimator-15ybsu0sjw.pdf

On the Least Trimmed Squares Estimator

Exact computation of the halfspace depth

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc.

/pdf/the-discrete-yet-ubiquitous-theorems-of-caratheodory-helly-z37zfviiiw.pdf

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

The Tukey depth (Proceedings of the International Congress of Mathematicians, vol. 2, pp. 523---531, 1975) of a point p with respect to a finite set S of points is the minimum number of elements of S contained in any closed halfspace that contains p. Algorithms for computing the Tukey depth of a point in various dimensions are considered. The running times of these algorithms depend on the value of the output, making them suited to situations, such as outlier removal, where the value of the output is typically small.

/pdf/output-sensitive-algorithms-for-tukey-depth-and-related-tnmo5vfo1a.pdf

Output-sensitive algorithms for Tukey depth and related problems

A Monte Carlo approximation algorithm for the Tukey depth problem in high dimensions is introduced. The algorithm is a generalization of an algorithm presented by Rousseeuw and Struyf (1998) [20]. The performance of this algorithm is studied both analytically and experimentally.

Absolute approximation of Tukey depth: Theory and experiments

A memoryless routing algorithm is one in which the decision about the next edge on the route to a vertex t for a packet currently located at vertex v is made based only on the coordinates of v, t, and the neighborhood, N(v), of v. The current paper explores the limitations of such algorithms by showing that, for any (randomized) memoryless routing algorithm A, there exists a convex subdivision on which A takes @W(n^2) expected time to route a message between some pair of vertices. Since this lower bound is matched by a random walk, this result implies that the geometric information available in convex subdivisions does not reduce the worst-case routing time for this class of routing algorithms. The current paper also shows the existence of triangulations for which the Random-Compass algorithm proposed by Bose et al. (2002, 2004) requires 2^@W^(^n^) time to route between some pair of vertices.

/pdf/memoryless-routing-in-convex-subdivisions-random-walks-are-278r4fydyh.pdf

Memoryless routing in convex subdivisions: Random walks are optimal

Oja depth (Oja 1983) is a generalization of the median to multivariate data that measures the centrality of a point x with respect to a set S of points in such a way that points with smaller Oja depth are more central with respect to S. Two relationships involving Oja depth and centers of mass are presented. The first is a form of Centerpoint Theorem which shows that the center of mass of the convex hull of a point set has low Oja depth. The second is an approximation result which shows that the center of mass of a point set approximates a point of minimum Oja depth.

Oja centers and centers of gravity

The majority depth of a point with respect to a point set is the number of major sides it is in. An algorithm for majority depth in R is given in this paper, and it is the first algorithm to compute the majority depth. This algorithm runs in O((n+m) log n) time with Brodal and Jacob’s data structure, and in O ( (n + m) log n log log n ) time in the word RAM model.

Dan Chen

Papers

Output-sensitive algorithms for Tukey depth and related problems

Absolute approximation of Tukey depth: Theory and experiments

Memoryless routing in convex subdivisions: Random walks are optimal

Oja centers and centers of gravity

Algorithms for Bivariate Majority Depth.