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**11 - 13**of**13**### On the Computational Complexity of some problems from Combinatorial Geometry

"... We study several canonical decision problems that arise from the most famous theorems from combinatorial geometry. We show that these are W[1]-hard (and NP-hard) if the dimension is part of the input by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Among others, we sh ..."

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We study several canonical decision problems that arise from the most famous theorems from combinatorial geometry. We show that these are W[1]-hard (and NP-hard) if the dimension is part of the input by fpt-reductions (which are actually ptime-reductions) from the d-Sum problem. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the Ham-Sandwich cut problem are W[1]-hard. Our reductions also imply that the problems we consider cannot be solved in time no(d) (where n is the size of the input), unless the Exponential-Time Hypothesis (ETH) is false. Our technique of embedding d-Sum into a geometric setting is conceptu-ally much simpler than direct fpt-reductions from purely combinatorial W[1]-hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems.

### Finding a largest empty convex subset in space is W[1]-hard

, 2014

"... We consider the following problem: Given a point set in space find a largest subset that is in convex position and whose convex hull is empty. We show that the (decision version of the) problem is W[1]-hard. 1 ..."

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We consider the following problem: Given a point set in space find a largest subset that is in convex position and whose convex hull is empty. We show that the (decision version of the) problem is W[1]-hard. 1

### Fixed Parameter Complexity and Approximability of Norm Maximization∗

, 2014

"... The problem of maximizing the p-th power of a p-norm over a halfspace-presented polytope in Rd is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in [19] that this problem is NP-hard for all values p ∈ N, if the dimension d of the ambient sp ..."

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The problem of maximizing the p-th power of a p-norm over a halfspace-presented polytope in Rd is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in [19] that this problem is NP-hard for all values p ∈ N, if the dimension d of the ambient space is part of the input. In this paper, we use the theory of parametrized complexity to analyze how heavily the hardness of norm maximization relies on the parameter d. More precisely, we show that for p = 1 the problem is fixed parameter tractable but that for all p ∈ N \ {1} norm maximization is W[1]-hard. Concerning approximation algorithms for norm maximization, we show that for fixed accuracy, there is a straightforward approximation algorithm for norm maximization in FPT running time, but there is no FPT approximation algorithm, the running time of which depends polynomially on the accuracy. As with the NP-hardness of norm maximization, the W[1]-hardness immediately carries over to various radius computation tasks in Computational Convexity. 1