Let $V$ be a multiset of $n$ points in $\mathbb{R}^d$, which we call voters, and let $k\geq 1$ and $\ell\geq 1$ be two given constants. We consider the following game, where two players $\mathcal{P}$ and $\mathcal{Q}$ compete over the voters in $V$: First, player $\mathcal{P}$ selects $k$ points in $\mathbb{R}^d$, and then player $\mathcal{Q}$ selects $\ell$ points in $\mathbb{R}^d$. Player $\mathcal{P}$ wins a voter $v\in V$ iff $\mathrm{dist}(v,P) \leq \mathrm{dist}(v,Q)$, where $\mathrm{dist}(v,P) := \min_{p\in P} \mathrm{dist}(v,p)$ and $\mathrm{dist}(v,Q)$ is defined similarly. Player $\mathcal{P}$ wins the game if he wins at least half the voters. The algorithmic problem we study is the following: given $V$, $k$, and $\ell$, how efficiently can we decide if player $\mathcal{P}$ has a winning strategy, that is, if $\mathcal{P}$ can select his $k$ points such that he wins the game no matter where $\mathcal{Q}$ places her points. Banik et al. devised a singly-exponential algorithm for the game in $\mathbb{R}^1$, for the case $k=\ell$. We improve their result by presenting the first polynomial-time algorithm for the game in $\mathbb{R}^1$. Our algorithm can handle arbitrary values of $k$ and $\ell$. We also show that if $d\geq 2$, deciding if player $\mathcal{P}$ has a winning strategy is $Σ_2^P$-hard when $k$ and $\ell$ are part of the input. Finally, we prove that for any dimension $d$, the problem is contained in the complexity class $\exists\forall \mathbb{R}$, and we give an algorithm that works in polynomial time for fixed $k$ and $\ell$.

We consider very natural ”fence enclosure” problems studied by Capoyleas, Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a set S of n points in the plane, we aim at finding a set of closed curves such that (1) each point is enclosed by a curve and (2) the total length of the curves is minimized. We consider two main variants. In the first variant, we pay a unit cost per curve in addition to the total length of the curves. An equivalent formulation of this version is that we have to enclose n unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most k closed curves and pay no cost per curve.

Let V be a set of n points in R d , which we call voters. A point p ∈ R d is preferred over another point p ′ ∈ R d by a voter υ ∈ V if dist(υ, p ) < dist(υ, p ′). A point p is called a plurality point if it is preferred by at least as many voters as any other point p ′. We present an algorithm that decides in O ( n log n ) time whether V admits a plurality point in the L 2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute a minimum-cost subset W ⊂ V such that V \ W admits a plurality point, and to compute a so-called minimum-radius plurality ball. Finally, we consider the problem in the personalized L 1 norm, where each point υ ∈ V has a preference vector 〈 w 1 (υ),…, w d (υ)〉 and the distance from υ to any point p ∈ R d is given by ∑ i =1 d w i (υ)· | x i (υ)− x i ( p )|. For this case we can compute in O ( n d −1 ) time the set of all plurality points of V . When all preference vectors are equal, the running time improves to O ( n ).

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time.

In a geometric k-clustering problem the goal is to partition a set of points in R^d into k subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering queries on a point set S: given a query box Q and an integer k > 2, compute an optimal k-clustering for the subset of S inside Q. We obtain the following results. * We present a general method to compute a (1+epsilon)-approximation to a range-clustering query, where epsilon>0 is a parameter that can be specified as part of the query. Our method applies to a large class of clustering problems, including k-center clustering in any Lp-metric and a variant of k-center clustering where the goal is to minimize the sum (instead of maximum) of the cluster sizes. * We extend our method to deal with capacitated k-clustering problems, where each of the clusters should not contain more than a given number of points. * For the special cases of rectilinear k-center clustering in R^1, and in R^2 for k = 2 or 3, we present data structures that answer range-clustering queries exactly.

Let V be a set of n points in R^d, which we call voters, where d is a fixed constant. A point p in R^d is preferred over another point p' in R^d by a voter v in V if dist(v,p) < dist(v,p'). A point p is called a plurality point if it is preferred by at least as many voters as any other point p'. We present an algorithm that decides in O(n log n) time whether V admits a plurality point in the L_2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute the smallest subset W of V such that V - W admits a plurality point, and to compute a so-called minimum-radius plurality ball. Finally, we consider the problem in the personalized L_1 norm, where each point v in V has a preference vector and the distance from v to any point p in R^d is given by sum_{i=1}^d w_i(v) cdot |x_i(v)-x_i(p)|. For this case we can compute in O(n^(d-1)) time the set of all plurality points of V. When all preference vectors are equal, the running time improves to O(n).