De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting in algorith
ms that use super-polynomial space. We introduce the notion of weighted treedepth and use it to refine the framework of de Berg et al. for obtaining polynomial space (with tight running times) on geometric graphs. As a result, we prove that for any fixed dimension $d ge 2$ on intersection graphs of similarly-sized fat objects many well-known graph problems including Independent Set, $r$-Dominating Set for constant $r$, Cycle Cover, Hamiltonian Cycle, Hamiltonian Path, Steiner Tree, Connected Vertex Cover, Feedback Vertex Set, and (Connected) Odd Cycle Transversal are solvable in time $2^{O(n^{1-1/d})}$ and within polynomial space.
In this paper, we consider the colorful $k$-center problem, which is a generalization of the well-known $k$-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the s
mallest radius $rho$, such that with $k$ balls of radius $rho$, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a pseudo-approximation algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs.
Several algorithms with an approximation guarantee of $O(log n)$ are known for the Set Cover problem, where $n$ is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements
are partitioned into $r$ emph{color classes}, and we are required to cover at least $k_t$ elements from each color class $mathcal{C}_t$, using the minimum number of sets. We give a randomized LP-rounding algorithm that is an $O(beta + log r)$ approximation for the Partition Set Cover problem. Here $beta$ denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard LP. As a corollary, we obtain improved approximation guarantees for various set systems for which $beta$ is known to be sublogarithmic in $n$. We also extend the LP rounding algorithm to obtain $O(log r)$ approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is NP-hard to obtain an $o(log r)$-approximation for any of these problems.
In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clients) in an arbitrary metric space $(X cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $alpha geq 1$.
Each server can have a single ball of arbitrary radius centered on it. Each client $x in X$ needs to be covered by at least $k$ such balls centered on servers. The objective function that we wish to minimize is the sum of the $alpha$-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the $t$-MMC, where we require the number of open servers to be at most some given integer $t$. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding $1$-covering problem, where the coverage demand of each client is $1$. Our reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for $1$-covering, we obtain the first constant approximations for the MMC and these generalizations.
