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For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. [ABSTRACT FROM AUTHOR]

Fair clustering is a constrained clustering problem where we need to partition a set of colored points. The fraction of points of each color in every cluster should be more or less equal to the fraction of points of this color in the dataset. The problem was recently introduced by Chierichetti et al. (2017) [1]. We propose a new construction of coresets for fair clustering for Euclidean and general metrics based on random sampling. For the Euclidean space R d , we provide the first coreset whose size does not depend exponentially on the dimension d. The question of whether such constructions exist was asked by Schmidt et al. (2019) [2] and Huang et al. (2019) [5]. For general metrics, our construction provides the first coreset for fair clustering. New coresets appear to be a handy tool for designing better approximation and streaming algorithms for fair and other constrained clustering variants. [ABSTRACT FROM AUTHOR]

Given a digraph G = (V , E) , the k -path partition problem aims to find a minimum collection of vertex-disjoint directed paths, of order at most k , to cover all the vertices. The problem has various applications. Its special case on undirected graphs is NP-hard when k ≥ 3 , and has received much study recently from the approximation algorithm perspective. However, the general problem on digraphs is seemingly untouched in the literature. We fill the gap with the first k / 2 -approximation algorithm, based on a novel concept of enlarging walk to minimize the number of singletons. Secondly, for k = 3 , we define a second novel kind of enlarging walks to greedily reduce the number of 2-paths in the 3-path partition and propose an improved 13/9-approximation algorithm. Lastly, for any k ≥ 7 , we present an improved (k + 2) / 3 -approximation algorithm built on the maximum path-cycle cover followed by a careful 2-cycle elimination process. [ABSTRACT FROM AUTHOR]

We consider Internet of Things (IoT) sensors deployed inside an area to be monitored. Drones can be used to collect the data from the sensors, but they are constrained in energy and storage. Therefore, all drones need to select a subset of sensors whose data are the most relevant to be acquired, modeled by assigning a reward. We present an optimization problem called Multiple-drone Data-collection Maximization Problem (MDMP) whose objective is to plan a set of drones' missions aimed at maximizing the overall reward from the collected data, and such that each individual drone's mission energy cost and total collected data are within the energy and storage limits, respectively. We optimally solve MDMP by proposing an Integer Linear Programming based algorithm. Since MDMP is NP -hard, we devise suboptimal algorithms for single- and multiple-drone scenarios. Finally, we thoroughly evaluate our algorithms on the basis of random generated synthetic data. [ABSTRACT FROM AUTHOR]

A garden is populated by n bamboos, each with its own daily growth rate. The Bamboo Garden Trimming Problem (BGT) is to design for a robotic gardener a perpetual schedule of cutting bamboos to keep the elevation of the garden as low as possible. The frequency of cutting is constrained by the time needed to move from one bamboo to the next, which is one day in Discrete BGT and is defined by the distance between the two bamboos in Continuous BGT. The bamboo garden is a metaphor for a collection of machines which have to be serviced, with different frequencies, by a robot which can service only one machine at a time. For Discrete BGT, we show tighter approximation algorithms, with one of them settling a long-standing conjecture about the related Pinwheel problem. For Continuous BGT, we propose approximation algorithms which achieve logarithmic approximation ratios. [ABSTRACT FROM AUTHOR]