Packing and Covering with Non-Piercing Regions

In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks). We show that the local search algorithm that yields PTASs when the regions are disks [Aschner/Katz/Morgenstern/Yuditsky, WALCOM 2013; Gibson/Pirwani, 2005; Mustafa/Raman/Ray, 2015] can be extended to work for non-piercing regions. While such an extension is intuitive and natural, attempts to settle this question have failed even for pseudodisks. The techniques used for analysis when the regions are disks rely heavily on the underlying geometry, and do not extend to topologically defined settings such as pseudodisks. In order to prove our results, we introduce novel techniques that we believe will find applications in other problems. We then consider the Capacitated Region Packing problem. Here, the input consists of a set of points with capacities, and a set of regions. The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity. We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded. Our result settles a conjecture of Har-Peled (see Conclusion of [Har-Peled, SoCG 2014]) in the affirmative. The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane. Finally, we consider the Capacitated Point Packing problem. In this setting, the regions have capacities, and our objective is to find a maximum cardinality subset of points such that no region has more points than its capacity. We show that this problem admits a PTAS when the capacity is unity, extending one of the results of Ene et al. [Ene/Har-Peled/Raichel, SoCG 2012]

On Range Summary Queries

We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter ? and a set P of n points in ?^d where each point is assigned a color from a set C, and the goal is to build a structure such that given any geometric range ?, we can efficiently find a list of approximate heavy hitters in ??P, i.e., colors that appear at least ? |??P| times in ??P, as well as their frequencies with an additive error of ? |??P|. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence S of 1+1/? weights such that the i-th weight in S has approximate rank i?|??P|, meaning, rank i?|??P| up to an additive error of ?|??P|. Previously, optimal results were only known in 1D [Wei and Yi, 2011] but a few sub-optimal methods were available in higher dimensions [Peyman Afshani and Zhewei Wei, 2017; Pankaj K. Agarwal et al., 2012]. We study the problems for two important classes of geometric ranges: 3D halfspace and 3D dominance queries. It is known that many other important queries can be reduced to these two, e.g., 1D interval stabbing or interval containment, 2D three-sided queries, 2D circular as well as 2D k-nearest neighbors queries. We consider the real RAM model of computation where integer registers of size w bits, w = ?(log n), are also available. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time O(log n + 1/?). Note that as the output size is 1/?, after investing the initial O(log n) searching time, our structure takes on average O(1) time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra log_w(1/?) (resp. log log_w(1/?)) factor in 3D (resp. 2D). By spending extra log^O(1)(1/?) factors in both time and space, we can also support quantile queries. We remark that it is hopeless to achieve a similar query bound for dimensions 4 or higher unless significant advances are made in the data structure side of theory of geometric approximations

The development and validation of a scoring tool to predict the operative duration of elective laparoscopic cholecystectomy

Background: The ability to accurately predict operative duration has the potential to optimise theatre efficiency and utilisation, thus reducing costs and increasing staff and patient satisfaction. With laparoscopic cholecystectomy being one of the most commonly performed procedures worldwide, a tool to predict operative duration could be extremely beneficial to healthcare organisations. Methods: Data collected from the CholeS study on patients undergoing cholecystectomy in UK and Irish hospitals between 04/2014 and 05/2014 were used to study operative duration. A multivariable binary logistic regression model was produced in order to identify significant independent predictors of long (> 90 min) operations. The resulting model was converted to a risk score, which was subsequently validated on second cohort of patients using ROC curves. Results: After exclusions, data were available for 7227 patients in the derivation (CholeS) cohort. The median operative duration was 60 min (interquartile range 45–85), with 17.7% of operations lasting longer than 90 min. Ten factors were found to be significant independent predictors of operative durations > 90 min, including ASA, age, previous surgical admissions, BMI, gallbladder wall thickness and CBD diameter. A risk score was then produced from these factors, and applied to a cohort of 2405 patients from a tertiary centre for external validation. This returned an area under the ROC curve of 0.708 (SE = 0.013, p  90 min increasing more than eightfold from 5.1 to 41.8% in the extremes of the score. Conclusion: The scoring tool produced in this study was found to be significantly predictive of long operative durations on validation in an external cohort. As such, the tool may have the potential to enable organisations to better organise theatre lists and deliver greater efficiencies in care

On Range Summary Queries

We study the query version of the approximate heavy hitter and quantile problems. In the former problem, the input is a parameter $\varepsilon$ and a set $P$ of $n$ points in $\mathbb{R}^d$ where each point is assigned a color from a set $C$, and we want to build a structure s.t. given any geometric range $\gamma$, we can efficiently find a list of approximate heavy hitters in $\gamma\cap P$, i.e., colors that appear at least $\varepsilon |\gamma \cap P|$ times in $\gamma \cap P$, as well as their frequencies with an additive error of $\varepsilon |\gamma \cap P|$. In the latter problem, each point is assigned a weight from a totally ordered universe and the query must output a sequence $S$ of $1+1/\varepsilon$ weights s.t. the $i$-th weight in $S$ has approximate rank $i\varepsilon|\gamma\cap P|$, meaning, rank $i\varepsilon|\gamma\cap P|$ up to an additive error of $\varepsilon|\gamma\cap P|$. Previously, optimal results were only known in 1D [WY11] but a few sub-optimal methods were available in higher dimensions [AW17, ACH+12]. We study the problems for 3D halfspace and dominance queries. We consider the real RAM model with integer registers of size $w=\Theta(\log n)$ bits. For dominance queries, we show optimal solutions for both heavy hitter and quantile problems: using linear space, we can answer both queries in time $O(\log n + 1/\varepsilon)$. Note that as the output size is $\frac{1}{\varepsilon}$, after investing the initial $O(\log n)$ searching time, our structure takes on average $O(1)$ time to find a heavy hitter or a quantile! For more general halfspace heavy hitter queries, the same optimal query time can be achieved by increasing the space by an extra $\log_w\frac{1}{\varepsilon}$ (resp. $\log\log_w\frac{1}{\varepsilon}$) factor in 3D (resp. 2D). By spending extra $\log^{O(1)}\frac{1}{\varepsilon}$ factors in time and space, we can also support quantile queries.Comment: To appear in ICALP'2

Packing and Covering with Non-Piercing Regions

In this paper, we design the first polynomial time approximation schemes for the Set Cover and Dominating Set problems when the underlying sets are non-piercing regions (which include pseudodisks). Earlier, PTASs were known only in the setting where the regions were disks. These techniques relied heavily on the circularity of the disks. We develop new techniques to show that a simple local search algorithm yields a PTAS for the problems on non-piercing regions. We then consider the Capacitated Region Packing problem. Here, the input consists of a set of points with capacities, and a set of regions. The objective is to pick a maximum cardinality subset of regions so that no point is covered by more regions than its capacity. We show that this problem admits a PTAS when the regions are k-admissible regions (pseudodisks are 2-admissible), and the capacities are bounded by some constant. Our result settles a conjecture of Har-Peled from 2014 in the affirmative. The conjecture was for a weaker version of the problem, namely when the regions are pseudodisks, the capacities are uniform, and the point set consists of all points in the plane. Finally, we consider the Capacitated Point Packing problem. In this setting, the regions have capacities, and our objective is to find a maximum cardinality subset of points such that no region has more points than its capacity. We show that this problem admits a PTAS when the capacity is unity. This extends a result of Ene et al. from 2012

Demand Hitting and Covering of Intervals

Hitting and Covering problems have been extensively studied in the last few decades and have applications in diverse areas. While the hitting and covering problems are NP-hard for most settings, they are polynomial solvable for intervals. Demand hitting is a generalization of the hitting problem, where there is an integer demand associated with each object, and the demand hitting set must contain at least as many points as the demand of each object. In this paper, we consider the demand hitting and covering problems for intervals that have no containment. For the unweighted setting, we give a simple greedy algorithm. In the weighted setting, we model this problem as a min-cost max flow problem using a non-trivial reduction and solve it using standard flow algorithms