Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17/9 + \epsilon) < 1.89 approximation, which improves to (558/325 + \epsilon) < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. As a second major and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also, in this case, the best-known polynomial-time approximation factor (even for the cardinality case) is (2 + \epsilon) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3/2 + \epsilon)-approximation for 2DKR, which improves to (4/3 + \epsilon) in the cardinality case.Comment: 64pages, full version of FOCS 2017 pape

Emergency response in complex buildings: Automated selection of safest and balanced routes

The extreme importance of emergency response in complex buildings during natural and human-induced disasters has been widely acknowledged. In particular, there is a need for efficient algorithms for finding safest evacuation routes, which would take into account the 3-D structure of buildings, their relevant semantics, and the nature and shape of hazards. In this article, we propose algorithms for safest routes and balanced routes in buildings, where an extreme event with many epicenters is occurring. In a balanced route, a trade-off between route length and hazard proximity is made. The algorithms are based on a novel approach that integrates a multiattribute decision-making technique, Dijkstra's classical algorithm and the introduced hazard proximity numbers, hazard propagation coefficient and proximity index for a route

Energy management of three-dimensional minimum-time intercept

A real-time computer algorithm to control and optimize aircraft flight profiles is described and applied to a three-dimensional minimum-time intercept mission

Green multimedia: informing people of their carbon footprint through two simple sensors

In this work we discuss a new, but highly relevant, topic to the multimedia community; systems to inform individuals of their carbon footprint, which could ultimately effect change in community carbon footprint-related activities. The reduction of carbon emissions is now an important policy driver of many governments, and one of the major areas of focus is in reducing the energy demand from the consumers i.e. all of us individually. In terms of CO2 generated from energy consumption, there are three predominant factors, namely electricity usage, thermal related costs, and transport usage. Standard home electricity and heating sensors can be used to measure the former two aspects, and in this paper we evaluate a novel technique to estimate an individual's transport-related carbon emissions through the use of a simple wearable accelerometer. We investigate how providing this novel estimation of transport-related carbon emissions through an interactive web site and mobile phone app engages a set of users in becoming more aware of their carbon emissions. Our evaluations involve a group of 6 users collecting 25 million accelerometer readings and 12.5 million power readings vs. a control group of 16 users collecting 29.7 million power readings

Minimax and Maximin Fitting of Geometric Objects to Sets of Points

This thesis addresses several problems in the facility location sub-area of computational geometry. Let S be a set of n points in the plane. We derive algorithms for approximating S by a step function curve of size k \u3c n, i.e., by an x-monotone orthogonal polyline ℜ with k \u3c n horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in S to the horizontal segment directly above or below it. We consider two types of problems: min-ε, where the goal is to minimize the error for a given number of horizontal segments k and min-#, where the goal is to minimize the number of segments for a given allowed error ε. After O(n) preprocessing time, we solve instances of the latter in O(min{k log n, n}) time per instance. We can then solve the former problem in O(min{n2, nk log n}) time. Both algorithms require O(n) space. The second contribution is a heuristic for the min-ε problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k - 1 segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in O(n log n) time and O(n) space. Then, we present an exact algorithm for the weighted version of this problem that runs in O(n2) time and generalize the heuristic to handle weights at the expense of an additional log n factor. At this point, a randomized algorithm that runs in O(n log2 n) expected time for the unweighted version is presented. It easily generalizes to the weighted case, though at the expense of an additional log n factor. Finally, we treat the maximin problem and present an O(n3 log n) solution to the problem of finding the furthest separating line through a set of weighted points. We conclude with solutions to the obnoxious wedge problem: an O(n2 log n) algorithm for the general case of a wedge with its apex on the boundary of the convex hull of S and an O(n2) algorithm for the case of the apex of a wedge coming from the input set S

Pedestrian Leadership and Egress Assistance Simulation Environment (PLEASE)

Over the past decade, researchers have been developing new ways to model pedestrian egress especially in emergency situations. The traditional methods of modeling pedestrian egress, including ow-based modeling and cellular automata, have been shown to be poor models of human behavior at an individual level, as well as failing to capture many important group social behaviors of pedestrians. This has led to the exploration of agent-based modeling for crowd simulations including those involving pedestrian egress. Using this model, we evaluate different heuristic functions for predicting good egress routes for a variety of real building layouts. We also introduce reinforcement learning as a means to represent individualized pedestrian route knowledge. Finally, we implement a group formation technique, which allows pedestrians in a group to share route knowledge and reach a consensus in route selection. Using the group formation technique, we consider the effects such knowledge sharing and consensus mechanisms have on pedestrian egress times