We introduce a novel coverage problem that arises in aerial surveying applications. The goal is to compute a shortest path that visits a given set of cones. The apex of each cone is restricted to lie on the ground pla...
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We introduce a novel coverage problem that arises in aerial surveying applications. The goal is to compute a shortest path that visits a given set of cones. The apex of each cone is restricted to lie on the ground plane. The common angle alpha of the cones represent the field of view of the onboard camera. The cone heights, which can be varying, correspond with the desired observation quality (e.g. resolution). This problem is a novel variant of the traveling salesman problem with neighborhoods (TSPN). We name it Cone-TSPN. Our main contribution is a polynomial time approximation algorithm for Cone-TPSN. We analyze its theoretical performance and show that it returns a solution whose length is at most O(1+log(hmax/hmin)) times the length of the optimal solution where hmax and hmin are the heights of the tallest and shortest input cones, *** demonstrate the use of our algorithm in a representative precision agriculture application. Wefurther study its performance in simulation using randomly generated cone sets. Our results indicate that the performance of our algorithm is superior to standard solutions.
We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-f...
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We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.
We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic ...
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We consider stochastic settings for clustering, and develop provably-good approximation algorithms for a number of these notions. These algorithms yield better approximation ratios compared to the usual deterministic clustering setting. Additionally, they offer a number of advantages including clustering which is fairer and has better long-term behavior for each user. In particular, they ensure that every user is guaranteed to get good service (on average). We also complement some of these with impossibility results.
Consider a set of n jobs and in uniform parallel machines, where each job has a length p(j )is an element of Q(+) and each machine has a speed s(i) is an element of Q(+). The goal of the graph balancing problem with s...
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Consider a set of n jobs and in uniform parallel machines, where each job has a length p(j )is an element of Q(+) and each machine has a speed s(i) is an element of Q(+). The goal of the graph balancing problem with speeds is to schedule each job j non-preemptively on one of a subset M-j of at most 2 machines so that the makespan is minimized. This is a NP-hard special case of the makespan minimization problem on unrelated parallel machines, where for the latter a longstanding open problem is to find an approximation algorithm with approximation ratio better than 2. Our main contribution is an approximation algorithm for the graph balancing problem with two speeds and two job lengths with approximation ratio (root 65+7)/8 approximate to 1.88278. In addition, we consider when every M-j has no cardinality constraints, this is the restricted assignment problem in the uniform parallel machine setting. We present a simple (2 - alpha/beta)-approximation algorithm in this case when every job has one of two job lengths p(j) is an element of {alpha, beta} where alpha < beta.
Millimeter-wave (mmWave) communication is a promising technology to cope with the exponential increase in 5G data traffic. Such networks typically require a very dense deployment of base stations. A subset of those, s...
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Millimeter-wave (mmWave) communication is a promising technology to cope with the exponential increase in 5G data traffic. Such networks typically require a very dense deployment of base stations. A subset of those, so-called macro base stations, feature high-bandwidth connection to the core network, while relay base stations are connected wirelessly. To reduce cost and increase flexibility, wireless backhauling is needed to connect both macro to relay as well as relay to relay base stations. The characteristics of mmWave communication mandates new paradigms for routing and scheduling. The paper investigates scheduling algorithms under different interference models. To showcase the scheduling methods, we study the maximum throughput fair scheduling problem. Yet the proposed algorithms can be easily extended to other problems. For a full-duplex network under the no interference model, we propose an efficient polynomial-time scheduling method, the schedule-oriented optimization. Further, we prove that the problem is NP-hard if we assume pairwise link interference model or half-duplex radios. Fractional weighted coloring based approximation algorithms are proposed for these NP-hard cases. Moreover, the approximation algorithm parallel data stream scheduling is proposed for the case of half-duplex network under the no interference model. It has better approximation ratio than the fractional weighted coloring based algorithms and even attains the optimal solution for the special case of uniform orthogonal backhaul networks.
The vertex k-center problem is a classical NP-Hard optimization problem with application to Facility Location and Clustering among others. This problem consists in finding a subset C subset of V of an input graph G = ...
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The vertex k-center problem is a classical NP-Hard optimization problem with application to Facility Location and Clustering among others. This problem consists in finding a subset C subset of V of an input graph G = (V, E), such that the distance from the farthest vertex in V to its nearest center in C is minimized, where vertical bar C vertical bar <= k, with k is an element of Z(+) as part of the input. Many heuristics, metaheuristics, approximation algorithms, and exact algorithms have been developed for this problem. This paper presents an analytical study and experimental evaluation of the most representative approximation algorithms for the vertex k-center problem. For each of the algorithms under consideration and using a common notation, we present proofs of their corresponding approximation guarantees as well as examples of tight instances of such approximation bounds, including a novel tight example for a 3-approximation algorithm. Lastly, we present the results of extensive experiments performed over de facto benchmark data sets for the problem which includes instances of up to 71009 vertices.
We consider the single-period joint assortment and inventory planning problem with stochastic demand and dynamic substitution across products, motivated by applications in highly differentiated markets, such as online...
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We consider the single-period joint assortment and inventory planning problem with stochastic demand and dynamic substitution across products, motivated by applications in highly differentiated markets, such as online retailing and airlines. This class of problems is known to be notoriously hard to deal with from a computational standpoint. In fact, prior to the present paper, only a handful of modeling approaches were shown to admit provably good algorithms, at the cost of strong restrictions on customers' choice outcomes. Our main contribution is to provide the first efficient algorithms with provable performance guarantees for a broad class of dynamic assortment optimization models. Under general rank-based choice models, our approximation algorithm is best possible with respect to the price parameters, up to lower-order terms. In particular, we obtain a constant-factor approximation under horizontal differentiation, where product prices are uniform. In more structured settings, where the customers' ranking behavior is motivated by price and quality cues, we derive improved guarantees through tailor-made algorithms. In extensive computational experiments, our approach dominates existing heuristics in terms of revenue performance, as well as in terms of speed, given the myopic nature of our methods. From a technical perspective, we introduce a number of novel algorithmic ideas of independent interest, and unravel hidden relations to submodular maximization.
We study the minimal decomposition of octilinear polygons with holes into octilinear triangles and rectangles. This new problem is relevant in the context of modern electronic CAD systems, where the generation and pro...
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We study the minimal decomposition of octilinear polygons with holes into octilinear triangles and rectangles. This new problem is relevant in the context of modern electronic CAD systems, where the generation and propagation of electromagnetic noise into multi layer PCBs has to be detected. It is a generalization of a problem deeply investigated: the minimal decomposition of rectilinear polygons into rectangles. We show that the new problem is NP-hard. We also show the NP-hardness of a related problem, that is the decomposition of an octilinear polygon with holes into octilinear convex polygons. For both problems, we propose efficient approximation algorithms. (C) 2019 Elsevier B.V. All rights reserved.
We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the 2-dimensional plane. Among other results, we give an O(root n)-approximation algorithm for the problem o...
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We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the 2-dimensional plane. Among other results, we give an O(root n)-approximation algorithm for the problem of fi nding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. We give an improved (O) over tilde (n(1/3)) approximation for the case of metrics induced by unweighted trees.
We study the covering-type k-violation linear program where at most k of the constraints can be violated. This problem is formulated as a mixed integer program and known to be strongly NP-hard. In this paper, we prese...
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We study the covering-type k-violation linear program where at most k of the constraints can be violated. This problem is formulated as a mixed integer program and known to be strongly NP-hard. In this paper, we present a simple (k + 1)approximation algorithm using a natural LP relaxation. We also show that the integrality gap of the LP relaxation is k + 1. This implies we can not get better approximation algorithms when we use the LP-relaxation as a lower bound of the optimal value.
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