scholarly journals Constant-factor approximation algorithms for the Traveling Salesperson Problem for Dubins' vehicle

2006 ◽  
Author(s):  
K. Savla ◽  
E. Frazzoli ◽  
F. Bullo
Keyword(s):  
Approximation Algorithms ◽  
Constant Factor ◽  
Traveling Salesperson Problem ◽  
Dubins Vehicle

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

scholarly journals A Unified Model and Algorithms for Temporal Map Labeling

Algorithmica ◽  
2020 ◽  
Vol 82 (10) ◽  
pp. 2709-2736
Author(s):  
Andreas Gemsa ◽  
Benjamin Niedermann ◽  
Martin Nöllenburg
Keyword(s):  
Approximation Algorithms ◽  
Traffic Control ◽  
Constant Factor ◽  
Maximization Problem ◽  
Fixed Position ◽  
Time Intervals ◽  
Map Labeling ◽  
Polynomial Time Algorithms ◽  
Dynamic Map ◽  
Flexible Model

Abstract We consider map labeling for the case that a map undergoes a sequence of operations such as rotation, zoom and translation over a specified time span. We unify and generalize several previous models for dynamic map labeling into one versatile and flexible model. In contrast to previous research, we completely abstract from the particular operations and express the labeling problem as a set of time intervals representing the labels’ presences, activities and conflicts. One of the model’s strength is manifested in its simplicity and broad range of applications. In particular, it supports label selection both for map features with fixed position as well as for moving entities (e.g., for tracking vehicles in logistics or air traffic control). We study the active range maximization problem in this model. We prove that the problem is -complete and [1]-hard, and present constant-factor approximation algorithms. In the restricted, yet practically relevant case that no more than k labels can be active at any time, we give polynomial-time algorithms as well as constant-factor approximation algorithms.

scholarly journals The Dubins Traveling Salesperson Problem With Stochastic Dynamics

2013 ◽  
Author(s):  
Ross P. Anderson ◽  
Dejan Milutinović
Keyword(s):  
Feedback Control ◽  
Bellman Equation ◽  
Stochastic Dynamics ◽  
Numerical Results ◽  
Traveling Salesperson Problem ◽  
Expected Time ◽  
Hamilton Jacobi Bellman Equation ◽  
Dubins Vehicle ◽  
Robotic Vehicle ◽  
Hamilton Jacobi Bellman

Motivated by applications in which a nonholonomic robotic vehicle should sequentially hit a series of waypoints in the presence of stochastic drift, we formulate a new version of the Dubins vehicle traveling salesperson problem. In our approach, we first compute the minimum expected time feedback control to hit one waypoint based on the Hamilton-Jacobi-Bellman equation. Next, minimum expected times associated with the control are used to construct a traveling salesperson problem based on a waypoint hitting angle discretization. We provide numerical results illustrating our solution and analyze how the stochastic drift affects the solution.

EFFICIENT APPROXIMATION ALGORITHMS FOR TWO-LABEL POINT LABELING

2001 ◽  
Vol 11 (04) ◽  
pp. 455-464 ◽  
Author(s):  
BINHAI ZHU ◽  
C. K. POON
Keyword(s):  
Approximation Algorithms ◽  
Mild Condition ◽  
Distinct Point ◽  
Constant Factor ◽  
Approximation Schemes ◽  
Uniform Size ◽  
Map Labeling ◽  
Label Point ◽  
Efficient Approximation

In this paper we propose and study two practical variations of the map labeling problem: Given a set S of n distinct (point) sites in the plane, label each site with: (1) a pair of non-intersecting squares of maximum possible size, (2) a pair of non-intersecting circles of maximum possible size (all the squares and circles are topologically open and are of uniform size). Almost nothing has been done before in this aspect, i.e., multi-label map labeling. We obtain constant-factor approximation algorithms for these problems. We also study bicriteria approximation schemes for these problems under a mild condition.

CUTTING OUT POLYGONS WITH LINES AND RAYS

2006 ◽  
Vol 16 (02n03) ◽  
pp. 227-248 ◽  
Author(s):  
OVIDIU DAESCU ◽  
JUN LUO
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Convex Polygon ◽  
Linear Time ◽  
Constant Factor ◽  
Open Problems ◽  
Cutting Out ◽  
Cutting Sequence

We present approximation algorithms for cutting out a polygon P with n vertices from another convex polygon Q with m vertices by line cuts and ray cuts. For line cuts we require both P and Q are convex while for ray cuts we require Q is convex and P is ray cuttable. Our results answer a number of open problems and are either the first solutions or significantly improve over previously known solutions. For the line cutting version, we prove a key property that leads to a simple, constant factor approximation algorithm. For the ray cutting version, we prove it is possible to compute in almost linear time a cutting sequence that is an O( log 2 n)-factor approximation of an optimal cutting sequence. No algorithms were previously known for the ray cutting version.

ON THE APPROXIMABILITY OF MAXIMUM AND MINIMUM EDGE CLIQUE PARTITION PROBLEMS

2007 ◽  
Vol 18 (02) ◽  
pp. 217-226 ◽  
Author(s):  
ANDERS DESSMARK ◽  
JESPER JANSSON ◽  
ANDRZEJ LINGAS ◽  
EVA-MARTA LUNDELL ◽  
MIA PERSSON
Keyword(s):  
Approximation Algorithms ◽  
Undirected Graph ◽  
Constant Factor ◽  
Graph Partition ◽  
Maximum Cliques ◽  
Graph Classes ◽  
Clustering Problems

We consider the following clustering problems: given an undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

scholarly journals Approximation Algorithms for Energy, Reliability, and Makespan Optimization Problems

2016 ◽  
Vol 26 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Guillaume Aupy ◽  
Anne Benoit
Keyword(s):  
Energy Consumption ◽  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Optimization Problems ◽  
Linear Chain ◽  
Constant Factor ◽  
Task Graph ◽  
Linear Chains ◽  
Energy Reliability ◽  
Independent Tasks

We consider the problem of scheduling an application on a parallel computational platform. The application is a particular task graph, either a linear chain of tasks, or a set of independent tasks. The platform is made of identical processors, whose speed can be dynamically modified. It is also subject to failures: if a processor is slowed down to decrease the energy consumption, it has a higher chance to fail. Therefore, the scheduling problem requires us to re-execute or replicate tasks (i.e., execute twice the same task, either on the same processor, or on two distinct processors), in order to increase the reliability. It is a tri-criteria problem: the goal is to minimize the energy consumption, while enforcing a bound on the total execution time (the makespan), and a constraint on the reliability of each task. Our main contribution is to propose approximation algorithms for linear chains of tasks and independent tasks. For linear chains, we design a fully polynomial-time approximation scheme. However, we show that there exists no constant factor approximation algorithm for independent tasks, unless P=NP, and we propose in this case an approximation algorithm with a relaxation on the makespan constraint.

Deals or No Deals: Contract Design for Online Advertising

2021 ◽  
Author(s):  
Anthony Kim ◽  
Vahab Mirrokni ◽  
Hamid Nazerzadeh
Keyword(s):  
Approximation Algorithms ◽  
Real Time ◽  
Online Advertising ◽  
Contract Design ◽  
Constant Factor ◽  
Common Value ◽  
Distributional Information ◽  
Using Data ◽  
Np Hardness ◽  
Better Than

We present a formal study of first-look and preferred deals that are a recently introduced generation of contracts for selling online advertisements, which generalize traditional reservation contracts and are suitable for advertisers with advanced targeting capabilities. Under these deals, one or more advertisers gain priority access to an inventory of impressions before others and can choose to purchase in real time. In particular, we propose constant-factor approximation algorithms for maximizing the revenue that can be obtained from these deals when offered to all or a subset of the advertisers, whose valuation distributions can be independent or correlated through a common value component. We evaluate our algorithms using data from Google’s ad exchange platform and show they perform better than the approximation guarantees and obtain significantly higher revenue than auctions; in certain cases, the observed revenue is 85%–96% of the optimal revenue achievable. We also prove the NP-hardness of designing deals when advertisers’ valuations are arbitrarily correlated and the optimality of menus of deals among a certain class of selling mechanisms in an incomplete distributional information setting.

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