scholarly journals Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

2021 ◽  
Author(s):  
Eswara Venkata Kumar Dhulipala
Keyword(s):  
Euclidean Distance ◽  
Travelling Salesman Problem ◽  
Dynamic Programming Algorithm ◽  
Constant Factor ◽  
Programming Algorithm ◽  
Worst Case ◽  
Angle Bisector ◽  
Set Of Points ◽  
The Given ◽  
Tour Length

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

scholarly journals Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

2021 ◽  
Author(s):  
Eswara Venkata Kumar Dhulipala
Keyword(s):  
Euclidean Distance ◽  
Travelling Salesman Problem ◽  
Dynamic Programming Algorithm ◽  
Constant Factor ◽  
Programming Algorithm ◽  
Worst Case ◽  
Angle Bisector ◽  
Set Of Points ◽  
The Given ◽  
Tour Length

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

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 Adding Edges for Maximizing Weighted Reachability

Algorithms ◽  
2020 ◽  
Vol 13 (3) ◽  
pp. 68 ◽  
Author(s):  
Federico Corò ◽  
Gianlorenzo D'Angelo ◽  
Cristina M. Pinotti
Keyword(s):  
Polynomial Time ◽  
Greedy Algorithms ◽  
Dynamic Programming Algorithm ◽  
Directed Acyclic Graphs ◽  
Constant Factor ◽  
Programming Algorithm ◽  
Single Source ◽  
Graph Augmentation ◽  
Set Size ◽  
Acyclic Graphs

In this paper, we consider the problem of improving the reachability of a graph. We approach the problem from a graph augmentation perspective, in which a limited set size of edges is added to the graph to increase the overall number of reachable nodes. We call this new problem the Maximum Connectivity Improvement (MCI) problem. We first show that, for the purpose of solve solving MCI, we can focus on Directed Acyclic Graphs (DAG) only. We show that approximating the MCI problem on DAG to within any constant factor greater than 1 − 1 / e is NP -hard even if we restrict to graphs with a single source or a single sink, and the problem remains NP -complete if we further restrict to unitary weights. Finally, this paper presents a dynamic programming algorithm for the MCI problem on trees with a single source that produces optimal solutions in polynomial time. Then, we propose two polynomial-time greedy algorithms that guarantee ( 1 − 1 / e ) -approximation ratio on DAGs with a single source, a single sink or two sources.

An Optimized Algorithm of Numerical Cutting-Path Control in Garment Manufacturing

2013 ◽  
Vol 796 ◽  
pp. 454-457 ◽  
Author(s):  
Jing Ye ◽  
Zhi Ge Chen
Keyword(s):  
Dynamic Programming ◽  
Ant Colony Algorithm ◽  
Travelling Salesman Problem ◽  
Dynamic Programming Algorithm ◽  
Ant Colony ◽  
Numerical Control ◽  
Programming Algorithm ◽  
Intelligence Algorithm ◽  
Hybrid Intelligence ◽  
Cutting Path

The garment cutting is a key process during the garment production. Most companies apply the manual labor or simple mechanical aids to achieve the goals. While these methods cost much time and labor. More and more automatic cutting equipment is applied to the garment cutting so as to save time, labor and materials. During the process of cutting, some problems are coming up, especially the cutting path. The cutting path of the garment numerical control cutter is regarded as generalized travelling salesman problem (GTSP). The garment contours can be regarded as the set of cities, and the nodes of a single contour can be regarded as cities. The cutter visits every contour exactly once. A hybrid intelligence algorithm was proposed to solve the problem. The ant colony algorithm was applied to a selected cutting path arbitrarily, an optimal contour sequence was found. Then the garment contour sequences shortest path was transformed into multi-segment graph shortest problem which is solved with the dynamic programming algorithm in order to optimize the knifes in-out point. The final optimal cutting path was constructed with ant colony optimization algorithm and dynamic programming algorithm. The practical application shows that the hybrid intelligence algorithm has satisfactory solution quality.

scholarly journals Automata Technique for The LCS Problem

2019 ◽  
Vol 35 (1) ◽  
pp. 21-37
Author(s):  
Trường Huy Nguyễn
Keyword(s):  
Dynamic Programming ◽  
Parallel Algorithms ◽  
Time Complexity ◽  
Dynamic Programming Algorithm ◽  
Longest Common Subsequence ◽  
Programming Algorithm ◽  
Worst Case ◽  
Common Subsequence ◽  
Input Strings ◽  
Upper Estimate

In this paper, we introduce two efficient algorithms in practice for computing the length of a longest common subsequence of two strings, using automata technique, in sequential and parallel ways. For two input strings of lengths m and n with m ≤ n, the parallel algorithm uses k processors (k ≤ m) and costs time complexity O(n) in the worst case, where k is an upper estimate of the length of a longest common subsequence of the two strings. These results are based on the Knapsack Shaking approach proposed by P. T. Huy et al. in 2002. Experimental results show that for the alphabet of size 256, our sequential and parallel algorithms are about 65.85 and 3.41m times faster than the standard dynamic programming algorithm proposed by Wagner and Fisher in 1974, respectively.

scholarly journals Data-Driven Distributionally Robust Stochastic Control of Energy Storage for Wind Power Ramp Management Using the Wasserstein Metric

Energies ◽  
2019 ◽  
Vol 12 (23) ◽  
pp. 4577
Author(s):  
Insoon Yang
Keyword(s):  
Energy Storage ◽  
Wind Power ◽  
Control Method ◽  
Empirical Distribution ◽  
Dynamic Programming Algorithm ◽  
Piecewise Linear Approximation ◽  
Programming Algorithm ◽  
Worst Case ◽  
Robust Solution ◽  
Distributionally Robust

The integration of wind energy into the power grid is challenging because of its variability, which causes high ramp events that may threaten the reliability and efficiency of power systems. In this paper, we propose a novel distributionally robust solution to wind power ramp management using energy storage. The proposed storage operation strategy minimizes the expected ramp penalty under the worst-case wind power ramp distribution in the Wasserstein ambiguity set, a statistical ball centered at an empirical distribution obtained from historical data. Thus, the resulting distributionally robust control policy presents a robust ramp management performance even when the future wind power ramp distribution deviates from the empirical distribution, unlike the standard stochastic optimal control method. For a tractable numerical solution, a duality-based dynamic programming algorithm is designed with a piecewise linear approximation of the optimal value function. The performance and utility of the proposed method are demonstrated and analyzed through case studies using the wind power data in the Bonneville Power Administration area for the year 2018.

scholarly journals Approximation Algorithms and an FPTAS for the Single Machine Problem with Biased Tardiness Penalty

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
G. Moslehi ◽  
K. Kianfar
Keyword(s):  
Dynamic Programming ◽  
Single Machine ◽  
Heuristic Algorithms ◽  
Dynamic Programming Algorithm ◽  
Performance Measure ◽  
Programming Algorithm ◽  
Scheduling Problems ◽  
Due Dates ◽  
Worst Case ◽  
Worst Case Ratio

This paper addresses a new performance measure for scheduling problems, entitled “biased tardiness penalty.” We study the approximability of minimum biased tardiness on a single machine, provided that all the due dates are equal. Two heuristic algorithms are developed for this problem, and it is shown that one of them has a worst-case ratio bound of 2. Then, we propose a dynamic programming algorithm and use it to design an FPTAS. The FPTAS is generated by cleaning up some states in the dynamic programming algorithm, and it requiresOn3/εtime.

scholarly journals Leadership in Congestion Games: Multiple User Classes and Non-Singleton Actions

2019 ◽  
Author(s):  
Alberto Marchesi ◽  
Matteo Castiglioni ◽  
Nicola Gatti
Keyword(s):  
Polynomial Time ◽  
Dynamic Programming Algorithm ◽  
Complete Characterization ◽  
Congestion Games ◽  
Mixed Integer ◽  
Programming Algorithm ◽  
Np Hard ◽  
Worst Case ◽  
Multiple User ◽  
Pure Strategies

We study the problem of finding Stackelberg equilibria in games with a massive number of players. So far, the only known game instances in which the problem is solved in polynomial time are some particular congestion games. However, a complete characterization of hard and easy instances is still lacking. In this paper, we extend the state of the art along two main directions. First, we focus on games where players' actions are made of multiple resources, and we prove that the problem is NP-hard and not in Poly-APX unless P = NP, even in the basic case in which players are symmetric, their actions are made of only two resources, and the cost functions are monotonic. Second, we focus on games with singleton actions where the players are partitioned into classes, depending on which actions they have available. In this case, we provide a dynamic programming algorithm that finds an equilibrium in polynomial time, when the number of classes is fixed and the leader plays pure strategies. Moreover, we prove that, if we allow for leader's mixed strategies, then the problem becomes NP-hard even with only four classes and monotonic costs. Finally, for both settings, we provide mixed-integer linear programming formulations, and we experimentally evaluate their scalability on both random game instances and worst-case instances based on our hardness reductions.

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