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.

scholarly journals Single-machine Pareto-scheduling with multiple weighting vectors for minimizing the total weighted late works

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Shuen Guo ◽  
Zhichao Geng ◽  
Jinjiang Yuan
Keyword(s):  
Polynomial Time ◽  
Single Machine ◽  
Approximation Scheme ◽  
Pareto Frontier ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Polynomial Time Approximation Scheme ◽  
Late Work ◽  
Weighting Vector ◽  
Late Works

<p style='text-indent:20px;'>In this paper, we study the single-machine Pareto-scheduling of jobs with multiple weighting vectors for minimizing the total weighted late works. Each weighting vector has its corresponding weighted late work. The goal of the problem is to find the Pareto-frontier for the weighted late works of the multiple weighting vectors. When the number of weighting vectors is arbitrary, it is implied in the literature that the problem is unary NP-hard. Then we concentrate on our research under the assumption that the number of weighting vectors is a constant. For this problem, we present a dynamic programming algorithm running in pseudo-polynomial time and a fully polynomial-time approximation scheme (FPTAS).</p>

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 Solving methods for the quay crane scheduling problem at port of Tripoli-Lebanon.

2020 ◽  
Author(s):  
Ali Skaf ◽  
Sid Lamrous ◽  
Zakaria Hammoudan ◽  
Marie-Ange Manier
Keyword(s):  
Dynamic Programming Algorithm ◽  
Heuristic Method ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Programming Algorithm ◽  
Scheduling Problem ◽  
Exact Methods ◽  
Three Solutions ◽  
Quay Crane Scheduling ◽  
Crane Scheduling

The quay crane scheduling problem (QCSP) is a global problem and all ports around the world seek to solve it, to get an acceptable time of unloading containers from the vessels or loading containers to the vessels and therefore reducing the docking time in the terminal. This paper proposes three solutions for the QCSP in port of Tripoli-Lebanon, two exact methods which are the mixed integer linear programming and the dynamic programming algorithm, to obtain the optimal solution and one heuristic method which is the genetic algorithm, to obtain near optimal solution within an acceptable CPU time. The main objective of these methods is to minimize the unloading or the loading time of the containers and therefore reduce the waiting time of the vessels in the terminals. We tested and validated our methods for small and large random instances. Finally, we compared the results obtained with these methods for some real instances in the port of Tripoli-Lebanon.

scholarly journals Dynamic Programming and Heuristic for Stochastic Uncapacitated Lot-Sizing Problems with Incremental Quantity Discount

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Yuli Zhang ◽  
Shiji Song ◽  
Cheng Wu ◽  
Wenjun Yin
Keyword(s):  
Dynamic Programming ◽  
Lot Sizing ◽  
Dynamic Programming Algorithm ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Programming Algorithm ◽  
Quantity Discount ◽  
Path Property ◽  
Commercial Solver

The stochastic uncapacitated lot-sizing problems with incremental quantity discount have been studied in this paper. First, a multistage stochastic mixed integer model is established by the scenario analysis approach and an equivalent reformulation is obtained through proper relaxation under the decreasing unit order price assumption. The proposed reformulation allows us to extend the production-path property to this framework, and furthermore we provide a more accurate characterization of the optimal solution. Then, a backward dynamic programming algorithm is developed to obtain the optimal solution and considering its exponential computation complexity in term of time stages, we design a new rolling horizon heuristic based on the proposed property. Comparisons with the commercial solver CPLEX and other heuristics indicate better performance of our proposed algorithms in both quality of solution and run 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.

scholarly journals AGV Scheduling Optimization for Medical Waste Sorting System

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Xueting He ◽  
Hao Quan ◽  
Wanlong Lin ◽  
Weiliang Deng ◽  
Zheyi Tan
Keyword(s):  
Optimization Problem ◽  
Variable Neighborhood Search ◽  
Programming Model ◽  
Dynamic Programming Algorithm ◽  
Medical Waste ◽  
Mixed Integer ◽  
Programming Algorithm ◽  
Neighborhood Search ◽  
Operational Flow ◽  
Sorting System

The dramatic increase in medical waste has put a severe strain on sorting operations. Traditional manual order picking is extremely susceptible to infection spread among workers and picking errors, while automated medical waste sorting systems can handle large volumes of medical waste efficiently and reliably. This paper investigates the optimization problem in the automated medical waste sorting system by considering the operational flow of medical waste. For this purpose, a mixed-integer programming model is developed to optimize the assignment among medical waste, presorting stations, and AGVs. An effective variable neighborhood search based on dynamic programming algorithm is proposed, and extensive numerical experiments are conducted. It is found that the proposed algorithm can efficiently solve the optimization problem, and the sensitivity analysis gives recommendations for the speed setting of the conveyor.

scholarly journals Parallel-Batch Scheduling and Transportation Coordination with Waiting Time Constraint

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Hua Gong ◽  
Daheng Chen ◽  
Ke Xu
Keyword(s):  
Waiting Time ◽  
Processing Time ◽  
Optimal Algorithm ◽  
Dynamic Programming Algorithm ◽  
Time Constraint ◽  
Batch Scheduling ◽  
Programming Algorithm ◽  
Np Hard ◽  
Processing Times ◽  
Equal Processing Times

This paper addresses a parallel-batch scheduling problem that incorporates transportation of raw materials or semifinished products before processing with waiting time constraint. The orders located at the different suppliers are transported by some vehicles to a manufacturing facility for further processing. One vehicle can load only one order in one shipment. Each order arriving at the facility must be processed in the limited waiting time. The orders are processed in batches on a parallel-batch machine, where a batch contains several orders and the processing time of the batch is the largest processing time of the orders in it. The goal is to find a schedule to minimize the sum of the total flow time and the production cost. We prove that the general problem is NP-hard in the strong sense. We also demonstrate that the problem with equal processing times on the machine is NP-hard. Furthermore, a dynamic programming algorithm in pseudopolynomial time is provided to prove its ordinarily NP-hardness. An optimal algorithm in polynomial time is presented to solve a special case with equal processing times and equal transportation times for each order.

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 A Heuristic for a Mixed Integer Program using the Characteristic Equation Approach

2017 ◽  
Vol 2 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Philimon Nyamugure ◽  
Elias Munapo ◽  
‘Maseka Lesaoana ◽  
Santosh Kumar
Keyword(s):  
Characteristic Equation ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Np Hard ◽  
Equation Approach ◽  
Mip Model ◽  
Polynomial Time Algorithms

While most linear programming (LP) problems can be solved in polynomial time, pure and mixed integer problems are NP-hard and there are no known polynomial time algorithms to solve these problems. A characteristic equation (CE) was developed to solve a pure integer program (PIP). This paper presents a heuristic that generates a feasible solution along with the bounds for the NP-hard mixed integer program (MIP) model by solving the LP relaxation and the PIP, using the CE.

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.

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