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 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 Inapproximability of Treewidth and Related Problems

2014 ◽  
Vol 49 ◽  
pp. 569-600 ◽  
Author(s):  
Y. Wu ◽  
P. Austrin ◽  
T. Pitassi ◽  
D. Liu
Keyword(s):  
Graphical Models ◽  
Fundamental Problem ◽  
Interval Graph ◽  
Directed Acyclic Graphs ◽  
Constant Factor ◽  
Linear Arrangement ◽  
Bounded Treewidth ◽  
Graph Problems ◽  
Acyclic Graphs ◽  
Small Set

Graphical models, such as Bayesian Networks and Markov networks play an important role in artificial intelligence and machine learning. Inference is a central problem to be solved on these networks. This, and other problems on these graph models are often known to be hard to solve in general, but tractable on graphs with bounded Treewidth. Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models. In this paper, we study the approximability of a number of graph problems: Treewidth and Pathwidth of graphs, Minimum Fill-In, One-Shot Black (and Black-White) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as Minimum Cut Linear Arrangement and Interval Graph Completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are NP-hard to approximate to within any constant factor in polynomial time.

scholarly journals Ubiquitous Wireless Power Transfer for Multiple Mobile Devices

2021 ◽  
Author(s):  
Alexander Decker de Souza ◽  
Luiz Filipe Menezes Vieira ◽  
Marcos Augusto Menezes Vieira
Keyword(s):  
Dynamic Programming ◽  
Mobile Devices ◽  
Linear Time ◽  
Greedy Algorithms ◽  
Dynamic Programming Algorithm ◽  
Wireless Power Transfer ◽  
Programming Algorithm ◽  
Power Transfer ◽  
Wireless Power ◽  
Programming Algorithms

We propose two new computational problems associated with the charging of mobile devices using wireless power transfer via magnetic induction. Algorithms for these problems may enable ubiquitous charging, meaning the user is no longer required to be aware of the devices charging processes. We prove both problems as being NP-Hard and propose three dynamic programming algorithms to solve them in linear time regarding the size of the time horizon. We also propose three greedy algorithms for the problems. Experiments indicate that the best dynamic-programming algorithm among those proposed reaches between 89% and 97% of effectiveness, while the best greedy reaches between 74% and 92%, depending on the considered scenario.

scholarly journals Different Approximation Algorithms for Channel Scheduling in Wireless Networks

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Qiufen Ni ◽  
Chuanhe Huang ◽  
Panos M. Pardalos ◽  
Jia Ye ◽  
Bin Fu
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Single Channel ◽  
Channel Allocation ◽  
Greedy Algorithms ◽  
Constant Factor ◽  
Scheduling Problem ◽  
Channel Scheduling ◽  
Speed Up

We introduce a new two-side approximation method for the channel scheduling problem, which controls the accuracy of approximation in two sides by a pair of parameters f , g . We present a series of simple and practical-for-implementation greedy algorithms which give constant factor approximation in both sides. First, we propose four approximation algorithms for the weighted channel allocation problem: 1. a greedy algorithm for the multichannel with fixed interference radius scheduling problem is proposed and an one side O 1 -IS-approximation is obtained; 2. a greedy O 1 , O 1 -approximation algorithm for single channel with fixed interference radius scheduling problem is presented; 3. we improve the existing algorithm for the multichannel scheduling and show an E O d / ε time 1 − ϵ -approximation algorithm; 4. we speed up the polynomial time approximation scheme for single-channel scheduling through merging two algorithms and show a 1 − ϵ , O 1 -approximation algorithm. Next, we study two polynomial time constant factor greedy approximation algorithms for the unweighted channel allocation with variate interference radius. A greedy O 1 -approximation algorithm for the multichannel scheduling problem and an O 1 , O 1 -approximation algorithm for single-channel scheduling problem are developed. At last, we do some experiments to verify the effectiveness of our proposed methods.

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 Prograph Based Analysis of Single Source Shortest Path Problem with Few Distinct Positive Lengths

2011 ◽  
Vol 1 (4) ◽  
pp. 90-97
Author(s):  
B. Bhowmik ◽  
S. Nag Chowdhury
Keyword(s):  
Shortest Path ◽  
Directed Graphs ◽  
Algorithm Design ◽  
Dynamic Programming Algorithm ◽  
Programming Algorithm ◽  
Single Source ◽  
Comparison Study ◽  
Design Technique ◽  
Shortest Path Problems ◽  
Finite Directed Graphs

In this paper we propose an experimental study model S3P2 of a fast fully dynamic programming algorithm design technique in finite directed graphs with few distinct nonnegative real edge weights. The Bellman-Ford’s approach for shortest path problems has come out in various implementations. In this paper the approach once again is re-investigated with adjacency matrix selection in associate least running time. The model tests proposed algorithm against arbitrarily but positive valued weighted digraphs introducing notion of  Prograph that speeds up finding the shortest path over previous implementations. Our experiments have established abstract results with the intention that the proposed algorithm can consistently dominate other existing algorithms for Single Source Shortest Path Problems. A comparison study is also shown among Dijkstra’s algorithm, Bellman-Ford algorithm, and our algorithm.

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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