APPROXIMATING THE SPANNING k-TREE FOREST PROBLEM

2012 ◽  
Vol 23 (07) ◽  
pp. 1543-1554
Author(s):  
CHUNG-SHOU LIAO ◽  
LOUXIN ZHANG
Keyword(s):  
Combinatorial Optimization ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Directed Graphs ◽  
Algorithmic Problem ◽  
Central Vertex ◽  
Weighted Distance ◽  
Spanning Forest ◽  
Distance Model ◽  
Tree Component

The spanning star forest problem is an interesting algorithmic problem in combinatorial optimization and finds different applications. We generalize it into the spanning k-tree forest problem, which is to find a maximum spanning forest in which each tree component has a central vertex and other vertices in the component have distance at most k away from the central vertex. We show that this new problem can be approximated with ratio [Formula: see text] in polynomial time for both undirected and directed graphs. In the weighted distance model, a ½-approximation algorithm is presented for it.

scholarly journals Inapproximability and Polynomial-Time Approximation Algorithm for UET Tasks on Structured Processor Networks

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
M. Bouznif ◽  
R. Giroudeau
Keyword(s):  
Approximation Algorithm ◽  
Large Class ◽  
Bipartite Graph ◽  
Polynomial Time ◽  
Performance Guarantee ◽  
Makespan Minimization ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Precedence Graph ◽  
Polynomial Time Approximation

We investigate complexity and approximation results on a processor networks where the communication delay depends on the distance between the processors performing tasks. We then prove that there is no heuristic with a performance guarantee smaller than 4/3 for makespan minimization for precedence graph on a large class of processor networks like hypercube, grid, torus, and so forth, with a fixed diameter . We extend complexity results when the precedence graph is a bipartite graph. We also design an efficient polynomial-time -approximation algorithm for the makespan minimization on processor networks with diameter .

scholarly journals Leafy spanning k-forests

2021 ◽  
Author(s):  
Cristina G. Fernandes ◽  
Carla N. Lintzmayer ◽  
Mário César San Felice
Keyword(s):  
Positive Integer ◽  
Approximation Algorithm ◽  
Best Approximation ◽  
Np Hard ◽  
Connected Graphs ◽  
Spanning Forest ◽  
The Best Approximation ◽  
Number Of Leaves

We denote by Maximum Leaf Spanning k-Forest the problem of, given a positive integer k and a graph G with at most k components, finding a spanning forest in G with at most k components and the maximum number of leaves. A leaf in a forest is defined as a vertex of degree at most one. The case k = 1 for connected graphs is known to be NP-hard, and is well studied in the literature, with the best approximation algorithm proposed more than 20 years ago by Solis-Oba. The best known approximation algorithm for Maximum Leaf Spanning k-Forest with a slightly different leaf definition is a 3-approximation based on an approach by Lu and Ravi for the k = 1 case. We extend the algorithm of Solis-Oba to achieve a 2-approximation for Maximum Leaf Spanning k-Forest.

scholarly journals Solving Multi-agent Path Finding on Strongly Biconnected Digraphs

2018 ◽  
Vol 62 ◽  
pp. 273-314
Author(s):  
Adi Botea ◽  
Davide Bonusi ◽  
Pavel Surynek
Keyword(s):  
Empirical Analysis ◽  
Polynomial Time ◽  
Directed Graphs ◽  
Path Finding ◽  
Undirected Graphs ◽  
Suboptimal Solutions ◽  
Polynomial Time Algorithms ◽  
Asymmetric Communication ◽  
Multi Agent

Much of the literature on suboptimal, polynomial-time algorithms for multi-agent path finding focuses on undirected graphs, where motion is permitted in both directions along a graph edge. Despite this, traveling on directed graphs is relevant in navigation domains, such as path finding in games, and asymmetric communication networks.We consider multi-agent path finding on strongly biconnected directed graphs. We show that all instances with at least two unoccupied positions have a solution, except for a particular, degenerate subclass where the graph has a cyclic shape. We present diBOX, an algorithm for multi-agent path finding on strongly biconnected directed graphs. diBOX runs in polynomial time, computes suboptimal solutions and is complete for instances on strongly biconnected digraphs with at least two unoccupied positions. We theoretically analyze properties of the algorithm and properties of strongly biconnected directed graphs that are relevant to our approach. We perform a detailed empirical analysis of diBOX, showing a good scalability. To our knowledge, our work is the first study of multi-agent path finding focused on directed graphs.

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