184 research outputs found
Complexity and approximation for Traveling Salesman Problems with profits
International audience; In TSP with profits we have to find an optimal tour and a set of customers satisfying a specific constraint. In this paper we analyze three different variants of TSP with profits that are NP-hard in general. We study their computational complexity on special structures of the underlying graph, both in the case with and without service times to the customers. We present polynomial algorithms for the polynomially solvable cases and fully polynomial time approximation schemes (FPTAS) for some NP-hard cases
Towards the solution of variants of Vehicle Routing Problem
Some of the problems that are used extensively in -real life are NP complete problems. There is no any algorithm which can give the optimal solution to NP complete problems in the polynomial time in the worst case. So researchers are applying their best efforts to design the approximation algorithms for these NP complete problems. Approximation algorithm gives the solution of a particular problem, which is close to the optimal solution of that problem. In this paper, a study on variants of vehicle routing problem is being done along with the difference in the approximation ratios of different approximation algorithms as being given by researchers and it is found that Researchers are continuously applying their best efforts to design new approximation algorithms which have better approximation ratio as compared to the previously existing algorithms
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
Revisiting Garg's 2-Approximation Algorithm for the k-MST Problem in Graphs
This paper revisits the 2-approximation algorithm for -MST presented by
Garg in light of a recent paper of Paul et al.. In the -MST problem, the
goal is to return a tree spanning vertices of minimum total edge cost. Paul
et al. extend Garg's primal-dual subroutine to improve the approximation ratios
for the budgeted prize-collecting traveling salesman and minimum spanning tree
problems. We follow their algorithm and analysis to provide a cleaner version
of Garg's result. Additionally, we introduce the novel concept of a kernel
which allows an easier visualization of the stages of the algorithm and a
clearer understanding of the pruning phase. Other notable updates include
presenting a linear programming formulation of the -MST problem, including
pseudocode, replacing the coloring scheme used by Garg with the simpler concept
of neutral sets, and providing an explicit potential function.Comment: Proceedings of SIAM Symposium on Simplicity in Algorithms (SOSA) 202
Approximation Algorithm for Unrooted Prize-Collecting Forest with Multiple Components and Its Application on Prize-Collecting Sweep Coverage
In this paper, we introduce a polynomial-time 2-approximation algorithm for
the Unrooted Prize-Collecting Forest with Components (URPCF) problem.
URPCF aims to find a forest with exactly connected components while
minimizing both the forest's weight and the penalties incurred by unspanned
vertices. Unlike the rooted version RPCF, where a 2-approximation algorithm
exists, solving the unrooted version by guessing roots leads to exponential
time complexity for non-constant . To address this challenge, we propose a
rootless growing and rootless pruning algorithm. We also apply this algorithm
to improve the approximation ratio for the Prize-Collecting Min-Sensor Sweep
Cover problem (PCMinSSC) from 8 to 5.
Keywords: approximation algorithm, prize-collecting Steiner forest, sweep
cover
Towards the solution of variants of Vehicle Routing Problem
Some of the problems that are used extensively in -real life are NP complete problems. There is no any algorithm which can give the optimal solution to NP complete problems in the polynomial time in the worst case. So researchers are applying their best efforts to design the approximation algorithms for these NP complete problems. Approximation algorithm gives the solution of a particular problem, which is close to the optimal solution of that problem. In this paper, a study on variants of vehicle routing problem is being done along with the difference in the approximation ratios of different approximation algorithms as being given by researchers and it is found that Researchers are continuously applying their best efforts to design new approximation algorithms which have better approximation ratio as compared to the previously existing algorithms
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