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

Charlemagne's challenge: the periodic latency problem.

Latency problems are characterized by their focus on minimizing the waiting time for all clients. We study periodic latency problems, a non-trivial extension of standard latency problems. In a periodic latency problem each client has to be visited regularly: there is a server traveling at unit speed, and there is a set of n clients with given positions. The server must visit the clients over and over again, subject to the constraint that successive visits to client i are at most qi time units away from each other. We investigate two main problems. In problem PLPP the goal is to find a repeatable route for the server visiting as many clients as possible, without violating their qi's. In problem PLP the goal is to minimize the number of servers needed to serve all clients. In dependence on the topol- ogy of the underlying network, we derive polynomial-time algorithms or hardness results for these two problems. Our results draw sharp separation lines between easy and hard cases.Latency problem; Periodicity; Complexity;

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

Prize-Collecting TSP with a Budget Constraint

We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint