imagine you have traveled to an unfamiliar city. before you start your daily tour around the city, you need to know a good route. in network theory (nt), this is the traveling salesman problem (tsp). a dynamic programming algorithm is often used for solving this problem. however, when the road network of the city is very complicated and dense, which is usually the case, it will take too long fo...

We consider the following natural heuristic for the Symmetric Traveling Salesman Problem: solve the subtour relaxation, yielding a solution x∗, and then find the best tour x̄ that is compatible with x∗, where compatible means that every subtour elimination constraint that is satisfied at equality at x∗ is also satisfied at equality at x̄ . We prove that finding the best compatible tour is NP-hard...

We consider the problem of searching for immobile friendly entities on an undirected network. The focus is on finding the first such entity. The search time is a random variable, whose probability density function (pdf) depends upon the path and also upon information about entity location. We seek a path choice that minimizes the expected search time. This problem differs from arc-covering prob...

A Constructive Algorithm to Prove P=NP Duan Wen-Qi ([email protected]) College of Economics and Management, Zhejiang Normal University, Jinhua, 321004, China Abstract: After reducing the undirected Hamiltonian cycle problem into the TSP problem with cost 0 or 1, we developed an effective algorithm to compute the optimal tour of the transformed TSP. Our algorithm is described as a growth pro...

The TSP (traveling salesman problem) is one of the typical NP-hard problems in combinatorial optimization problem. The fast and effective approximate algorithms are needed to solve the large-scale problem in reasonable computing time. The known approximate algorithm can not give a good enough tour for the larger instance in reasonable time. So an algorithm called multilevel reduction algorithm ...

TripBuilder is an unsupervised system helping tourists to build their own personalized sightseeing tour [1, 3, 2]. Given a target touristic city, the time available for the visit, and the tourist’s profile, TripBuilder provides a time-budgeted tour that maximizes tourist’s interests and takes into account both the time needed to enjoy the attractions and to move from one Point of Interest (PoI)...

The goal of this paper is to develop a Decision Support System (DSS) as a journey planner in complex and large multimodal urban network called Rahyar. Rahyar attempts to identify the most desirable itinerary among all feasible alternatives. The desirability of an itinerary is measured by a disutility function, which is defined as a weighted sum of some criteria. The weight...

In this paper, three types of (philosophical, optimistic and pessimistic) multigranulation single valued neutrosophic (SVN) covering-based rough set models are presented, and these three models are applied to the problem of multi-criteria group decision making (MCGDM).Firstly, a type of SVN covering-based rough set model is proposed.Based on this rough set model, three types of mult...

The Conditional Covering Problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facilit...

In the previous lecture, we covered a series of online/offline edge-weighted Steiner tree/forest problems. This lecture extends the discussion to the node-weighted scope. In particular, we will study the nodeweighted Steiner tree/forest problem and introduce an offline O(logn)−approximation polynomial-time algorithm[KR95]. It is well known there is no polynormial-time algorithm that achieves o(...