Discounted Reward TSP

Consider a rescue plan after a major disaster such as an earthquake, where the objective is to find and rescue as many survivors as possible. The rescue team has to decide where to search for survivors, and as time progress the number of survivors in each location decreases. This problem can be modeled as Discounted Reward TSP on a graph $$G=(V,E)$$G=(V,E) where each node $$v\in V$$v∈V represents a potential place for searching survivors, and the length of an edge represents the time it takes to travel from one place to another. Each node has an initial prize $$\pi (v)$$π(v) (that represents the number of survivors in it) and this prize deteriorates exponentially. Therefore, the prize collected from node $$v\in V$$v∈V is $$\pi (v) \lambda ^t$$π(v)λt, where $$\lambda $$λ is the deterioration rate and t is the first time v was visited. The objective is to find a path that maximizes the total prize collected from the nodes of G. We present two different algorithms for Discounted Reward TSP, each improves the previously best known approximation ratio of $$0.1481-\delta $$0.1481-δ shown by Blum et al. (SIAM J Comput 37(2):653–670, 2007). Our better algorithm is a $$(0.1929-\delta )$$(0.1929-δ)-approximation algorithm.