Approximating Routing and Connectivity Problems with Multiple Distances

Abstract

We consider routing and connectivity problems for which the input includes a complete graph G and multiple edge-weight functions $d_1, d_2, \dots , d_r$. In each case, a solution is a minimum-cost subgraph H satisfying the constraints of the particular problem. The cost of each edge of H is determined by any chosen function $d_i$, but there is a service fee $g \ge 0$ for each maximal connected component formed by edges associated with the same function. This is motivated by applications for which a solution can be split between multiple providers, each corresponding to a distance $d_i$. One example is the Traveling Car Renter Problem (CaRS), which is a generalization of the Traveling Salesman Problem (TSP) whose goal is to visit a set of cities by renting cars from multiple companies. In this paper, we give $\mathcal {O}(\log n)$-approximations for the generalizations with multiple distances of several problems (Steiner TSP, Profitable Tour Problem, and Constrained Forest Problem). This factor is the best-possible unless $\mathrm {P}= \mathrm {NP}$.

Supported by grant #2015/11937-9, São Paulo Research Foundation (FAPESP), grants #425340/2016-3, #313026/2017-3, #422829/2018-8, National Council for Scientific and Technological Development (CNPq), and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.