Airports and railways with unsplittable demand

Abstract

In the problem of airports and railways with unsplittable demand (ARUD), we are given a complete graph $G=(V,E)$ with weights on the vertices $a:V\to R^{+}$, and the length of the edges $\ell :V\times V\to R^{+}$. Additionally, a positive integer k serves as the capacity parameter. We are also provided with a function $b:V\to N$ that defines a non-zero demand for each city. The goal is to compute a spanning forest R of G and a subset $A\subseteq V$ of minimum cost such that each component in R has one open facility and the total demand in each component is at most k (the capacity constraint). The cost of the solution $(A,R)$ is defined as ${\sum }_{v\in A}a\left(v\right)+{\sum }_{e\in E\left(R\right)}\ell \left(e\right)$. This problem is a generalization of the Airport and Railways (AR) problem introduced by Adamaszek et al. (STACS 2016). In Adamaszek et al. version, each vertex has a unit demand.

This paper presents a bi-criteria approximation algorithm for the metric ARUD problem in the sense that the algorithm is allowed to exceed the capacity constraints by $O\left(k\right)$ while the cost of the solution is compared with the cost of an optimal solution that does not violate the capacity constraint. Our approach builds upon an existing approximation algorithm for the metric AR problem, developed by Adamaszek et al. (STACS 2018), and further leverages the well-known rounding algorithm of Shmoys and Tardos for the Generalized Assignment Problem (GAP). Assuming the total demand is polynomially bounded in the number of vertices, our algorithm runs in polynomial time. We also show that it is NP-hard to find an approximate solution for ARUD within any factor without violating the capacity constraints. This is the case even when each demand is polynomially bounded in the number of vertices. Furthermore, we determine the complexity of ARUD for some fixed values of k.