Facility Location for Congesting Commuters and Generalizing the Cost-Distance Problem

TL;DR

This paper introduces a new facility location problem where connection costs depend on congestion and commute costs, providing approximation algorithms for different cost function scenarios and extending existing problems.

Contribution

It defines the Facility Location for Congesting Commuters problem, employing novel approximation techniques and generalizing the Cost-Distance problem for nonincreasing cost functions.

Findings

Approximate solutions are achievable despite inapproximability results.

Novel use of approximate Caratheodory's Theorem for nondecreasing costs.

Algorithm achieves known approximation guarantees for generalized problem.

Abstract

In Facility Location problems there are agents that should be connected to facilities and locations where facilities may be opened so that agents can connect to them. We depart from Uncapacitated Facility Location and by assuming that the connection costs of agents to facilities are congestion dependent, we define a novel problem, namely, Facility Location for Congesting (Selfish) Commuters. The connection costs of agents to facilities come as a result of how the agents commute to reach the facilities in an underlying network with cost functions on the edges. Inapproximability results follow from the related literature and thus approximate solutions is all we can hope for. For when the cost functions are nondecreasing we employ in a novel way an approximate version of Caratheodory's Theorem [5] to show how approximate solutions for different versions of the problem can be derived. For when the cost functions are nonincreasing we show how this problem generalizes the Cost-Distance problem [38] and provide an algorithm that for this more general case achieves the same approximation guarantees.