Optimizing Budget Allocation in Graphs

TL;DR

This paper introduces a novel framework for facility location problems on graphs where edge weights are assignable, providing efficient algorithms for trees and approximation methods for general graphs.

Contribution

It proposes a new framework for budget allocation in facility location problems, with linear-time algorithms for trees and approximation algorithms for general graphs.

Findings

Linear-time algorithms for trees in center and median problems.

Hardness results for general graph cases.

An $O( log^2(n))$ approximation algorithm for general graphs.

Abstract

In the classical facility location problem we consider a graph $G$ with fixed weights on the edges of $G$. The goal is then to find an optimal positioning for a set of facilities on the graph with respect to some objective function. We introduce a new framework for facility location problems, where the weights on the graph edges are not fixed, but rather should be assigned. The goal is to find a valid assignment for which the resulting weighted graph optimizes the facility location objective function. We present algorithms for finding the optimal {\em budget allocation} for the center point problem and for the median point problem on trees. Our algorithms run in linear time, both for the case where a candidate vertex is given as part of the input, and for the case where finding a vertex that optimizes the solution is part of the problem. We also present a hardness result for the general graph case of the center point problem, followed by an $O(\log^2(n))$ approximation algorithm on graphs - with general metric spaces.