On the complexity of the upgrading version of the maximal covering location problem

TL;DR

The paper studies Up-MCLP, where one chooses $p$ facilities and edge-length upgrades within budget $B$ to maximize covered demand within radius $R$ on networks. It delivers a nuanced complexity landscape: polynomial-time/greedy solutions for star networks with uniform weights, and a $O(n^3)$ dynamic approach for the single-facility case on path networks, contrasted by NP-hardness results for non-uniform weights on stars and for $p$-facility variants on paths. For trees, it provides a pseudo-polynomial algorithm via a leaves-to-root DP on a transformed binary tree, under integer upgrade assumptions, and extends these ideas to the general $p$-Up-MCLP setting. The work advances understanding of upgrading variants of classical location problems, with practical implications for infrastructure and service delivery planning under budget constraints.

Abstract

In this article, we study the complexity of the upgrading version of the maximal covering location problem with edge length modifications on networks. This problem is NP-hard on general networks. However, in some particular cases, we prove that this problem is solvable in polynomial time. The cases of star and path networks combined with different assumptions for the model parameters are analysed. In particular, we obtain that the problem on star networks is solvable in O(nlogn) time for uniform weights and NP-hard for non-uniform weights. On paths, the single facility problem is solvable in O(n^3) time, while the p-facility problem is NP-hard even with uniform costs and upper bounds (maximal upgrading per edge), as well as, integer parameter values. Furthermore, a pseudo-polynomial algorithm is developed for the single facility problem on trees with integer parameters.

Figures (6)

• Figure 1: Illustration for the uniform case in Theorem \ref{['theo:NPhard:1_facility_stars1_uni']}, with edge labels $(\ell_j,u_j,c_j)$.
• Figure 2: Illustration for the proof of Theorem \ref{['theo:NPhard:1_facility_stars1_nonuni']}, with edge labels $(\ell_j,u,c)$.
• Figure 3: Notation used in Section \ref{['sec:facility:path']}
• Figure 4: Notation used in Theorem \ref{['theo:NPhard:p_facility_path']}, with edge labels $(\ell_j,u_j)$.
• Figure 6: Binary tree with node weights in brackets.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Corollary 1
• Theorem 7
• Theorem 8
• Example 1