Best weighted-sum approximation of a given weighted-maximum solution

Determine, for a multi-objective shortest-path problem on a graph G=(V,E) with objective costs F_1,...,F_n and a fixed evaluation weight vector w in the unit simplex defining WM(P)=max_i w_i F_i(P), a weight vector \hat{w} in the unit simplex such that the path P that minimizes the weighted-sum cost \sum_i \hat{w}_i F_i(P) yields the smallest possible weighted-maximum value WM(P) with respect to w. In other words, select weighted-sum scalarization weights that minimize the approximation error to the weighted-maximum objective for the given w.

Background

The paper studies scalarizations for multi-objective path planning on graphs, contrasting weighted sum (WS) and weighted maximum (WM). WS is efficient but cannot find solutions in non-convex regions of the Pareto front, whereas WM is Pareto-complete but NP-hard in discrete domains.

After establishing an n-factor worst-case bound on using WS to approximate WM for fixed weights, the authors pose the question of selecting WS weights that best approximate a given WM solution. They formalize this as the Best WS Approximation problem and subsequently prove it is NP-hard, underscoring the difficulty of the task despite its importance for practical approximations.