On the Strength of Linear Relaxations in Ordered Optimization
Víctor Blanco, Diego Laborda, Miguel Martínez-Antón
TL;DR
This work investigates when the linear programming relaxation of discrete ordered median location problems recovers integral solutions, revealing that sorting substantially increases combinatorial complexity and weakens exactness except in cases closely related to the non-ordered $p$-median problem. It introduces an ordered-contribution framework and primal–dual certificates to characterize LP-recovery, develops notions such as strongly sortable solutions, and analyzes how conic combinations of ${\boldsymbol{\lambda}}$-weights influence integrality. The paper provides tight theoretical conditions for integrality recovery and heaps of empirical evidence showing LP relaxations are strong for the $p$-median but markedly weaker for other ordered variants, with clusterability of the input data significantly impacting relaxation quality. These insights clarify when LP-based approaches can solve ordered location problems in polynomial time and motivate the search for stronger convex relaxations and data-geometric conditions to improve practical solvability.
Abstract
We study the conditions under which the convex relaxation of a mixed-integer linear programming formulation for ordered optimization problems, where sorting is part of the decision process, yields integral optimal solutions. Thereby solving the problem exactly in polynomial time. Our analysis identifies structural properties of the input data that influence the integrality of the relaxation. We show that incorporating ordered components introduces additional layers of combinatorial complexity that invalidate the exactness observed in classical (non-ordered) settings. In particular, for certain ordered problems such as the min--max case, the linear relaxation never recovers the integral solution. These results clarify the intrinsic hardness introduced by sorting and reveal that the strength of the relaxation depends critically on the ``proximity'' of the ordered problem to its classical counterpart: problems closer to the non-ordered case tend to admit tighter relaxations, while those further away exhibit substantially weaker behavior. Computational experiments on benchmark instances confirm the predictive value of the integrality conditions and demonstrate the practical implications of exact relaxations for ordered location problems.