On a Linear Program for Minimum-Weight Triangulation

TL;DR

This paper establishes a novel connection between a classical linear program and approximation algorithms for the NP-hard problem of minimum-weight triangulation, showing the LP's integrality gap is bounded and aligning LP solutions with heuristics.

Contribution

It demonstrates for the first time that a known LP formulation for MWT has a bounded integrality gap and aligns with existing heuristics in finding the MWT.

Findings

The integrality gap of the LP is bounded by a constant.

LP solutions coincide with heuristics when they find the MWT.

Establishes a new link between LP formulations and approximation algorithms for MWT.

Abstract

Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial- time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: for an LP formulation due to Dantzig et al. (1985): (i) the integrality gap is bounded by a constant; (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP.