Solving the geometric firefighter routing problem via integer programming

Zambon, Maurício J. O.; Rezende, Pedro J. de; Souza, Cid C. de

doi:10.1016/j.ejor.2018.10.037

Cited by 5 publications

(1 citation statement)

References 14 publications

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“…Also, since the number of variables in the new proposed model is exponential in the cardinality of the input point set, we resort to the use of column generation, which leads to the development of a branch-and-price algorithm. Since ILP is often applied in Operations Research, but much less frequently in Computational Geometry, this article can be seen as a further contribution towards bridging these two communities [8,16,10,12,13,26,30].…”

Section: Our Contributionmentioning

confidence: 99%

Solving the Minimum Convex Partition of Point Sets with Integer Programming

Sapucaia¹,

Rezende²,

Souza³

2020

Preprint

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The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map -which represents, say, a geographical area -into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (mcpp). A convex partition of a point set P in the plane is a subdivision of the convex hull of P whose edges are segments with both endpoints in P and such that all internal faces are empty convex polygons. The mcpp is an NP-hard problem where one seeks to find a convex partition with the least number of faces.We present a novel polygon-based integer programming formulation for the mcpp, which leads to better dual bounds than the previously known edge-based model. Moreover, we introduce a primal heuristic, a branching rule and a pricing algorithm. The combination of these techniques leads to the ability to solve instances with twice as many points as previously possible while constrained to identical computational resources. A comprehensive experimental study is presented to show the impact of our design choices.

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