On the complexity of solving a decision problem with flow-depending costs: the case of the IJsselmeer dikes

TL;DR

This paper analyzes the complexity of an integer programming model for flood protection cost-benefit analysis in the Netherlands, revealing non-integrality of the associated polytope and conditions for polynomial-time solvability.

Contribution

It establishes the non-integrality of the polytope for the model and identifies conditions under which the LP relaxation is integral, with implications for solving the model efficiently.

Findings

Polytope associated with the IP model is non-integral.

Certain natural conditions guarantee LP relaxation integrality.

IP model solvable in polynomial time with fixed number of dike segments or barrier heights.

Abstract

We consider a fundamental integer programming (IP) model for cost-benefit analysis flood protection through dike building in the Netherlands, due to Verweij and Zwaneveld. Experimental analysis with data for the Ijsselmeer lead to integral optimal solution of the linear programming relaxation of the IP model. This naturally led to the question of integrality of the polytope associated with the IP model. In this paper we first give a negative answer to this question by establishing non-integrality of the polytope. Second, we establish natural conditions that guarantee the linear programming relaxation of the IP model to be integral. We then test the most recent data on flood probabilities, damage and investment costs of the IJsselmeer for these conditions. Third, we show that the IP model can be solved in polynomial time when the number of dike segments, or the number of feasible barrier heights, are constant.