A Multi-period Vertex Cover Problem and Application to Fuel Management

Abstract

We consider a generalisation of M IN WEIGHTED VERTEX COVER motivated by a problem in wildfire prevention. The problem is defined for a fixed number of time periods and we have to choose, at each period, some vertices to be deleted such that we never have two adjacent remaining vertices. The specificity is that whenever a vertex is deleted it reappears after a given number of periods. Consequently we may need to delete a single vertex several times. The objective is to minimise the total weight (cost) of deleted vertices. The considered application motivates the case of planar graphs. While similar problems have been mainly solved using mixed integer linear models (MIP) we investigate a graph approach that allows to take into account the structure of the underlying graph. We use a reduction to the usual M IN WEIGHTEDVERTEX COVER to devise efficient approximation algorithms and to raise some polynomial classes.