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

### Marc Demange, Cerasela Tanasescu

#### Abstract

We consider a generalisation of MIN 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 MIN WEIGHTED VERTEX COVER to devise efficient approximation algorithms and to raise some polynomial classes.

#### References

- Ager, A., Vaillant, N., and Finney, M. (2010). A comparison of landscape fuel treatment strategies to mitigate wildland fire risk in the urban interface and preserve old forest structure. Forest Ecology and Management, 259:1556-1570.
- Ausiello, G., G., M. P., Crescenzi, P., and Kann, V. (1999). Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer-Verlag New York, Inc., 1st edition.
- Baker, B. (1994). Approximation algorithms for npcomplete problems on planar graphs. Journal of the ACM, 41(1):153-180.
- Boer, M. M., Sadler, R. J., Wittkuhn, R. S., McCaw, L., and Grierson, P. F. (2015). Longterm impacts of prescribed burning on regional extent and incidence of wildfiresevidence from 50 years of active fire management in sw australian forests. Forest Ecology and Management, 259:132-142.
- Chung, W. (2015). Optimizing fuel treatments to reduce wildland fire risk. Current Forestry Reports, 1:44-51.
- Demange, M. and Ekim, T. (2013). A note on the nphardness of two matching problems in induced subgrids. Discrete Mathematics & Theoretical Computer Science, 15(2):233-242.
- Diestel, R. (2012). Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer.
- Edwards, K. J. and Farr, G. E. (2001). Fragmentability of graphs. Journal of Combinatorial Theory, 82:30-37.
- Garey, M. and Johnson, D. S. (1979). Computers and Intractability: A Guide to the Theory of NPCompleteness. W. H. Freeman & Co., New York, NY, USA.
- Hof, J., Omi, P., Bevers, M., and Laven, R. (2000). A timing-oriented approach to spatial allocation of fire management effort. Forest Science, 46(3):442-451.
- Kim, Y., Bettinger, P., and Finney, M. (2009). Spatial optimization of the pattern of fuel management activities and subsequent effects on simulated wildfires. European Journal of Operational Research, 197:253-265.
- Minas, J., Hearne, J., and Handmer, J. (2012). A review of operations research methods applicable to wildfire management. International Journal of Wildland Fire, 21(3):189-196.
- Minas, J., Hearne, J., and Martell, D. (2014). A spatial optimisation model for multi-period landscape level fuel management to mitigate wildfire impacts. European Journal of Operational Research, 232(2):412-422.
- Rachmawati, R.and Ozlen, M., Reinke, K., and Hearne, J. (2015). A model for solving the prescribed burn planning problem. SpringerPlus, 4:1-21.
- Wei, Y., Rideout, D., and Kirsch, A. (2008). An optimization model for locating fuel treatments across a landscape to reduce expected fire losses. Canadian Journal of Forest Research, 38(4):868-877.

#### Paper Citation

#### in Harvard Style

Demange M. and Tanasescu C. (2016). **A Multi-period Vertex Cover Problem and Application to Fuel Management** . In *Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-171-7, pages 51-57. DOI: 10.5220/0005708900510057

#### in Bibtex Style

@conference{icores16,

author={Marc Demange and Cerasela Tanasescu},

title={A Multi-period Vertex Cover Problem and Application to Fuel Management},

booktitle={Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2016},

pages={51-57},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005708900510057},

isbn={978-989-758-171-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

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

SN - 978-989-758-171-7

AU - Demange M.

AU - Tanasescu C.

PY - 2016

SP - 51

EP - 57

DO - 10.5220/0005708900510057