# Approximation Algorithms for Graph Burning

• Anthony Bonato
• Shahin Kamali
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)

## Abstract

Numerous approaches study the vulnerability of networks against social contagion. Graph burning studies how fast a contagion, modeled as a set of fires, spreads in a graph. The burning process takes place in synchronous, discrete rounds. In each round, a fire breaks out at a vertex, and the fire spreads to all vertices that are adjacent to a burning vertex. The selection of vertices where fires start defines a schedule that indicates the number of rounds required to burn all vertices. Given a graph, the objective of an algorithm is to find a schedule that minimizes the number of rounds to burn graph. Finding the optimal schedule is known to be NP-hard, and the problem remains NP-hard when the graph is a tree or a set of disjoint paths. The only known algorithm is an approximation algorithm for disjoint paths, which has an approximation ratio of 1.5.

We present approximation algorithms for graph burning. For general graphs, we introduce an algorithm with an approximation ratio of 3. When the graph is a tree, we present another algorithm with approximation ratio 2. Moreover, we consider a setting where the graph is a forest of disjoint paths. In this setting, when the number of paths is constant, we provide an optimal algorithm which runs in polynomial time. When the number of paths is more than a constant, we provide two approximation schemes: first, under a regularity condition where paths have asymptotically equal lengths, we show the problem admits an approximation scheme which is fully polynomial. Second, for a general setting where the regularity condition does not necessarily hold, we provide another approximation scheme which runs in time polynomial in the size of the graph.

## Keywords

Approximation algorithms Graph algorithms Graph burning problem Information dissemination Social contagion

## References

1. 1.
Anshelevich, E., Chakrabarty, D., Hate, A., Swamy, C.: Approximability of the firefighter problem - computing cuts over time. Algorithmica 62(1–2), 520–536 (2012)
2. 2.
Bessy, S., Bonato, A., Janssen, J.C.M., Rautenbach, D., Roshanbin, E.: Burning a graph is hard. Discret. Appl. Math. 232, 73–87 (2017)
3. 3.
Bessy, S., Bonato, A., Janssen, J.C.M., Rautenbach, D., Roshanbin, E.: Bounds on the burning number. Discret. Appl. Math. 235, 16–22 (2018)
4. 4.
Bonato, A., Gunderson, K., Shaw, A.: Burning the plane: densities of the infinite cartesian grid. Preprint (2019)Google Scholar
5. 5.
Bonato, A., Janssen, J., Roshanbin, E.: Burning a graph as a model of social contagion. In: Bonato, A., Graham, F.C., Prałat, P. (eds.) WAW 2014. LNCS, vol. 8882, pp. 13–22. Springer, Cham (2014).
6. 6.
Bonato, A., Janssen, J., Roshanbin, E.: How to burn a graph. Internet Math. 12(1–2), 85–100 (2016)
7. 7.
Bonato, A., Lidbetter, T.: Bounds on the burning numbers of spiders and path-forests. ArXiv e-prints, July 2017Google Scholar
8. 8.
Bond, R.M., et al.: A 61-million-person experiment in social influence and political mobilization. Nature 489(7415), 295–298 (2012)Google Scholar
9. 9.
Cai, L., Verbin, E., Yang, L.: Firefighting on trees: (1$$-1$$/e)-approximation, fixed parameter tractability and a subexponential algorithm. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 258–269. Springer, Heidelberg (2008). Google Scholar
10. 10.
Chen, N., Gravin, N., Lu, P.: On the approximability of budget feasible mechanisms. In: Proceedings of Annual ACM-SIAM Symposium on Discrete Algorithms SODA, pp. 685–699 (2011)Google Scholar
11. 11.
Chen, W., et al.: Influence maximization in social networks when negative opinions may emerge and propagate. In: Proceedings of SIAM International Conference on Data Mining, SDM, pp. 379–390 (2011)Google Scholar
12. 12.
Chen, W., Wang, Y., Yang, S.: Efficient influence maximization in social networks. In: Proceedings of ACM International Conference on Knowledge Discovery and Data Mining (SIGKDD), pp. 199–208 (2009)Google Scholar
13. 13.
Czumaj, A., Rytter, W.: Broadcasting algorithms in radio networks with unknown topology. J. Algorithms 60(2), 115–143 (2006)
14. 14.
Domingos, P.M., Richardson, M.: Mining the network value of customers. In: Proceedings of ACM International Conference on Knowledge Discovery and Data Mining (SIGKDD), pp. 57–66 (2001)Google Scholar
15. 15.
Elkin, M., Kortsarz, G.: Sublogarithmic approximation for telephone multicast. J. Comput. Syst. Sci. 72(4), 648–659 (2006)
16. 16.
Fajardo, D., Gardner, L.M.: Inferring contagion patterns in social contact networks with limited infection data. Netw. Spat. Econ. 13(4), 399–426 (2013)
17. 17.
Assmann, S.F.: Problems in discrete applied mathematics. Ph.D. thesis, MIT (1983)Google Scholar
18. 18.
Finbow, S., King, A.D., MacGillivray, G., Rizzi, R.: The firefighter problem for graphs of maximum degree three. Discret. Math. 307(16), 2094–2105 (2007)
19. 19.
Fitzpatrick, S.L., Li, Q.: Firefighting on trees: how bad is the greedy algorithm? Congr. Numer. 145, 187–192 (2000)
20. 20.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, Stuttgart (1979)
21. 21.
Ghaffari, M., Haeupler, B., Khabbazian, M.: Randomized broadcast in radio networks with collision detection. Distrib. Comput. 28(6), 407–422 (2015)
22. 22.
Hedetniemi, S.M., Hedetniemi, S.T., Liestman, A.L.: A survey of gossiping and broadcasting in communication networks. Networks 18(4), 319–349 (1988)
23. 23.
Jansen, K., Solis-Oba, R.: An asymptotic fully polynomial time approximation scheme for bin covering. Theor. Comput. Sci. 306(1–3), 543–551 (2003)
24. 24.
Kempe, D., Kleinberg, J.M., Tardos, É: Maximizing the spread of influence through a social network. In: Proceedings of the International Conference on Knowledge Discovery and Data Mining (SIGKDD), pp. 137–146 (2003)Google Scholar
25. 25.
Kempe, D., Kleinberg, J., Tardos, É.: Influential nodes in a diffusion model for social networks. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1127–1138. Springer, Heidelberg (2005). Google Scholar
26. 26.
Kempe, D., Kleinberg, J.M., Tardos, É.: Maximizing the spread of influence through a social network. Theory Comput. 11, 105–147 (2015)
27. 27.
Kleinberg, J.M.: Cascading behavior in social and economic networks. In: Proceedings of ACM Conference on Electronic Commerce (EC), pp. 1–4 (2013)Google Scholar
28. 28.
Kowalski, D.R., Pelc, A.: Optimal deterministic broadcasting in known topology radio networks. Distrib. Comput. 19(3), 185–195 (2007)
29. 29.
Kramer, A.D.I.: The spread of emotion via Facebook. In: CHI Conference on Human Factors in Computing Systems, (CHI), pp. 767–770 (2012)Google Scholar
30. 30.
Kramer, A.D.I., Guillory, J.E., Hancock, J.T.: Experimental evidence of massive-scale emotional contagion through social networks. In: Proceedings of the National Academy of Sciences, pp. 8788–8790 (2014)Google Scholar
31. 31.
Land, M.R., Lu, L.: An upper bound on the burning number of graphs. In: Bonato, A., Graham, F.C., Prałat, P. (eds.) WAW 2016. LNCS, vol. 10088, pp. 1–8. Springer, Cham (2016). Google Scholar
32. 32.
Mitsche, D., Pralat, P., Roshanbin, E.: Burning graphs: a probabilistic perspective. Graphs Comb. 33(2), 449–471 (2017)
33. 33.
Mitsche, D., Pralat, P., Roshanbin, E.: Burning number of graph products. Theor. Comput. Sci. 746, 124–135 (2018)
34. 34.
Nikzad, A., Ravi, R.: Sending secrets swiftly: approximation algorithms for generalized multicast problems. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8573, pp. 568–607. Springer, Heidelberg (2014). Google Scholar
35. 35.
Peleg, D.: Time-efficient broadcasting in radio networks: a review. In: Janowski, T., Mohanty, H. (eds.) ICDCIT 2007. LNCS, vol. 4882, pp. 1–18. Springer, Heidelberg (2007). Google Scholar
36. 36.
Ravi, R.: Rapid rumor ramification: approximating the minimum broadcast time (extended abstract). In: Proceedings of Symposium on Foundations of Computer Science (FOCS), pp. 202–213 (1994)Google Scholar
37. 37.
Richardson, M., Domingos, P.M.: Mining knowledge-sharing sites for viral marketing. In: Proceedings of the ACM International Conference on Knowledge Discovery and Data Mining (SIGKDD), pp. 61–70 (2002)Google Scholar
38. 38.
Schindelhauer, C.: On the inapproximability of broadcasting time. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 226–237. Springer, Heidelberg (2000). Google Scholar
39. 39.
Sim, K.A., Tan, T.S., Wong, K.B.: On the burning number of generalized petersen graphs. Bull. Malays. Math. Sci. Soc. 6, 1–14 (2017)Google Scholar
40. 40.
Slater, P.J., Cockayne, E.J., Hedetniemi, S.T.: Information dissemination in trees. SIAM J. Comput. 10(4), 692–701 (1981)
41. 41.
Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001).