Algorithmica

, Volume 52, Issue 2, pp 177–202

# Improved Algorithms and Complexity Results for Power Domination in Graphs

Article

## Abstract

The NP-complete Power Dominating Set problem is an “electric power networks variant” of the classical domination problem in graphs: Given an undirected graph G=(V,E), find a minimum-size set PV such that all vertices in V are “observed” by the vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “bounded-treewidth dynamic programs.” For treewidth being upper-bounded by a constant, we achieve a linear-time algorithm. In particular, we present a simplified linear-time algorithm for Power Dominating Set in trees. Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by |P| is W[2]-hard and it cannot be better approximated than Dominating Set.

### Keywords

Design and analysis of algorithms Computational complexity Parameterized complexity Fixed-parameter algorithms Graph algorithms Graphs of bounded treewidth (Power) domination in graphs

## Preview

### References

1. 1.
Aazami, A., Stilp, M.D.: Approximation algorithms and hardness for domination with propagation. In: Proc. 10th APPROX/11th RANDOM. Lecture Notes in Computer Science, vol. 4627, pp. 1–15. Springer, Berlin (2007) Google Scholar
2. 2.
Adjih, C., Jacquet, P., Viennot, L.: Computing connected dominating sets with multipoint relays. Ad Hoc Sens. Wirel. Netw. 1(1–2), 27–39 (2005) Google Scholar
3. 3.
Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for Dominating Set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002)
4. 4.
Alber, J., Fan, H., Fellows, M.R., Fernau, H., Niedermeier, R., Rosamond, F., Stege, U.: A refined search tree technique for Dominating Set on planar graphs. J. Comput. Syst. Sci. 71(4), 385–405 (2005)
5. 5.
Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial time data reduction for Dominating Set. J. ACM 51(3), 363–384 (2004)
6. 6.
Alber, J., Niedermeier, R.: Improved tree decomposition based algorithms for domination-like problems. In: Proc. 5th LATIN. Lecture Notes in Computer Science, vol. 2286, pp. 613–628. Springer, Berlin (2002) Google Scholar
7. 7.
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation—Combinatorial Optimization Problems and Their Approximability Properties. Springer, Berlin (1999)
8. 8.
Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209, 1–45 (1998)
9. 9.
Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: Proc. 32nd WG. Lecture Notes in Computer Science, vol. 4271, pp. 1–14. Springer, Berlin (2006) Google Scholar
10. 10.
Booth, K.S., Johnson, J.H.: Dominating sets in chordal graphs. SIAM J. Comput. 11, 191–199 (1982)
11. 11.
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. SIAM Monographs on Discrete Mathematics and Applications (1999) Google Scholar
12. 12.
Brueni, D.J., Heath, L.S.: The PMU placement problem. SIAM J. Discrete Math. 19(3), 744–761 (2005)
13. 13.
Demaine, E.D., Hajiaghayi, M.: Bidimensionality: New connections between FPT algorithms and PTASs. In: Proc. 16th SODA, pp. 590–601. ACM/SIAM, New York (2005) Google Scholar
14. 14.
Dewdney, A.K.: Fast Turing reductions between problems in NP: chap. 4; reductions between NP-complete problems. Technical report, Department of Computer Science, University of Western Ontario, Canada, 1981 Google Scholar
15. 15.
Dorfling, M., Henning, M.A.: A note on power domination in grid graphs. Discrete Appl. Math. 154(6), 1023–1027 (2006)
16. 16.
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) Google Scholar
17. 17.
Feige, U.: A threshold of ln n for approximating Set Cover. J. ACM 45(4), 634–652 (1998)
18. 18.
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
19. 19.
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
20. 20.
Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: distance from triviality. In: Proc. 1st IWPEC. Lecture Notes in Computer Science, vol. 3162, pp. 162–173. Springer, Berlin (2004) Google Scholar
21. 21.
Haynes, T.W., Hedetniemi, S.M., Hedetniemi, S.T., Henning, M.A.: Domination in graphs: applied to electric power networks. SIAM J. Discrete Math. 15(4), 519–529 (2002)
22. 22.
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Pure and Applied Mathematics, vol. 209. Dekker, New York (1998)
23. 23.
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs, Pure and Applied Mathematics, vol. 208. Dekker, New York (1998) Google Scholar
24. 24.
Haynes, T.W., Henning, M.A.: Domination in graphs. In: Gross, J.L., Yellen, J. (eds.) Handbook of Graph Theory, pp. 889–909. CRC Press, Boca Raton (2004) Google Scholar
25. 25.
Keil, J.M.: The complexity of domination problems in circle graphs. Discrete Appl. Math. 36, 25–34 (1992)
26. 26.
Kloks, T.: Treewidth: Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994)
27. 27.
Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Parameterized power domination complexity. Inf. Process. Lett. 98(4), 145–149 (2006)
28. 28.
Kratsch, D.: Algorithms. In: Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.) Domination in Graphs: Advanced Topics, pp. 191–231. Dekker, New York (1998) Google Scholar
29. 29.
Liao, C.S., Lee, D.T.: Power dominating problem in graphs. In: Proc. 11th COCOON. Lecture Notes in Computer Science, vol. 3595, pp. 818–828. Springer, Berlin (2005) Google Scholar
30. 30.
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006)
31. 31.
Raible, D.: Algorithms and complexity results for power domination in networks. Master’s thesis, Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany (2005). (In German) Google Scholar
32. 32.
Reed, B.A.: Algorithmic aspects of tree width. In: Reed, B.A., Sales, C.L. (eds.) Recent Advances in Algorithms and Combinatorics, pp. 85–107. Springer, Berlin (2003)
33. 33.
Telle, J.A., Proskurowski, A.: Practical algorithms on partial k-trees with an application to domination-like problems. In: Proc. 3rd WADS. Lecture Notes in Computer Science, vol. 709, pp. 610–621. Springer, Berlin (1993) Google Scholar
34. 34.
Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math. 10(4), 529–550 (1997)
35. 35.
Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001) Google Scholar