# The Complexity of Finding (Approximate Sized) Distance-d Dominating Set in Tournaments

• Conference paper
• First Online:
Frontiers in Algorithmics (FAW 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10336))

Included in the following conference series:

• 659 Accesses

## Abstract

A tournament is an orientation of a complete graph. For a positive integer d, a distance-d dominating set in a tournament is a subset S of vertices such that every vertex in $$V{\setminus }S$$ is reachable by a path of length at most d from one of the vertices in S. When $$d=1$$, the set is simply called a dominating set. While the complexity of finding a k-sized dominating set is complete for the complexity class LOGSNP and the parameterized complexity class W[2], it is well-known that every tournament on n vertices has a dominating set of size $$g(n) = \lg n - \lg \lg n + 2$$ that can be found in $${{\mathrm{O}}}\left( n^2 \right)$$ time. We first show that for any k, one can find a dominating set of size at most $$k + g(n)$$ in $${{\mathrm{O}}}\left( n^2/k \right)$$ time, and prove an (unconditional) lower bound of $$\varOmega (n^2/k)$$ for any $$k > \epsilon \lg n$$ for any $$\epsilon >0$$. Hence in particular, we can find a $$(1+\epsilon ) \lg n$$ sized dominating set in the optimal $$\varTheta ({n^2/\lg n})$$ time.

For distance-d dominating sets, it is known that any tournament has a distance-2 dominating set consisting of a single vertex. Such a vertex is called a king or a 2-king and can be found in $${{\mathrm{O}}}\left( n \sqrt{n} \right)$$ time. It follows that there is a vertex, from which every other vertex is reachable by a path of length at most d for any $$d \ge 2$$ and such a vertex is called a d-king. A d-king can be found in $${{\mathrm{O}}}\left( n^{1+1/2^{d-1}} \right)$$ for any $$d \ge 2$$ [3]. We generalize our result for dominating set to show that for $$d \ge 2$$,

• we can find a k-sized distance-d dominating set in a tournament in $${{\mathrm{O}}}\left( k(n/k)^{1+1/2^{d-1}} \right)$$ time for any $$k \ge 1$$, and

• we can find a $$(g(n) + k)$$-sized distance-d dominating set in a tournament using $${{\mathrm{O}}}\left( k(n/k)^{1+1/(2^d-1)} \right)$$ time for any $$k \ge 1$$.

The second algorithm provides a faster runtime for finding distance-d dominating sets that are larger than g(n). We also show that the second algorithm is optimal whenever $$k \ge \epsilon \lg n$$ for any $$\epsilon > 0$$.

Thus our algorithms provide tradeoffs between the (additive) approximation factor and the complexity of finding distance-d dominating sets in tournaments. For the problem of finding a d-king, we show some additional results.

This is a preview of subscription content, log in via an institution to check access.

## Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
• Available as PDF
• Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
• Available as EPUB and PDF
• Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
• Compact, lightweight edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

## References

1. Abboud, A., Williams, V.V., Wang, J.R.: Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, 10–12 January 2016, pp. 377–391 (2016)

2. Acharya, J., Falahatgar, M., Jafarpour, A., Orlitksy, A., Suresh, A.T.: Maximum selection and sorting with adversarial comparators and an application to density estimation. Comput. Res. Repos. abs/1606.02786, 1–24 (2016)

3. Ajtai, M., Feldman, V., Hassidim, A., Nelson, J.: Sorting and selection with imprecise comparisons. ACM Trans. Algorithms 12(2), 19:1–19:19 (2016)

4. Alon, N., Spencer, J.: The Probabilistic Method. Wiley, Hoboken (1992)

5. Balasubramanian, R., Raman, V., Srinivasaragavan, G.: Finding scores in tournaments. J. Algorithms 24(2), 380–394 (1997)

6. Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999)

7. Garg, S., Philip, G.: Raising the bar for vertex cover: fixed-parameter tractability above a higher guarantee. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, 10–12 January 2016, pp. 1152–1166 (2016)

8. Giannopoulou, A.C., Mertzios, G.B., Niedermeier, R.: Polynomial fixed-parameter algorithms: a case study for longest path on interval graphs. In: Husfeldt, T., Kanj, I. (eds.) 10th International Symposium on Parameterized and Exact Computation (IPEC 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 43, pp. 102–113. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2015)

9. Graham, R.L., Spencer, J.H.: A constructive solution to a tournament problem. Canad. Math. Bull. 14, 45–48 (1971)

10. Gutin, G., Yeo, A.: Constraint satisfaction problems parameterized above or below tight bounds: a survey. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) The Multivariate Algorithmic Revolution and Beyond. LNCS, vol. 7370, pp. 257–286. Springer, Heidelberg (2012). doi:10.1007/978-3-642-30891-8_14

11. Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15:1–15:31 (2014)

12. Lu, X., Wang, D., Wong, C.K.: On the bounded domination number of tournaments. Discret. Math. 220(1–3), 257–261 (2000)

13. Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)

14. Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Comput. Syst. Sci. 75(2), 137–153 (2009)

15. Maurer, S.B.: The king chicken theorems. Math. Mag. 53(2), 67–80 (1980)

16. Megiddo, N., Vishkin, U.: On finding a minimum dominating set in a tournament. Theor. Comput. Sci. 61, 307–316 (1988)

17. Moon, J.: Topics on tournaments. In: Selected Topics in Mathematics. Athena series. Holt, Rinehart and Winston (1968)

18. Papadimitriou, C.H., Yannakakis, M.: On limited nondeterminism and the complexity of the V-C dimension. J. Comput. Syst. Sci. 53(2), 161–170 (1996). http://dx.doi.org/10.1006/jcss.1996.0058

19. Shen, J., Sheng, L., Wu, J.: Searching for sorted sequences of kings in tournaments. SIAM J. Comput. 32(5), 1201–1209 (2003)

## Author information

Authors

### Corresponding author

Correspondence to Varunkumar Jayapaul .

## Rights and permissions

Reprints and permissions

© 2017 Springer International Publishing AG

### Cite this paper

Biswas, A., Jayapaul, V., Raman, V., Satti, S.R. (2017). The Complexity of Finding (Approximate Sized) Distance-d Dominating Set in Tournaments. In: Xiao, M., Rosamond, F. (eds) Frontiers in Algorithmics. FAW 2017. Lecture Notes in Computer Science(), vol 10336. Springer, Cham. https://doi.org/10.1007/978-3-319-59605-1_3

• DOI: https://doi.org/10.1007/978-3-319-59605-1_3

• Published:

• Publisher Name: Springer, Cham

• Print ISBN: 978-3-319-59604-4

• Online ISBN: 978-3-319-59605-1

• eBook Packages: Computer ScienceComputer Science (R0)