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

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.