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In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D cup {u,v}$ is at most $alpha$ times that in $G$. For general graphs, the only known previous approximability result is an $O(log n)$-approximation algorithm ($n=|V|$) for $alpha = 1$ by Ding et al. For any constant $alpha > 1$, we give an $O(n^{1-frac{1}{alpha}}(log n)^{frac{1}{alpha}})$-approximation algorithm. When $alpha geq 5$, we give an $O(sqrt{n}log n)$-approximation algorithm. Finally, we prove that, when $alpha =2$, unless $NP subseteq DTIME(n^{polylog n})$, for any constant $epsilon > 0$, the problem admits no polynomial-time $2^{log^{1-epsilon}n}$-approximation algorithm, improving upon the $Omega(log n)$ bound by Du et al. (albeit under a stronger hardness assumption).
We show that there is no deterministic local algorithm (constant-time distributed graph algorithm) that finds a $(7-epsilon)$-approximation of a minimum dominating set on planar graphs, for any positive constant $epsilon$. In prior work, the best low
Given a graph $G=(V,E)$, the dominating set problem asks for a minimum subset of vertices $Dsubseteq V$ such that every vertex $uin Vsetminus D$ is adjacent to at least one vertex $vin D$. That is, the set $D$ satisfies the condition that $|N[v]cap D
We investigate the polynomial-time approximability of the multistage version of Min-Sum Set Cover ($mathrm{DSSC}$), a natural and intriguing generalization of the classical List Update problem. In $mathrm{DSSC}$, we maintain a sequence of permutation
This paper is devoted to the online dominating set problem and its variants. We believe the paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future,
In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a