Online Connected Dominating Set Leasing
We introduce the \emph{Online Connected Dominating Set Leasing} problem
(OCDSL) in which we are given an undirected connected graph $G = (V, E)$, a set
$\mathcal{L}$ of lease types each characterized by a duration and cost, and a
sequence of subsets of $V$ arriving over time. A node can be leased using lease
type $l$ for cost $c_l$ and remains active for time $d_l$. The adversary gives
in each step $t$ a subset of nodes that need to be dominated by a connected
subgraph consisting of nodes active at time $t$. The goal is to minimize the
total leasing costs. OCDSL contains the \emph{Parking Permit
Problem}~\cite{PPP} as a special subcase and generalizes the classical offline
\emph{Connected Dominating Set} problem~\cite{Guha1998}. It has an $\Omega(\log
^2 n + \log |\mathcal{L}|)$ randomized lower bound resulting from lower bounds
for the \emph{Parking Permit Problem} and the \emph{Online Set Cover}
problem~\cite{Alon:2003:OSC:780542.780558,Korman}, where $|\mathcal{L}|$ is the
number of available lease types and $n$ is the number of nodes in the input
graph. We give a randomized $\mathcal{O}(\log ^2 n + \log |\mathcal{L}| \log
n)$-competitive algorithm for OCDSL. We also give a deterministic algorithm for
a variant of OCDSL in which the dominating subgraph need not be connected, the
\emph{Online Dominating Set Leasing} problem. The latter is based on a simple
primal-dual approach and has an $\mathcal{O}(|\mathcal{L}| \cdot
\Delta)$-competitive ratio, where $\Delta$ is the maximum degree of the input
graph.