Online Connected Dominating Set Leasing

TL;DR

This paper introduces the Online Connected Dominating Set Leasing problem, a new online problem combining leasing costs with dominating set requirements, and provides competitive algorithms with proven bounds.

Contribution

It formalizes the OCDSL problem, relates it to existing problems, and offers the first randomized and deterministic algorithms with competitive ratios.

Findings

Provides a randomized O(log^2 n + log |L| log n)-competitive algorithm.

Establishes an Omega(log^2 n + log |L|) lower bound on competitiveness.

Introduces a primal-dual based deterministic algorithm for a related variant.

Abstract

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.