Online Connected Dominating Set Leasing
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We introduce the Online Connected Dominating Set Leasing problem (OCDSL) in which we are given an undirected connected graph G = (V, E), a set 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 Parking Permit Problem PPP as a special subcase and generalizes the classical offline Connected Dominating Set problem Guha1998. It has an Ω( ^2 n + |L|) randomized lower bound resulting from lower bounds for the Parking Permit Problem and the Online Set Cover problem Alon:2003:OSC:780542.780558,Korman, where |L| is the number of available lease types and n is the number of nodes in the input graph. We give a randomized O( ^2 n + |L| 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 Online Dominating Set Leasing problem. The latter is based on a simple primal-dual approach and has an O(|L| ·Δ)-competitive ratio, where Δ is the maximum degree of the input graph.
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