An Optimal Algorithm for Online Prize-Collecting Node-Weighted Steiner Forest

We study the Online Prize-collecting Node-weighted Steiner Forest problem (OPC-NWSF) in which we are given an undirected graph \(G=(V, E)\) with \(|V| = n\) and node-weight function \(w: V \rightarrow \mathcal {R}^+\). A sequence of k pairs of nodes of G, each associated with a penalty, arrives online. OPC-NWSF asks to construct a subgraph H such that each pair \(\{s, t\}\) is either connected (there is a path between s and t in H) or its associated penalty is paid. The goal is to minimize the weight of H and the total penalties paid. The current best result for OPC-NWSF is a randomized \(\mathcal {O}(\log ^4 n)\)-competitive algorithm due to Hajiaghayi et al. (ICALP 2014). We improve this by proposing a randomized \(\mathcal {O}(\log n \log k)\)-competitive algorithm for OPC-NWSF, which is optimal up to constant factor since OPC-NWSF has a randomized lower bound of \(\varOmega (\log ^2 n)\) due to Korman [11]. Moreover, our result also implies an improvement for two special cases of OPC-NWSF, the Online Prize-collecting Node-weighted Steiner Tree problem (OPC-NWST) and the Online Node-weighted Steiner Forest problem (ONWSF). In OPC-NWST, there is a distinguished node which is one of the nodes in each pair. In ONWSF, all penalties are set to infinity. The currently best known results for OPC-NWST and ONWSF are a randomized \(\mathcal {O}(\log ^3 n)\)-competitive algorithm due to Hajiaghayi et al. (ICALP 2014) and a randomized \(\mathcal {O}(\log n \log ^2 k)\)-competitive algorithm due to Hajiaghayi et al. (FOCS 2013), respectively.