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

Abstract

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.

This work was partially supported by the Federal Ministry of Education and Research (BMBF) as part of the project ‘Resilience by Spontaneous Volunteers Networks for Coping with Emergencies and Disaster’ (RESIBES), (grant no. 13N13955 to 13N13957).