Online node-weighted steiner forest and extensions via disk paintings

We give the first polynomial-time online algorithm for the node-weighted Steiner forest problem with a poly-logarithmic competitive ratio. The competitive ratio of our algorithm is optimal up to a logarithmic factor. For the special case of graphs with an excluded fixed minor (e.g., planar graphs), we obtain a logarithmic competitive ratio, which is optimal up to a constant, using a different online algorithm. Both these results are obtained as special cases of generic results for a large class of problems that can be encoded as online {0, 1}-proper functions. Our results are obtained by using a new framework for online network design problems that we call disk paintings. The central idea in this technique is to amortize the cost of primal updates to a set of carefully selected mutually disjoint fixed-radius dual disks centered at a subset of terminals. We hope that this framework will be useful for other online network design problems

Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering

We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest (EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF, we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance, we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal, we need to augment our solution subgraph to connect the new terminal to the root. The goal is to minimize the total weight of the solution while respecting the degree bounds on the vertices. In the offline setting, edge-weighted degree-bounded Steiner tree (EW-DB-ST) and its many variations have been extensively studied since early eighties. Unfortunately, the recent advancements in the online network design problems are inherently difficult to adapt for degree-bounded problems. In particular, it is not known whether the fractional solution obtained by standard primal-dual techniques for mixed packing/covering LPs can be rounded online. In contrast, in this paper we obtain our result by using structural properties of the optimal solution, and reducing the EW-DB-SF problem to an exponential-size mixed packing/covering integer program in which every variable appears only once in covering constraints. We then design a generic integral algorithm for solving this restricted family of IPs. As mentioned above, we demonstrate a new technique for solving mixed packing/covering integer programs. Define the covering frequency k of a program as the maximum number of covering constraints in which a variable can participate. Let m denote the number of packing constraints. We design an online deterministic integral algorithm with competitive ratio of O(k*log(m)) for the mixed packing/covering integer programs. We prove the tightness of our result by providing a matching lower bound for any randomized algorithm. We note that our solution solely depends on m and k. Indeed, there can be exponentially many variables. Furthermore, our algorithm directly provides an integral solution, even if the integrality gap of the program is unbounded. We believe this technique can be used as an interesting alternative for the standard primal-dual techniques in solving online problems