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

## TL;DR

This paper introduces a novel online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest problem, leveraging a new approach for mixed packing/covering integer programs.

## Contribution

It presents the first online algorithm with poly-logarithmic competitive ratio for EW-DB-SF, using a structural reduction to mixed packing/covering IPs and a new integral solving technique.

## Key findings

- Achieved poly-logarithmic competitive ratio for EW-DB-SF.
- Developed a generic integral algorithm for mixed packing/covering IPs.
- Introduced a new technique for solving mixed packing/covering integer programs.

## Abstract

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 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. 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 believe this technique can be used as an interesting alternative for the standard primal-dual techniques in solving online problems.