Online Degree-Bounded Steiner Network Design

Abstract

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem -which asks for a subgraph with minimum degree that connects a given set of vertices -is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance.We design a simple greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.Fürer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the natural family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems.

The multi-budgeted and weighted bounded degree metric Steiner network problem

“…Then, a polyhedral characterization of any extreme point solution 1 is given i.e., that every extreme point solution of the bounded polyhedron defined by the above linear program (after we drop the integrality constraints) satisfies the following property: ∃e ∈ E such that x e ≥ 1 2 . In other words, any basic (extreme point) feasible solution x is always sparse (a few number of non-zero entries) and thus must always have at least one variable with relatively large fractional value (in particular greater than 1 2 ). If, for every subset S ⊆ V , we define max s i ∈S,t i / ∈S {r i } = f (S), then the constraint (1) of the above LP becomes e∈δ (S) x e ≥ f (S).…”

“…By Jain's result [8] each of y, z has a variable with fractional value greater than half. This means that at least one variable in one of the λy or (1 − λ)z has value ≥ 1 4 . Apply Jain's algorithm and a (4, 4) bi-criteria algorithm is immediate.…”

“…Apply Jain's algorithm and a (4, 4) bi-criteria algorithm is immediate. Moreover, in that paper it was shown that for every basic feasible solution x of the generalized LP (that spells out the connectivity requirements plus the linear budget), there is always a coordinate with fractional value greater than 1 3 . This is proved by the following way: a basic feasible solution is obtained for the relaxed LP + lin.…”

“…∈S {r i } which is at least 1, we have that there must be an edge crossing S with fractional value ≥ 1 3 . This is done by considering only the tight constraints of subsets of vertices defining the obtained basic feasible solution x (by dropping the linear budget).…”

“…The problem was studied in the directed graph setting in [20,19]. See also [1,18] for some on-line special cases/variants of the problem and see also [22] for other potential case in relevant scenarios.…”