Steiner Forest Orientation Problems

Abstract

Abstract. We consider connectivity problems with orientation constraints. Given a directed graph D and a collection of ordered node pairs P let P [D] = {(u, v) ∈ P : D contains a uv-path}. In the Steiner Forest Orientation problem we are given an undirected graph G = (V, E) with edge-costs and a set P ⊆ V × V of ordered node pairs. The goal is to find a minimum-cost subgraph H of G and an orientation D of H such that P [D] = P . We give a 4-approximation algorithm for this problem. In the Maximum Pairs Orientation problem we are given a graph G and a multi-collection of ordered node pairs P on V . The goal is to find an orientation D of G such that |P [D]| is maximum. Generalizing the result of Arkin and Hassin [DAM'02] for |P | = 2, we will show that for a mixed graph G (that may have both directed and undirected edges), one can decide in n O(|P |) time whether G has an orientation D with P [D] = P (for undirected graphs this problem admits a polynomial time algorithm for any P , but it is NP-complete on mixed graphs). For undirected graphs, we will show that one can decide whether G admits an orientation D withhence this decision problem is fixed-parameter tractable, which answers an open question from Dorn et al. [AMB'11]. We also show that Maximum Pairs Orientation admits ratio O(log |P |/ log log |P |), which is better than the ratio O(log n/ log log n) of Gamzu et al. [WABI'10] when |P | < n.

A Tight Lower Bound for Steiner Orientation

“…They also gave a polynomial time algorithm for the special case when k = 2. Cygan, Kortsarz and Nutov [7] generalized this by giving an n O(k) algorithm for all k ≥ 1, i.e., STEINER ORIENTATION is in XP parameterized by k. Although the algorithm of Cygan et al is polynomial time for fixed k, the degree of the polynomial changes as k changes. This left open the question of whether one could design an FPT algorithm for STEINER ORIENTATION parameterized by k, i.e., an algorithm which runs in time f (k) • n O (1) for some computable function f independent of n.…”