On the Bidirected Cut Relaxation for Steiner Forest

TL;DR

This paper advances the understanding of the Bidirected Cut Relaxation for Steiner Forest by proposing a half-integral rounding scheme with a provable approximation guarantee: from any half-integral forest-BCR solution, one can obtain a feasible Steiner Forest with cost at most $(16/9)\cdot c(x)$. The core technique is a novel recursive densest-subgraph contraction algorithm that, together with MST augmentation on the densest subgraph and careful LP restructuring, yields a polynomial-time rounding. The work also analyzes structural properties such as representation dependence, proves a lower bound of $\tfrac{3}{2}$ on the integrality gap, and shows the Steiner Tree case (single nontrivial demand component) aligns with existing Steiner Tree relaxations, achieving a gap strictly below 2. Overall, the results contribute a tighter understanding of LP relaxations for Steiner Forest and point toward potential improvements beyond the classic 2-approximation, including insights into when representations influence LP values.

Abstract

The Steiner Forest problem is an important generalization of the Steiner Tree problem. We are given an undirected graph with nonnegative edge costs and a collection of pairs of vertices. The task is to compute a cheapest forest with the property that the elements of each pair belong to the same connected component of the forest. The current best approximation factor for Steiner Forest is 2, which is achieved by the classical primal-dual algorithm; improving on this factor is a big open problem in the area. Motivated by this open problem, we study an LP relaxation for Steiner Forest that generalizes the well-studied Bidirected Cut Relaxation for Steiner Tree. We prove that this relaxation has several promising properties. Among them, it is possible to round any half-integral LP solution to a Steiner Forest instance while increasing the cost by at most a factor 16/9. To prove this result we introduce a novel recursive densest-subgraph contraction algorithm.

Figures (6)

• Figure 1: The left part of figure illustrates an LP solution for an instance with pairs $\{a_1,a_2\}$, $\{b_1,b_2\}$, and $\{c_1,c_2\}$. Every edge $e$ with $x^r_e =\frac{1}{2}$ is drawn in the same color as the vertex $r$. In the middle, we see the LP solution in the recursive call of Algorithm \ref{['algorithm']} (applied to the instance arising from the contraction of $W=\{b_1,b_2\}$) after rerouting and splitting-off. The solution returned by the algorithm is shown on the right.
• Figure 2: The instance and LP solution we use to establish an integrality gap lower bound for $q=4$. The figure shows the graph $G=(V,E)$ (left), the demand graph of the instance (middle), and an illustration of the values of $x_e^r$ in the LP solution (right). For an edge $e\in \overrightarrow{E}$ we have $x^r_e = \frac{1}{q} = \frac{1}{4}$ if $e$ is drawn in the same color as the vertex $r$. All other variables $x^r_e$ are zero. In particular, for every non-colored vertex $r$, we have $x^r_e=0$ for all $e\in \overrightarrow{E}$.
• Figure 3: Illustration of the graph $G=(V,E)$ (top), the demand graph for the pairs $\mathcal{P}_1$ (middle), and the demand graph for a different representation $\mathcal{P}_2$ of $\mathcal{P}_1$ (bottom). All edges have cost one.
• Figure 4: Optimal solutions to \ref{['eq:forest-bcr']} for the two different instances $((V,E), c, \mathcal{P}_1)$ (top) and $((V,E), c, \mathcal{P}_2)$ (bottom). For all red edges we have $x^{b_1}_e = \frac{1}{2}$, for all green edges we have $x^{c_2}_e = \frac{1}{2}$, for all blue edges we have $x^{d_1}_e = \frac{1}{2}$, and for all orange edges, we have $x^{e_2}_e = \frac{1}{2}$. All other variables $x^r_e$ are zero. For the variables $z^r_P$, all positive values are shown. The solution shown for the instance $((V,E), c, \mathcal{P}_1)$ has value $12$ and the solution shown for the instance $((V,E), c, \mathcal{P}_2)$ has value $13$.
• Figure 5: The picture illustrates the values $y^r_{U,P}$ for $r=b_1$ (top), $r=a_1$ (middle) and $r=a_2$ (bottom) for an optimal dual solution for $((V,E), c, \mathcal{P}_1)$. The colored sets indicate non-zero variables $y^r_{U,P}$, where the color of the drawn set $U$ indicates the pair $P$. For every drawn set $U$, we have a variable $y^r_{U,P}$ with value $1$, where $P=\{a_1,a_2\}$ for darkblue sets, $P=\{a_2, a_3\}$ for orange sets, $P=\{b_1,b_2\}$ for red sets, $P=\{c_1,c_2\}$ for green sets, $P=\{d_1,d_2\}$ for lightblue sets, and $P=\{e_1,e_2\}$ for violet sets.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof