The Bidirected Cut Relaxation for Steiner Tree: Better Integrality Gap Bounds and the Limits of Moat Growing

TL;DR

It is proved that the integrality gap of BCR is at most 1.898, improving significantly on the previous bound of 1.9988, and it is proved that no moat-growing algorithm yields dual solutions certifying an integrality gap below 12/7.

Abstract

The Steiner Tree problem asks for the cheapest way of connecting a given subset of the vertices in an undirected graph. One of the most prominent linear programming relaxations for Steiner Tree is the Bidirected Cut Relaxation (BCR). Determining the integrality gap of this relaxation is a long-standing open question. For several decades, the best known upper bound was 2, which is achievable by standard techniques. Only very recently, Byrka, Grandoni, and Traub [FOCS 2024] showed that the integrality gap of BCR is strictly below 2. We prove that the integrality gap of BCR is at most 1.898, improving significantly on the previous bound of 1.9988. For the important special case where a terminal minimum spanning tree is an optimal Steiner tree, we show that the integrality gap is at most 12/7, by providing a tight analysis of the dual-growth procedure by Byrka et al. To obtain the general bound of 1.898 on the integrality gap, we generalize their dual growth procedure to a broad class of moat-growing algorithms. Moreover, we prove that no such moat-growing algorithm yields dual solutions certifying an integrality gap below 12/7. Finally, we observe an interesting connection to the Hypergraphic Relaxation.

Figures (17)

• Figure 1: An example of the moat growing process for the undirected LP. The instance has four terminals $a,d,e,f$ (shown as squares), and initially, we grow a set around each of these terminals (left picture). Once the sets $U_a$ (red) and $U_e$ (green) meet, we stop growing their corresponding dual variables and instead grow the dual variable $U_S$ (brown) for the set $S=\{a,e\}$. Once, the vertex $b$ is reachable from $S=\{a,e\}$ by tight edges, it is included in the set $U_S$. The same applies symmetrically for the sets $U_d$ (blue) and $U_f$ (orange) and the vertex $c$. The colors on the edges indicate the contributions of the sets $S\subseteq R$ to the tightness of the edges. An edge is tight once it is fully colored.
• Figure 2: An example of (a part of) an instance with two terminals $s_1$ and $s_2$, indicated as squares. The green and blue color indicate the contributions of the sets growings around $s_1$ (green) and $s_2$ (blue) to the tightness of the edges. To the edge $(m,v)$ both sets contribute equally. The edges between $s_i$ and $m$ have length $\frac{6}{7} \, t$ each, and the two terminals can reach the vertex $m$ at time $\frac{6}{7} \, t$. However, they are merged only at time $t = \frac{7}{12} \cdot \textnormal{dist}_{}\left(s_1,s_2\right)$. By this time they have reached the vertex $v$, which is at distance $\frac{2}{7} \, t$ from $m$. Thus, the edge from $m$ to $v$ effectively gets tight at speed $2$.
• Figure 3: An example of two terminals interacting. The top left picture shows (a part of) an instance with two terminals $s_1$ and $s_2$, where the numbers indicate the cost/length of the edges. The other pictures indicate the contributions of the sets growing around $s_1$ (green) and $s_2$ (blue) to the tightness of the edges at different times $t$. The terminals $s_1$ and $s_2$ have distance $36$ and have thus a merge time of (at most) $21$. Observe that at time $t=18$ the two sets meet at the vertex $m$, which is at distance $18$ from $s_1$ (and from $s_2$). At time $t=21$, the vertex $v$ is reachable from $s_1$ by a path of tight edges. The vertex $v$ has distance $27$ from $s_1$ and thus between the meeting time $t=18$ until the merge time $t=21$, the maximum distance of vertices reachable from $s_1$ grows effectively at speed $3$. (This maximum distance is not continuous and there are "jumps", but on average between the meeting time and the merge time it increases at speed $3$.)
• Figure 4: The left part of the figure shows (a part of) an instance with $k$ terminals interacting. If they meet at the vertex $m$ at time $t=1$, then the edge $(m,v)$ afterwards becomes tight at speed $k$. However, this instance is not MST-optimal, because the star centered at $m$ connecting the $k$ terminals would be an improving component (for $k>2$). (If the drop of this terminal set was smaller than $k$, then the terminals would have been merged much earlier and would not continue growing each on their own.) The example in the right part of the figure shows that even the interaction of just three terminals can lead to reaching far away vertices. In this example, the edges incident to the terminals are long, e.g. they could have length $1$ each, whereas the other edges are short, e.g. they have length $\varepsilon$ each for some small $\varepsilon > 0$. The figure shows the contributions of sets growing around the three terminals at time $t=1+\varepsilon$. The vertex $w$ is reachable from $s_1$ by a path of tight edges, but this path could be arbitrarily long. (The only bound on the distance of $s_1$ and $w$ that we can give in this example is $3+3\varepsilon$ by considering a non-tight path via one of the terminals $s_2$ or $s_3$. However, this bound is too weak to be useful.) Again, this instance is not MST-optimal, where a star centered at $v$ connecting $s_1, s_2, s_3$ is an improving component.
• Figure 5: An abstract drawing of the component $K$ in our invariant for $X=\{s_1,s_2\}$. The component shown here corresponds to the example from \ref{['fig:zig_zag_example']}. Both $s_1$ and $s_2$ reach the vertex $m$ at time $t = 18 = \frac{1}{2}\, \textnormal{dist}_{}\left(s_1,s_2\right) \geq \frac{1}{2}\, \textnormal{drop}_{G}\left(\{s_1,s_2\}\right)$. At time $t=21=\frac{7}{12}\, \textnormal{drop}_{G}\left(\{s_1,s_2\}\right)$ the two terminals are merged and become part of the same set $S$, which at this time has not interacted with any sets disjoint from $S$. From time $t=18$ until time $t=21$ the $m$-$v$ path becomes tight effectively at speed $3$. Thus, if $v$ is reached at time $21$, it can have distance $9$ from $m$.

Theorems & Definitions (130)

• Theorem 1.1
• Definition 1.2: MST-optimal instances
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: Merge Plan
• Definition 2.2
• Definition 2.3: canonical merge plan
• Definition 2.4: feasible merge plan
• Definition 2.6: components, full components