A Strong Linear Programming Relaxation for Weighted Tree Augmentation

Abstract

The Weighted Tree Augmentation Problem (WTAP) is a fundamental network design problem where the goal is to find a minimum-cost set of additional edges (links) to make an input tree 2-edge-connected. While a 2-approximation is standard and the integrality gap of the classic Cut LP relaxation is known to be at least 1.5, achieving approximation factors significantly below 2 has proven challenging. Recent advances of Traub and Zenklusen using local search culminated in a ratio of $1.5+ε$, establishing the state-of-the-art. In this work, we present a randomized approximation algorithm for WTAP with an approximation ratio below 1.49. Our approach is based on designing and rounding a strong linear programming relaxation for WTAP which incorporates variables that represent subsets of edges and the links used to cover them, inspired by lift-and-project methods like Sherali-Adams.

Figures (12)

• Figure 1: Blue edges are correlated, and the remaining edges are not. Here we show a possible split Structured LP solution. The links of $\tilde{z}$ are in red and all have value $\frac{1}{2}$. There will be three sets $V_i$ for which $E(V_i)$ is non-empty. Two of the sets are shown boxed. The third set contains all vertices not in the two displayed partitions, plus the top vertex of each of the displayed boxed sets. Observe that the sets partition the edges and links, while there can be some overlap in the vertex sets.
• Figure 2: Shown is an example of a tree $Z_i^+$, where in bold blue is the subtree $A_i^\mathscr{E}$. $\mathcal{Q}_i$ has four members, $Q_1,\dots,Q_4$, labeled here. Note that $Q_2$ and $Q_3$ are treated as different subtrees.
• Figure 3: Example of an event $\mathscr{F} = (R, R_\mathsf{small}, L_\mathsf{small})$ for a subtree $R$ defined by edges $\{e,f,g,h\}$ (depicted by thick lines), with its leaves in $V(R_\mathsf{small})$ in blue. The set $L_\mathsf{small}$ is depicted by dashed lines and note that the edges in $R_\mathsf{small} = \{e,f\}$ are each covered by at most $2 \leq \rho$ links in $L_\mathsf{small}$. We remark that the tree edges in $R \setminus R_\mathsf{small} = \{f, h\}$ are covered by more than $\rho$ links, which are not depicted in the figure.
• Figure 4: Examples of extensions. Left: An event $\mathscr{F}$ defined on subtree $R$ (edges $e_1, e_2, e_3$ in red) is extended to a larger subtree $Q$ which also includes edges in green ($f_1,f_2, \ldots, f_6$). Here, the unique extension of $\mathscr{F}$ that is consistent is found by growing outwards from the center $c=v$. Right: An example where the event $\mathscr{F}$ is defined on a subtree $R$ with a single edge $e_1$ and $Q$ consists additionally of $f_1, f_2$. Here we have to be careful to only extend $R$ in one direction to make sure that there is a unique extension consistent with $\mathscr{F}$ and the intended solution. As we prioritize the root of $R$ if it is an internal vertex of $Q$, we get that $c=u$ in this case, and we extend $R$ upwards.
• Figure 5: Left: An event $\mathscr{F}$ defined on subtree $R(\mathscr{F})$ (edges $f, g, h, i, j, k$) with $R_\mathsf{small}(\mathscr{F})$ being the thick edges $(f, g, i, k)$ and $L_\mathsf{small}(\mathscr{F})$ the dashed links ($\ell_1, \ell_2, \ell_3)$. Right: Expanding the supernode $s$ we obtain the tree $\mathsf{expand}(R(\mathscr{F}))$, where $R_u$ consists of $e$ plus the blue edges and $R_v$ consists of $e$ plus the red edges. Suitable choices of $S$ include $\{\ell_1, \ell_2\}$ and $\{\ell_1, \ell_2, \ell_4\}$, so that $\mathscr{E}_S^w$ will be well-defined for $w \in \{u, v\}$.

Theorems & Definitions (68)

• Theorem 1.1.1: label=thm:main
• Theorem 2.2.1: FGKS18
• Theorem 3.0.1: label=thm:reduction_to_structured
• Remark 3.1.1
• Theorem 3.2.1
• Theorem 3.2.2: label=thm:structured_LP_round
• Theorem 3.2.3: label=thm:round_odd_cut
• proof
• Lemma 3.3.2
• proof