Improved Approximation Algorithms for Path and Forest Augmentation via a Novel Relaxation

TL;DR

The paper addresses the Forest Augmentation Problem (FAP) and Path Augmentation Problem (PAP), seeking minimum edge additions to achieve 2-edge-connectivity. It introduces a novel reduction to structured PAP instances with a $(\frac{7}{4}+\varepsilon)$-approximation preservation, a new relaxation called 2-Edge Cover with Path Constraints (2ECPC), and a packing counterpart Track Packing Problem (TPP); a factor-revealing LP yields a strong starting solution that is then refined via bridge-covering and gluing within the structured setting. The combination yields a concrete $1.9412$-approximation for PAP and, together with prior reductions, a $1.9955$-approximation for FAP, breaking the long-standing 2-approximation barrier for these problems. The techniques—the reduction to structured instances, the 2ECPC/TPP relaxation pair, and the tailored credit-based augmentation—may be of independent interest for broader connectivity augmentation challenges and could inform future breakthroughs beyond TAP-like methods. Overall, the work significantly advances survivable network design by delivering tighter guarantees and new methodological tools for augmentation problems.

Abstract

The Forest Augmentation Problem (FAP) asks for a minimum set of additional edges (links) that make a given forest 2-edge-connected while spanning all vertices. A key special case is the Path Augmentation Problem (PAP), where the input forest consists of vertex-disjoint paths. Grandoni, Jabal Ameli, and Traub [STOC'22] recently broke the long-standing 2-approximation barrier for FAP, achieving a 1.9973-approximation. A crucial component of this result was their 1.9913-approximation for PAP; the first better-than-2 approximation for PAP. In this work, we improve these results and provide a 1.9412-approximation for PAP, which implies a 1.9955-approximation for FAP. One of our key innovations is a $(\frac{7}{4} + \varepsilon)$-approximation preserving reduction to so-called structured instances, which simplifies the problem and enables our improved approximation. Additionally, we introduce a new relaxation inspired by 2-edge covers and analyze it via a corresponding packing problem, where the relationship between the two problems is similar to the relationship between 2-edge covers and 2-matchings. Using a factor-revealing LP, we bound the cost of our solution to the packing problem w.r.t. the relaxation and derive a strong initial solution. We then transform this solution into a feasible PAP solution, combining techniques from FAP and related connectivity augmentation problems, along with new insights. A key aspect of our approach is leveraging the properties of structured PAP instances to achieve our final approximation guarantee. Our reduction framework and relaxation may be of independent interest in future work on connectivity augmentation problems.

Figures (4)

• Figure 1: Non-structured instance of PAP. Dashed edges are in $E(\mathcal{P})$, solid edges are links of $L$. Filled vertices are interior vertices of paths from $\mathcal{P}\xspace$, empty vertices are endvertices of paths in $\mathcal{P}\xspace$. The path $P = b_1 b_2 b_3 b_4 \in \mathcal{P}$ is a separator, the cycle $a_2 a_3 a_4 a_2$ is a 1-contractible subgraph, the path $c_1 c_2 c_3 d_4 d_3 d_2 d_1$ is a $P^2$-separator, the cycle $e_1 e_2 e_3 f_2 f_1 e_1$ is a $C^2$-separator, and the path $j_1 j_2 j_3$ is a degenerate path.
• Figure 2: Dashed edges are in $E(\mathcal{P})$, solid edges are links of $L$. Filled vertices are interior vertices of paths from $\mathcal{P}\xspace$, empty vertices are endvertices of paths in $\mathcal{P}\xspace$. Left: Feasible solution $Y$ for 2ECPC, since $E(\mathcal{P}\xspace) \cup Y$ is a 2-edge cover with $\delta_Y(P) \geq 2$ for each $P \in \mathcal{P}\xspace$. Right: Solution $X = \{T_1, T_2, T_3, T_4, T_5\}$ for the corresponding TPP instance with two tracks containing 1 link ($T_1 = \{ u_7u_8 \}$ and $T_2 = \{ v_5 u_9 \}$), two tracks containing 3 links ($T_3 = \{u_2w_1, u_1v_1, w_1 v_2\}$ and $T_4=\{v_9 w_{11}, u_{11} v_{11}, w_{11} v_{10}\}$) and one track containing 5 links ($T_5=\{u_5 w_4, u_4 v_4, w_4 w_3, u_3 v_3, w_3 u_6\}$). Note that $|Y| = 17 = 2 |\mathcal{P}\xspace| - 5 = 2|\mathcal{P}\xspace| - |X|$.
• Figure 3: Example for the bridge-covering process: a pseudo-ear covering the witness path from $u$ to $w$. $H$ consists of the dashed edges (in $E(\mathcal{P})$) and solid black edges (links of $H$). Filled vertices are interior vertices of paths from $\mathcal{P}\xspace$, empty vertices are endvertices of paths in $\mathcal{P}\xspace$. The big circles represent 2EC blocks or components. The blue edges form a pseudo-ear $Q^{uw}$ from $u$ to $w$. All bridges, vertices and blocks on the path from $u$ to $w$ in $H$ will be in a 2EC block in $H \cup Q^{uw}$.
• Figure 4: Example for the gluing algorithm and a good cycle. The bridgeless 2-edge-cover consists of large components (big circles) and small components (short cycles). $H$ consists of the dashed edges (in $E(\mathcal{P})$) and solid black edges (links of $H$). Blue edges are edges in $E(G) \setminus E(H)$. The cycle $e_4 e_5 e_6$ is a good cycle in the components graph as it contains 2 large components. The cycle $e_4 e_1 e_7$ is also a good cycle as it shortcuts the lower small components by using the Hamiltonian Path through $g$.

Theorems & Definitions (103)

• Theorem 1
• Theorem 2
• Definition 1: $\alpha$-contractable subgraph
• Definition 2: Degenerate Path
• Definition 3: ($\alpha, \eps)$-structured Instance
• Lemma 1
• Lemma 2
• Lemma 3
• Definition 4: Pseudo-ear and Witness Path
• Lemma 4