Model-Agnostic Approximation of Constrained Forest Problems

TL;DR

The paper presents a model-agnostic shell-decomposition framework that achieves a $(2+\varepsilon)$-approximation for Constrained Forest Problems (CFPs) across multiple computational models by reducing CFP subproblems to four core building blocks: (i) (1+\varepsilon)-approximate Set-Source Shortest-Path Forest (aSSSP), (ii) Minimum Spanning Tree (MST), (iii) Root-Path Selection (RPS), and (iv) Forest-Function Evaluation (FFE). It unifies classic survivable network design problems such as Steiner Forest (SF), Point-to-Point Connection (PPC), and Facility Placement and Connection (FPC) under a single framework, with runtime depending on the efficiencies of these subroutines and the degree of shortcut knowledge, via Partwise Aggregation (PA) and related notions. The authors demonstrate instantiations in Congest, PRAM, and Multi-Pass Streaming (MPS) models, showing how to replace the typical $\sqrt{n}+D$ bottleneck with topology-aware parameters like $T^{PA}$ and $T^{DA}(p)$, and they explore randomized equality testing to achieve near-universal time for certain SF input representations. The work offers a principled path toward universal-optimal CFP algorithms, highlights the fundamental role of PA and DA subroutines, and suggests several open questions on extending to broader function classes and achieving tighter universal bounds in various models.

Abstract

Constrained Forest Problems (CFPs) as introduced by Goemans and Williamson in 1995 capture a wide range of network design problems with edge subsets as solutions, such as Minimum Spanning Tree, Steiner Forest, and Point-to-Point Connection. While individual CFPs have been studied extensively in individual computational models, a unified approach to solving general CFPs in multiple computational models has been lacking. Against this background, we present the shell-decomposition algorithm, a model-agnostic meta-algorithm that efficiently computes a $(2+ε)$-approximation to CFPs for a broad class of forest functions. To demonstrate the power and flexibility of this result, we instantiate our algorithm for 3 fundamental, NP-hard CFPs in 3 different computational models. For example, for constant $ε$, we obtain the following $(2+ε)$-approximations in the Congest model: 1. For Steiner Forest specified via input components, where each node knows the identifier of one of $k$ disjoint subsets of $V$, we achieve a deterministic $(2+ε)$-approximation in $O(\sqrt{n}+D+k)$ rounds, where $D$ is the hop diameter of the graph. 2. For Steiner Forest specified via symmetric connection requests, where connection requests are issued to pairs of nodes, we leverage randomized equality testing to reduce the running time to $O(\sqrt{n}+D)$, succeeding with high probability. 3. For Point-to-Point Connection, we provide a $(2+ε)$-approximation in $O(\sqrt{n}+D)$ rounds. 4. For Facility Placement and Connection, a relative of non-metric Facility Location, we obtain a $(2+ε)$-approximation in $O(\sqrt{n}+D)$ rounds. We further show how to replace the $\sqrt{n}+D$ term by the complexity of solving Partwise Aggregation, achieving (near-)universal optimality in any setting in which a solution to Partwise Aggregation in near-shortcut-quality time is known.

Figures (4)

• Figure 1: Operation of our shell-decomposition algorithm (\ref{['alg:gw-clean']}) on an $s$-$t$-shortest-path instance seeking to connect the red nodes, starting with $r\xspace_0 = 1/2$ (cf. Line \ref{['line:init-radius']}), and working over phases $0$, $1$, and $2$. Panels are labeled with their phase number and the illustrated step of \ref{['alg:gw-clean']}. Nodes absorbed by the SSSP forest are drawn in orange, edge-cost reduction is indicated in purple, edges selected into the SSSP forest are marked in green, and edges selected into the solution are marked in black. Distance approximations and deferred forest-function evaluation are not shown.
• Figure 2: Illustration of our shell-decomposition argument on a small instance of Steiner Forest (SF--IC). Gray lines indicate original edges, black lines indicate (parts of) selected edges, black circle linings indicate active components, and node colors indicate input components. The illustrations provided by goemans1995approximation (Figs. 2--5) are stylistically similar, but our visualization clarifies the phase-wise charging argument underlying our shell-decomposition algorithm.
• Figure 3: Overview of our model-agnostic shell-decomposition algorithm for approximating Constrained Forest Problems. Main tasks are colored; simple book-keeping operations are shaded in gray.
• Figure 4: Example of the lower-bound graph used to reduce Equality Testing to Steiner Forest, for an instance with $n = 8$. Node colors indicate input components (SF--CIC) resp. connection requests (SF--SRC), and node shapes indicate which nodes are simulated by which player.

Theorems & Definitions (53)

• Theorem 1: Model-Agnostic Complexity of Constrained Forest Problems
• Definition 1: Steiner Forest (SF)
• Definition 2: Point-to-Point Connection (PPC)
• Definition 3: FPC [intuitive]
• Definition 4: FPC [rephrased]
• Definition 5: Partwise Aggregation ghaffari2016algorithms
• Definition 6: Shortcut Quality
• Theorem 2: Original GW Algorithm goemans1995approximation
• Theorem 3
• Lemma 1