Model-Agnostic Approximation of Constrained Forest Problems

TL;DR

This paper introduces a model-agnostic meta-algorithm called shell-decomposition that efficiently approximates a broad class of constrained forest problems across multiple computational models, including distributed settings.

Contribution

The paper presents the shell-decomposition algorithm, a novel, unified approach for approximating constrained forest problems in various computational models, with specific algorithms for multiple NP-hard problems.

Findings

Achieves a (2+ε)-approximation for CFPs in different models.

Provides efficient distributed algorithms with near-optimal round complexities.

Demonstrates flexibility by instantiating the algorithm for multiple NP-hard problems.

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 $(2+\epsilon)$-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 $\epsilon$, we obtain the following $(2+\epsilon)$-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 $(2+\epsilon)$-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 $(2+\epsilon)$-approximation in $O(\sqrt{n}+D)$ rounds. 4. For Facility Placement and Connection, a relative of non-metric Facility Location, we obtain a $(2+\epsilon)$-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.