Approximation and parameterized algorithms for covering disjointness-compliable set families

2025-12-23

AI Summary

This paper studies the set-family covering problem under “disjoint compatibility”—a structural condition that generalizes symmetry—and unifies classical network design problems including k-MST, Generalized Point-to-Point (G-P2P) connectivity, and multi-root Covering Steiner. We propose the first generic $O(alpha log au)$-approximation framework for this class. Technically, we integrate structural analysis of compatible set families, an extended primal-dual method, parameterized search trees, and recursive contraction. We prove that an $alpha$-to-$(alpha+1)$ approximation is achievable in $O^*(3^ au)$ time, establishing a fundamental trade-off between fixed-parameter tractability and approximation ratio. Our contributions include: (i) the first deterministic $O(log n)$-approximation algorithm for G-P2P; (ii) improving the approximation ratio for multi-root Covering Steiner to $O(log^4 n)$; and (iii) achieving exact $O^*(3^ au)$-time solvability for the proper case.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

A set-family ${cal F}$ is disjointness-compliable if $A' subseteq A in {cal F}$ implies $A' in {cal F}$ or $A setminus A' in {cal F}$; if ${cal F}$ is also symmetric then ${cal F}$ is proper. A classic result of Goemans and Williamson [SODA 92:307-316] states that the problem of covering a proper set-family by a min-cost edge set admits approximation ratio $2$, by a classic primal-dual algorithm. However, there are several famous algorithmic problems whose set-family ${cal F}$ is disjointness-compliable but not symmetric -- among them $k$-Minimum Spanning Tree ($k$-MST), Generalized Point-to-Point Connection (G-P2P), Group Steiner, Covering Steiner, multiroot versions of these problems, and others. We will show that any such problem admits approximation ratio $O(αlog τ)$, where $τ$ is the number of inclusion-minimal sets in the family ${cal F}$ that models the problem and $α$ is the best known approximation ratio for the case when $τ=1$. This immediately implies several results, among them the following two. (i) The first deterministic polynomial time $O(log n)$-approximation algorithm for the G-P2P problem. Here the $τ=1$ case is the $k$-MST problem. (ii) Approximation ratio $O(log^4 n)$ for the multiroot version of the Covering Steiner problem, where each root has its own set of groups. Here the $τ=1$ case is the Covering Steiner problem. We also discuss the parameterized complexity of covering a disjointness-compliable family ${cal F}$, when parametrized by $τ$. We will show that if ${cal F}$ is proper then the problem is fixed parameter tractable and can be solved in time $O^*(3^τ)$. For the non-symmetric case we will show that the problem admits approximation ratio between $α$ and $α+1$ in time $O^*(3^τ)$, which is essentially the best possible.

Problem

• Develop approximation algorithms for covering disjointness-compliable set families.
• Apply results to problems like G-P2P and multiroot Covering Steiner.
• Analyze parameterized complexity with respect to inclusion-minimal sets.

Innovation

• Develops approximation algorithms for disjointness-compliable set families.
• Uses parameter τ to bound approximation ratios and complexity.
• Applies primal-dual methods to asymmetric covering problems.