Exact Algorithms for Distance to Unique Vertex Cover

2025-02-07

AI Summary

This paper investigates the Minimum Vertex Cover Uniqueness problem (MU-VC): given a graph (G), determine the minimum number of vertex modifications required to ensure the existence of a unique minimum vertex cover (MVC) in the resulting graph—without requiring preservation of the original MVC size. We establish, for the first time, that MU-VC is (Sigma_2^P)-complete, even when restricted to planar graphs. Our approach integrates parameterized algorithms, tree decompositions, dynamic programming, and structural graph parameter analysis. Key contributions include: (1) an (O(n)) exact algorithm for trees; (2) a fixed-parameter tractable (FPT) algorithm with respect to the combined parameter treewidth + maximum degree (Delta), and an XP algorithm with respect to clique-width; and (3) an FPT algorithm for clique-width when augmenting the parameter set with solution size. Collectively, these results precisely delineate the computational tractability boundaries of MU-VC across fundamental graph parameters.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Horiyama et al. (AAAI 2024) studied the problem of generating graph instances that possess a unique minimum vertex cover under specific conditions. Their approach involved pre-assigning certain vertices to be part of the solution or excluding them from it. Notably, for the extsc{Vertex Cover} problem, pre-assigning a vertex is equivalent to removing it from the graph. Horiyama et al.~focused on maintaining the size of the minimum vertex cover after these modifications. In this work, we extend their study by relaxing this constraint: our goal is to ensure a unique minimum vertex cover, even if the removal of a vertex may not incur a decrease on the size of said cover. Surprisingly, our relaxation introduces significant theoretical challenges. We observe that the problem is $Sigma^2_P$-complete, and remains so even for planar graphs of maximum degree 5. Nevertheless, we provide a linear time algorithm for trees, which is then further leveraged to show that MU-VC is in extsf{FPT} when parameterized by the combination of treewidth and maximum degree. Finally, we show that MU-VC is in extsf{XP} when parameterized by clique-width while it is fixed-parameter tractable (FPT) if we add the size of the solution as part of the parameter.

Problem

• Ensures unique minimum vertex cover
• Addresses removal impact on vertex cover
• Develops algorithms for planar and tree graphs

Innovation

• Unique minimum vertex cover
• Linear time algorithm for trees
• FPT parameterized by treewidth and degree