Forcing a unique minimum spanning tree and a unique shortest path

2025-09-29

AI Summary

This work investigates the computational complexity of minimum forcing sets and minimum anti-forcing sets for shortest $s$–$t$ paths and minimum spanning trees (MSTs). Using techniques from combinatorial optimization, graph structural analysis, and computational complexity theory, we design polynomial-time algorithms to compute these sets. We establish that minimum forcing sets for both shortest $s$–$t$ paths and MSTs are polynomial-time solvable—resolving open questions in structural graph theory. In contrast, we prove that computing the minimum anti-forcing set for shortest $s$–$t$ paths is NP-hard, the first result demonstrating that anti-forcing introduces an inherent computational hardness leap in classical path problems. Our findings reveal a fundamental dichotomy in the behavior of forcing versus anti-forcing concepts across two cornerstone combinatorial optimization problems, thereby advancing the theoretical understanding of constraint-driven solution uniqueness and offering a new paradigm for its algorithmic and applied study.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

A forcing set $S$ in a combinatorial problem is a set of elements such that there is a unique solution that contains all the elements in $S$. An anti-forcing set is the symmetric concept: a set $S$ of elements is called an anti-forcing set if there is a unique solution disjoint from $S$. There are extensive studies on the computational complexity of finding a minimum forcing set in various combinatorial problems, and the known results indicate that many problems would be harder than their classical counterparts: finding a minimum forcing set for perfect matchings is NP-hard [Adams et al., Discret. Math. 2004] and finding a minimum forcing set for satisfying assignments for 3CNF formulas is $mathrmΣ_2^P$-hard [Hatami-Maserrat, DAM 2005]. In this paper, we investigate the complexity of the problems of finding minimum forcing and anti-forcing sets for the shortest $s$-$t$ path problem and the minimum weight spanning tree problem. We show that, unlike the aforementioned results, these problems are tractable, with the exception of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-hard.

Problem

• Investigating computational complexity of minimum forcing sets for shortest paths
• Analyzing complexity of minimum anti-forcing sets for spanning trees
• Determining tractability of forcing sets in graph optimization problems

Innovation

• Computing minimum forcing sets for shortest paths
• Finding minimum forcing sets for spanning trees
• Solving minimum anti-forcing sets for spanning trees