Forcing a unique minimum spanning tree and a unique shortest path
Tatsuya Gima, Yasuaki Kobayashi, Yota Otachi, Takumi Sato
TL;DR
The paper analyzes the computational complexity of minimum forcing and anti-forcing sets for two classical problems: shortest $s$-$t$ paths and minimum-weight spanning trees. It leverages a matroid-centric view to obtain polynomial-time algorithms for the MST variants and reveals a complexity gap where forcing SP is poly-time but anti-forcing SP is NP-complete. For shortest paths, it provides a DAG-based dynamic programming approach for forcing sets, and shows NP-hardness for anti-forcing sets, with tractable exceptions on fixed paths and bounded-treewidth graphs via advanced logic techniques. The results establish clear tractability boundaries for forcing-type questions in fundamental combinatorial problems and extend matroid methods to these decision problems. Overall, the work highlights when forcing constructions remain manageable and when they inherit intractability from parallel decision problems like Vertex Cover, offering both theoretical and practical insights for combinatorial optimization.
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: the decision version of finding a minimum forcing set for perfect matchings is NP-complete [Adams et al., Discret. Math. 2004] and that of finding a minimum forcing set for satisfying assignments for 3CNF formulas is $Σ_2^{\mathrm{P}}$-complete [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 the decision version of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-complete.