Lower Bounds for the Minimum Spanning Tree Cycle Intersection Problem
Manuel Dubinsky, Kun-Mao Chao, César Massri, Gabriel Taubin
TL;DR
The paper investigates the Minimum Spanning Tree Cycle Intersection problem, formalizing the intersection number $\cap(G)$ as the count of non-empty intersections among tree-cycles arising from a spanning tree. It first proves a general lower bound $\frac{1}{2}\left(\frac{\nu^2}{n-1} - \nu\right)$ with $\nu = m - n + 1$, derived via a convex-optimization view over bond contributions, yielding $l_{n,m}$. It then conjectures a tighter bound for graphs with a universal vertex, expressed in terms of $2\nu = q(n-1) + r$ as $\hat{l}_{n,m} = (n-1)\binom{q}{2} + q r \leq \cap(G)$, grounded in the notion of $\nu$-regular graphs. The authors support the conjecture with experiments and discuss implications for the sparsity of the cycle-intersection matrix and potential benefits for fast linear solvers on large graphs. Overall, the work provides foundational lower bounds and a conjectured tight form that could guide both theoretical and practical approaches to MSTCI and related discrete differential forms computations.
Abstract
Minimum spanning trees are important tools in the analysis and design of networks. Many practical applications require their computation, ranging from biology and linguistics to economy and telecommunications. The set of cycles of a network has a vector space structure. Given a spanning tree, the set of non-tree edges defines cycles that determine a basis. The intersection of two such cycles is the number of edges they have in common and the intersection number -- denoted $\cap(G)$ -- is the number of non-empty pairwise intersections of the cycles of the basis. The Minimum Spanning Tree Cycle Intersection problem consists in finding a spanning tree such that the intersection number is minimum. This problem is relevant in order to integrate discrete differential forms. In this paper, we present two lower bounds of the intersection number of an arbitrary connected graph $G=(V,E)$. In the first part, we prove the following statement: $$\frac{1}{2}\left(\frac{ν^2}{n-1} - ν\right) \leq \cap(G),$$ where $n = |V|$ and $ν$ is the \emph{cyclomatic number} of $G$. In the second part, based on some experimental results and a new observation, we conjecture the following improved tight lower bound: $$(n-1) \binom{q}{2} + q \ r\leq \cap(G),$$ where $2 ν= q (n-1) + r$ is the integer division of $2 ν$ and $n-1$. This is the first result in a general context, that is for an arbitrary connected graph.