Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees

TL;DR

The paper resolves the fine-grained complexity of computing minimum-cost spanning trees with vertex degree constraints across multiple graph-structure parameters. It develops a dense-graph algorithmic framework (NLC-/clique-width) and width-parametrized DP techniques (treewidth, pathwidth, cutwidth), combined with Cut&Count and the Isolation Lemma, to achieve ETH/SETH-tight results: SETH-tight upper and lower bounds for pathwidth and cutwidth, ETH-tight algorithms for clique-width, and nearly SETH-tight bounds for treewidth. The core contributions include a novel pattern-based approach for clique-width, randomized single-exponential DP for pathwidth/cutwidth, and tight lower bounds via CSP-based reductions, illustrating a precise landscape for degree-constrained spanning trees. Collectively, these results illuminate the feasibility limits of exact degree-constrained MST computations on structured graphs and have implications for related graph-factor problems and parameterized algorithm design.

Abstract

We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$ achieves a degree specified by $D(v)$. Specifically, we consider three kinds of constraint functions ordered by their generality -- $D$ may either assign each vertex to a list of admissible degrees, an upper bound on the degrees, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical graph parameters of the input graph $G$. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by the pathwidth and cutwidth, an ETH-tight algorithm parameterized by the cliquewidth, and a nearly SETH-tight algorithm parameterized by treewidth.

Figures (4)

• Figure 1: Let the pattern $A = \langle v, u \rangle$, and $A_2 = \pi_2(A, v, u, i = 2)$ for $v, u$ as given in the figure, and $(v', u') = \pi_2(2,v,u)$. For some edge-mapping $\alpha$, fixed forest $(\tilde{F}, \tilde{g})$, and $\alpha$-spanning tree $T_A$ for $(F_A, f_A)$ and $(\tilde{F}, \tilde{g})$, we illustrate how to obtain an $\alpha$-spanning tree $T_{A_2}$ for $(F_{A_2}, f_{A_2})$ and $(\tilde{F}, \tilde{g})$ by "moving" the blue edge, certifying that $A\preceq A_2$. Observe that if the dashed path were to connect with the $i$'th vertex corresponding to $u$ instead, we could certify $A \preceq A_1 \coloneqq \pi_1(A, v, u, i = 2)$ in a similar way; this is the intuitive reason for $\{ A \} \preceq \{A_1, A_2\}$, i.e., the forward direction of \ref{['cor:cw_equiv_rrule12']}.
• Figure 2: Example illustrating how $T_{A_1}^*$ is derived from $T_{A_1}$ in the proof of Lemma \ref{['lem:cw_equiv_rrule_1']}.
• Figure 3: Gadgets for the proof of \ref{['thm:tw_lb']}
• Figure 4: Gadgets for the proof of \ref{['thm:ctw_lb']}

Theorems & Definitions (61)

• Definition : Nice tree decomposition
• Theorem 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Lemma 9