Exact Exponential Time Algorithms for Max Internal Spanning Tree

We consider the NP-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O*(c^n) (c <= 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O*(1.8669^n) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364^k n^O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analyses for parameterized algorithms.

💡 Research Summary

The paper addresses the Maximum Internal Spanning Tree (MIST) problem, which asks for a spanning tree of a given connected undirected graph that maximizes the number of internal vertices (vertices of degree at least two in the tree). Since a Hamiltonian path is a spanning tree with exactly |V|‑2 internal vertices, MIST generalizes the Hamiltonian‑Path problem and is therefore NP‑hard. The authors contribute three families of exact exponential‑time algorithms, each improving on previously known bounds.

First, they present dynamic‑programming (DP) schemes for general graphs and for graphs with bounded maximum degree. The DP state records a subset of vertices together with a “frontier” that captures how the partial tree connects to the rest of the graph. By iterating over all possible extensions of the frontier, the algorithm enumerates all feasible partial trees. The number of states grows as O*(cⁿ) where the base c depends on the maximum degree Δ. For unrestricted graphs c is at most 3, and for Δ≤4 it drops slightly below 3. This establishes that, even without any structural restriction, MIST can be solved in O*(3ⁿ) time, matching the best known exponential‑time bounds for many related spanning‑tree optimisation problems.

The core of the paper, however, is a branching‑and‑reduction algorithm tailored to graphs whose maximum degree is three. The algorithm applies a set of simple reduction rules repeatedly: (i) vertices of degree one must become leaves and can be removed together with their incident edge, (ii) vertices of degree two force the inclusion of one of their incident edges, which in turn determines the status of the other edge. After exhaustive application of these rules the remaining “core” graph consists solely of degree‑three vertices.

At this point the algorithm selects an arbitrary edge e = {u, v} incident to a degree‑three vertex and creates two recursive branches: one where e is forced into the spanning tree, and one where e is forbidden. In the “include” branch u’s degree drops to two, making u an internal vertex; in the “exclude” branch the removal of e may create new degree‑one or degree‑two vertices, triggering further reductions. The recursion proceeds until the graph becomes empty, at which point the number of internal vertices accumulated along the path is the solution.

To analyse the running time the authors employ a Measure‑and‑Conquer (M&C) framework. They define a measure µ = |V₃| + α·|V₂| + β·|V₁|, where Vₖ denotes the set of vertices of degree k in the current instance, and α, β are constants in (0,1) chosen to balance the contributions of degree‑two and degree‑one vertices. Each reduction rule strictly decreases µ, and each branching step reduces µ by two different amounts δ₁ and δ₂ in the two branches. By solving the recurrence T(µ) ≤ T(µ‑δ₁) + T(µ‑δ₂) and optimising α and β numerically, the authors obtain the smallest possible exponential base, which turns out to be 1.8669. Consequently the algorithm runs in O*(1.8669ⁿ) time while using only polynomial space (the recursion needs to store the current graph and the measure value).

The paper also investigates a parameterised version of MIST: given an integer k, decide whether a spanning tree with at least k internal vertices exists. Using essentially the same reduction rules, the authors redefine the measure as the remaining “budget” k′ = k – (already secured internal vertices). The branching analysis is repeated with respect to k′, yielding a recurrence whose worst‑case root is 2.1364. Hence the parameterised algorithm runs in O*(2.1364ᵏ·n^{O(1)}) time, which is the first instance of a Measure‑and‑Conquer analysis applied to a parameterised graph problem.

The contributions are significant for several reasons. The DP results show that bounded‑degree graphs admit a modest improvement over the trivial O*(2ⁿ) bound, while the branching algorithm dramatically lowers the exponent for the important class of subcubic graphs. The polynomial‑space requirement makes the algorithm practical for moderate‑size instances where exponential‑space DP would be infeasible. Moreover, the parameterised result provides a fast exact method when the desired number of internal vertices is small, a scenario that arises in network design and phylogenetic reconstruction.

In the discussion, the authors outline future directions: extending the branching framework to graphs with Δ=4 or Δ=5, refining the reduction rules to obtain even smaller bases, performing empirical studies to validate the theoretical constants, and applying the Measure‑and‑Conquer technique to related optimisation problems such as Maximum Internal Path or Maximum Internal Cycle. Overall, the paper advances the state of the art in exact exponential‑time algorithms for spanning‑tree optimisation, introduces a novel use of Measure‑and‑Conquer in the parameterised setting, and opens several avenues for further research.