Adaptive Vertex Reduction

• Adaptive vertex reduction is an algorithmic framework that dynamically removes vertices from geometric and combinatorial structures based on criteria like curvature or LP integrality gaps.
• It enables effective tradeoffs between reduced complexity and approximation error, as seen in mesh simplification where removal thresholds directly affect vertex count and accuracy.
• In graph optimization, the approach detects c-essential vertices via v-avoiding LP relaxations, streamlining vertex-hitting set problems and enhancing computational efficiency.

Adaptive vertex reduction refers to algorithmic strategies for systematically eliminating vertices from combinatorial or geometric structures—such as graphs, mesh triangulations, or optimization problem instances—based on dynamically computed criteria. Two principal instantiations of adaptive vertex reduction are documented: (1) intrinsic mesh simplification via curvature-based filtering and barycentric tracking, and (2) search-space reduction in graph optimization problems via detection of “c-essential” vertices through LP integrality-gap analysis. Both paradigms aim to minimize complexity while controlling—by parameter or threshold—the degree of approximation or structural error introduced.

1. Intrinsic Vertex Reduction in Mesh Simplification

Intrinsic mesh simplification, as detailed by Shoemaker et al., operates on triangle meshes $M = (V, E, F)$ embedded in $\mathbb{R}^3$ by constructing an intrinsic triangulation $(V', E', F')$—a $\Delta$-complex equipped with positive edge lengths $\ell_{ij}$ for each $(i, j) \in E'$ (Shoemaker et al., 2023). The metric is intrinsic: edge lengths prescribe the unique Euclidean geometry within each triangle. The simplification procedure is governed by the notion of discrete Gaussian curvature $K_i = 2\pi - \sum_{ijk \in F'} \theta^i_{jk}$, where $\theta^i_{jk}$ is computed from the intrinsic edge lengths via the law of cosines.

The algorithm proceeds as follows:

• A priority queue $P$ contains all intrinsic vertices $i \in V'$ ordered by $|K_i|$.
• Vertices $i$ with $|K_i| < \tau$ (user-selected threshold $\tau$) are considered “nearly developable” and eligible for removal.
• To maintain consistency, the valence of $i$ is first reduced to 3 (internal) or 2 (boundary) using local intrinsic edge flips. Each such flip modifies mesh connectivity without altering intrinsic edge lengths, thus preserving the intrinsic metric.
• Upon reaching minimal valence, $i$ can be removed if the resulting local mesh satisfies triangle inequalities. The geometric location of the eliminated vertex is recorded by barycentric coordinates $(c_i^j, c_i^k, c_i^l)$ in the final 1-ring triangle $jkl$.
• Barycentric records are updated through further flips or vertex removals by means of explicit algebraic substitutions or two-dimensional transformations.

The full algorithm operates in near-linear time $O(N\log N + F)$, with most runtime devoted to barycentric-coordinate tracking (∼80%), preserves the Euler characteristic, and thus leaves global topological invariants and total curvature unchanged.

2. Experimental Evaluation and Error–Complexity Tradeoffs

Evaluation on the Thingi10k dataset (≈7,000 manifold meshes) demonstrates the method’s parameterized behavior (Shoemaker et al., 2023):

• For thresholds $\tau$ ranging from $10^{-9}$ to $\pi$, the fraction of removable vertices increases from $5.57\%\pm 13.84\%$ up to $88.75\%\pm 22.34\%$.
• Proportion of those actually removed (after local validation): $99.56\%\pm 4.73\%$ at $\tau=10^{-9}$, $91.41\%\pm 10.12\%$ at $\tau = 10^{-2}$, and $94.57\%\pm 9.17\%$ at $\tau = \pi$.
• Mean mesh processing times: $0.15$–$0.45$ s for removal only; $0.61$–$2.70$ s for barycentric tracking; total $0.76$–$3.15$ s per mesh, scaling with $\tau$.
• Mean-squared error (MSE) in PDE solution (Laplace on simplified mesh, spike at vertex $v$, interpolation via barycentrics): $\tau=10^{-4}$ yields $\operatorname{MSE}\approx 1.8 \times 10^{-5}$, increasing to $\operatorname{MSE}\approx 4.9 \times 10^{-3}$ for $\tau=10^{-2}$. Errors concentrate in regions of nonzero curvature.

This methodology achieves a smooth tradeoff between mesh complexity and metric error, governed by $\tau$—with higher $\tau$ permitting more aggressive reduction at the cost of fidelity.

3. Essential-Vertex–Based Reduction in Graph Optimization

In the domain of vertex-hitting set problems, adaptive reduction has been formalized through the concept of $c$-essential vertices (Jansen et al., 2024). For a minimization problem $\Pi$ on graphs, a vertex $v\in V(G)$ is $c$-essential if $v$ appears in every feasible solution of size at most $c \cdot \operatorname{opt}_\Pi(G)$. The search for adaptive reduction then becomes the computational detection of such essential vertices.

A canonical approach uses a “$v$-avoiding” LP relaxation:

$$\text{(LP}_v\text{)}\quad \min \sum_{u \in V(G)} x_u \quad \text{s.t.}\ \sum_{u \in O} x_u \geq 1\ \forall O,\ x_v = 0,\ 0 \leq x_u \leq 1$$

Here, obstacles $O$ express the vertex-hitting requirements. If, for each feasible singleton $\{v\}$, the $v$-avoiding LP has integrality gap at most $c$ and can be solved in polynomial time, then all $(c+1)$-essential vertices can be detected efficiently.

In the case of undirected Vertex Multicut, the $v$-avoiding LP has integrality gap $\leq 2$ when $\{v\}$ is a multicut, enabling polynomial-time detection of all 3-essential vertices, which must be present in any optimum of size $\leq k$. The same paradigm extends—given a bounded-gap $v$-avoiding LP and efficient solution algorithm—to Cograph Deletion (gap $\leq 2.5$ for certain inputs), again enabling adaptive reduction by essential-vertex detection prior to FPT branching.

4. Algorithmic Workflow and Pseudocode Summaries

In intrinsic mesh simplification (Shoemaker et al., 2023), the key routines are:

IntrinsicSimplify(M, τ): IntrinsicDelaunayRetriangulation(M) compute curvatures K[i] P ← priority queue by |K[i]| Removed ← ∅ while ¬P.empty(): i ← P.pop_min() if |K[i]| ≥ τ: break Q ← empty stack success ← TryRemove(i, Q) if success: update curvatures for neighbors reinsert eligible neighbors return M, Removed

For essential-vertex detection in graph optimization (Jansen et al., 2024), the routine is:

Input: G, terminal-pairs T, integer k S ← ∅ for each v ∈ V(G): solve (VM_v) to optimal λ_v if λ_v > k: S ← S ∪ {v} return S

In both frameworks, the adaptive nature is encoded: the selection or removal of vertices is guided step-by-step by dynamically computed objective quantities—curvature in geometric settings or LP-optimal values in combinatorial optimization.

5. Complexity, Lower Bounds, and Generalizations

Intrinsic reduction for mesh simplification operates in $O(n \log n + F)$ time, where the log-factor arises from the priority queue, and the bottleneck is barycentric-coordinate bookkeeping (Shoemaker et al., 2023).

For essential-vertex detection, per-vertex LPs can be solved in polynomial time; the overall runtime for Vertex Multicut, combining the essential detector with the Marx–Razgon FPT branching, achieves $2^{O(\ell^3)} n^{O(1)}$, where $\ell$ is the number of nonessential vertices in an optimal solution (Jansen et al., 2024). For Cograph Deletion, combining the detector with a branching algorithm yields $3.115^\ell n^{O(1)}$.

However, for certain problems (e.g., Directed Feedback Vertex Set), detecting $(2 - \varepsilon)$-essential vertices is NP-hard under the Unique Games Conjecture, establishing hardness of stronger reductions in these contexts. The improvement in Cograph Deletion arises specifically on instances where the trivial singleton solution is available; for general inputs, the standard LP relaxation has larger integrality gap.

6. Applicability and Limitations

Adaptive vertex reduction is robustly applicable to many mesh processing and graph-optimization problems whenever local “removability” can be evaluated efficiently—by curvature, valence, or LP gap. The essential-vertex pipeline—identify a suitable $v$-avoiding LP, analyze its gap, invoke general detection theorems, and combine with best-known FPT subroutines—extends to a broad class of vertex-hitting problems provided the requisite LP relaxations admit bounded integrality gaps and separation oracles.

A plausible implication is that the effectiveness of adaptive vertex reduction, whether geometric or combinatorial, is typically governed by an underlying tradeoff: aggressive reduction controlled by a parameter ($\tau$ or $c$) inexorably degrades solution accuracy or completeness, but enables tractable subproblem size for downstream algorithms. The choice of reduction parameter remains both the critical practical control and the main axis of algorithmic design.