A Weight Function Lemma Heuristic for Graph Pebbling

TL;DR

This paper analyzes the Weight Function Lemma (WFL) for graph pebbling, focusing on its dual formulation which constructs certificates from $r$-strategies with vertex weights. It identifies bottlenecks in weight distribution, particularly the tension between maximizing peripheral weight and minimizing surplus in closer neighborhoods, and proposes a heuristic that builds one strategy per neighbor of the target, using trunks of shortest paths to $r$-peripheral vertices to set a uniform baseline. The authors prove lower bounds on the best dual bounds via a detailed analysis of surplus weight and neighborhood structure, and demonstrate the heuristic's effectiveness by improving upper bounds for the Blanuša $B_2$ and Flower snarks $J_m$, including exact results in some cases. These results highlight the value of structured strategy selection within WFL for challenging graph families, and point to future work in automating steps and extending the approach to broader graph classes.

Abstract

Graph pebbling is a problem in which pebbles are distributed across the vertices of a graph and moved according to a specific rule: two pebbles are removed from a vertex to place one on an adjacent vertex. The goal is to determine the minimum number of pebbles required to ensure that any target vertex can be reached, known as the pebbling number. Computing the pebbling number lies beyond NP in the polynomial hierarchy, leading to bounding methods. One of the most prominent techniques for upper bounds is the Weight Function Lemma (WFL), which relies on costly integer linear optimization. To mitigate this cost, an alternative approach is to consider the dual formulation of the problem, which allows solutions to be constructed by hand through the selection of strategies given by subtrees with associated weight functions. To improve the bounds, the weights should be distributed as uniformly as possible among the vertices, balancing their individual contribution. However, despite its simplicity, this approach lacks a formal framework. To fill this gap, we introduce a novel heuristic method that refines the selection of balanced strategies. The method is motivated by our theoretical analysis of the limitations of the dual approach, in which we prove lower bounds on the best bounds achievable. Our theoretical analysis shows that the bottleneck lies in the farthest vertices from the target, forcing surplus weight onto the closer neighborhoods. To minimize surplus weight beyond the theoretical minimum, our proposed heuristic prioritizes weight assignment to the farthest vertices, building the subtrees starting from the shortest paths to them and then filling in the weights for the remaining vertices. Applying our heuristic to Flower snarks and Blanuša snarks, we improve the best-known upper bounds, demonstrating the effectiveness of a structured strategy selection when using the WFL.

Figures (8)

• Figure 1: Adapted representation from Ref. WFL of the optimal $r$-strategies highlighted in solid lines (a) $T_1$, (b) $T_2$, and (c) $T_3$ for the Petersen graph $P$. The Petersen graph is vertex-transitive, allowing us to consider the vertex $r$ as the only considered target. Note that $|\omega_\mathcal{T}| = 36$, $\omega_{\text{min}} = 4$, and $\lambda_\mathcal{T} = 9$. Consequently, $\pi(P) \leq 10$. Since $n(P) = 10$, it follows that $\pi(P) = 10$ and $\lambda_P = 9$.
• Figure 2: Application of the heuristic method to the target $x_3$ of (a) Blanuša 2 graph. We have $e(x_3) = 4$, $N(x_3) = \{z_3, x_1, x_5 \}$, and $P(x_3) = \{x_2', z_1', z_2'\}$. The neighbors $z_3$, $x_1$, and $x_5$ of $x_3$ lead to the strategies (b) $T_1$, (c) $T_2$, (d) $T_3$. Certificate $\mathcal{T} = \{T_1, T_2, T_3\}$ gives the WFL ratio $\lambda_\mathcal{T} = 29.25$. We omit the root $x_3$ in the representation of the subtrees. The edges of the trunk are represented by double lines, while the branches are represented by standard lines. The minimum weight, calculated by Eq. \ref{['eq:min']}, is $\omega_{\text{min}} = 2$ and its minimizer is $P_{\text{min}}(x_3) = \{x_2' \}$. Therefore, we have surplus weight in $z_1'$ and $z_2'$ to work with. For $z_2'$, the shortest path from $x_5$ to $z_2'$, indicated by dashed edges and vertices in subtree $T_3$, is replaced by the non-shortest path $x_5$, $z_5$, $z_5'$, $z_1'$, $z_2'$ in order to reduce the surplus weight. For $z_1'$, on the other hand, we take half of the weight of $z_1'$ in $T_1$, as well as of its two closest ancestors, eliminating the surplus weight on the vertex $x_3'$. The weights are updated with the original value crossed out with a single strikethrough. Observe that we have no surplus weight beyond the theoretical limit of the algorithm. However, our choice for the trunks of $\mathcal{T}$ is not the only one with this property. Specifically, in the subtree $T_1$, we could reconfigure the subgraph induced by the vertices $x_1'$ and $z_1'$ by placing $x_1'$ as a child of $x_4'$ and eliminating the vertex $x_3'$. For step $4$, the vertices that have not yet reached the minimum weight are $x_5'$, $x_4$, $x_2$, and $x_1'$. The vertices $x_5'$, $x_4$, and $x_2$ are solved directly by adding branches. For $x_1'$, which already has weight $1$ in the trunk of $T_1$, we can reach a total of $1.75$ by adding branches in $T_2$ and $T_3$. To reach the minimum of $2$ for the vertex $x_1'$, we update the weight of $x_1'$ to $5/4$ in $T_1$, forcing us to create surplus weight in its ancestor $x_3'$. In this case, the weights are updated with the original value crossed out with a double strikethrough.
• Figure 3: Application of the heuristic method to the target $z_0$ of Flower graph $J_m$. We start by building the strategies of (a) $J_3$ and (b) $J_5$ to arrive at the pattern of the strategies of (c) $J_m$, with $m = 2k + 1, m > 5$. For each graph, we show only one strategy ($T_1$). The other two strategies are obtained by swapping the vertex labels $v$ with $x$ ($T_2$) and $y$ ($T_3$), respectively. Certificate $\mathcal{T} = \{T_1, T_2, T_3\}$ gives the WFL ratio $\lambda_\mathcal{T} = 2^{k + 2} 3/2 + 2k - 2$ for $m \geq 3$, particularly resulting in a WFL ratio of $12$ and $26$ for the graphs $J_3$ and $J_5$, respectively. Similarly to the case of Blanuša 2, we omit the root $z_0$ in the representation of the subtrees, and we represent the trunk edges and branches by double lines and standard lines, respectively. For any $m \geq 3$, $e(z_0) = k + 2$, $N(z_0) = \{v_0, x_0, y_0 \}$, $P(z_0) = \{z_k, z_{-k} \}$, and the minimum weight, calculated by Eq. \ref{['eq:min']}, is $\omega_{\text{min}} = 3$. Furthermore, $P_{\text{min}}(z_0) = P(z_0)$, and the shortest path in $G - \{ z_0 \}$ between each neighbor of $z_0$ and each $z_0$-peripheral vertex is unique. Therefore, each set of trunks is unique to its respective graph. For the branches added in step $4$, observe that there is always a missing unit in the weight of the vertices $v_k$, $v_{-k}$, $x_k$, $x_{-k}$, $y_k$ and $y_{-k}$. We solve these vertices by adding two branches at each of the vertices $z_k$ and $z_{-k}$, assigning weight $1/2$ to the new vertices. This is enough to finalize the certificate of the $J_3$ graph but observe that vertices $z_1$ and $z_{-1}$ have no weight after step $3$ in graph $J_5$. We solve this case by adding the branches $v_{-1}z_{-1}$ and $v_1z_1$, with weight $1$ on both $z_{-1}$ and $z_1$. In general, the vertices $z_{j}$, $z_{-j}$, for each $1 \leq j \leq k - 1$, have no weight in $J_m$ after step $3$ and to them we add the branches $v_{-j}z_{-j}$ and $v_jz_j$, with weight $1$ on both $z_{-j}$ and $z_j$, finally getting for the $z_0$-strategy shown in (c).
• Figure 4: Flower graphs (a) $J_3$ and (b) $J_5$.
• Figure 5: Set $\mathcal{T} = \{T_1, T_2, T_3 \}$ of $z_0$-strategies for the Flower graph $J_3$, where (a) shows $T_1$, (b) $T_2$, and (c) $T_3$. We omit the root $z_0$ in the representation of the subtrees, and we represent the trunk edges and branches by double lines and standard lines, respectively.

Theorems & Definitions (7)

• Lemma 1.1: Weight Function Lemma WFL
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof