带折扣的图上打击支配：次模性、复杂性与精确算法
Discounted Hitting Domination on Graphs with Submodularity, Complexity and Exact Algorithms
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中文总结 AI 辅助
本文提出折扣打击支配数，证明其次模性，给出与距离支配数的等价条件及复杂度结果，并为蜘蛛图设计多项式算法和精确MILP求解。
中文摘要 AI 辅助
在一个具有固定已验证源集合的网络上，折扣平均诱导出一个均衡支撑$h_i^S=\mathbb{E}_i[\lambda^{T_S}]$，即随机游走在衰减之前到达$S$的折扣概率。我们定义\emph{折扣打击支配数}$\delta_{\lambda,\tau}(G)$为保证每个顶点处$h_i^S\ge\tau$所需的最小源数量。尽管该势能已通过惩罚和组打击概率为人所知，但相关的最小基数均匀覆盖问题似乎是新的。聚合支撑是单调次模的，而均匀下限问题是一个精确的次模覆盖问题。此外，若$\lambda^{r+1}<\tau\le\left(\frac{\lambda}{\Delta}\right)^r$，则$\delta_{\lambda,\tau}(G)$等于距离$r$支配数。这导致在最大度为三的图上，当$(\lambda,\tau)=(1/4,1/14)$时问题具有NP完全性和APX完全性。对于蜘蛛图，我们获得了精确的有限状态刻画，并对每个固定的有理数对$(\lambda,\tau)$给出了多项式时间算法，同时证明了分支顶点不必属于最小源集合。最后，一个精确的混合整数线性规划公式在真实网络和合成图上验证了最优放置，并展示了与度、接近度和经典支配的显著差异。
英文摘要
On a network with a fixed set of verified sources, discounted averaging induces an equilibrium support $h_i^S=\mathbb{E}_i[λ^{T_S}]$, the discounted probability that a random walk reaches $S$ before attenuation. We define the \emph{discounted hitting domination number} $δ_{λ,τ}(G)$ as the minimum number of sources required to guarantee $h_i^S\geτ$ at every vertex. Although this potential is known through penalized and group hitting probabilities, the associated minimum-cardinality uniform-coverage problem appears to be new. Aggregate support is monotone submodular, while the uniform-floor problem is an exact submodular-cover problem. Moreover, if $λ^{r+1}<τ\le\left(\fracλΔ\right)^r$, then $δ_{λ,τ}(G)$ equals the distance-$r$ domination number. This yields NP-completeness and APX-completeness at $(λ,τ)=(1/4,1/14)$ on graphs of maximum degree three. For spiders, we obtain an exact finite-state characterization and a polynomial-time algorithm for every fixed rational pair $(λ,τ)$, and show that the branching vertex need not belong to a minimum source set. Finally, an exact mixed-integer linear formulation certifies optimal placements on a real network and a synthetic graph and demonstrates substantial differences from degree, closeness, and classical domination.
发表机构
- University of Maryland Eastern Shore（马里兰东部大学）
- Universidad Nacional de San Luis（圣路易斯国立大学）
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
