图中的松弛装填函数
Released packing functions in graphs
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中文总结 AI 辅助
该研究提出图的松弛装填函数变体，定义相关判定问题RPP，证明其NP难性，给出多项式规模整数线性规划模型并开展多面体研究初步工作。
中文摘要 AI 辅助
我们提出并开始研究图中装填函数的一个变体。给定顶点集为$V$的图$G$，以及非负整数向量$\boldsymbol{k}=(k_v)_{v\in V}$、$\boldsymbol{\ell}=(l_v)_{v\in V}$和$\boldsymbol{u}=(u_v)_{v\in V}$，若函数$f:V\rightarrow\mathbb{Z}_0^+$满足：对每个$v\in V$有$l_v\leq f(v)\leq u_v$，且所有满足$f(v)=u_v$的顶点$v$的闭邻域上的$f$值之和不超过$k_v$，则称$f$为$G$的松弛$(\boldsymbol{k},\boldsymbol{\ell},\boldsymbol{u})$-装填函数。$f$的权重为$f(V)=\sum_{v\in V}f(v)$。我们研究了相关的判定问题（RPP）：给定$G$、$\boldsymbol{k}$、$\boldsymbol{\ell}$、$\boldsymbol{u}$和整数$x$，判断$G$是否存在权重至少为$x$的松弛$(\boldsymbol{k},\boldsymbol{\ell},\boldsymbol{u})$-装填函数。我们将RPP与$r$-依赖集问题关联起来，推导了若干NP难性结果，将RPP建模为紧凑（规模为多项式级）的整数线性规划，并开展了多面体研究的初步工作。
英文摘要
We introduce and start the study of a variant of packing functions in graphs. Given a graph $G$ with vertex set $V$ and nonnegative integer vectors $\mathbf{k}=(k_v)_{v\in V}$, $\boldsymbol\ell=(l_v)_{v\in V}$ and $\mathbf{u}=(u_v)_{v\in V}$, a function $f : V \rightarrow \mathbb{Z}_0^+$ is a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of $G$ if $l_v\leq f(v)\leq u_v$ for every $v\in V$ and the sum of the values of $f$ over the closed neighborhood of vertices $v$ with $f(v) = u_v$ is at most $k_v$. The weight of $f$ is the value $f(V) = \sum_{v\in V} f(v)$. We study the associated decision problem (RPP), which asks, given $G$, $\mathbf{k}$, $\boldsymbol\ell$, $\mathbf{u}$ and an integer number $x$, whether $G$ admits a Released $( \mathbf{k}, \boldsymbol\ell, \mathbf{u})$-packing function of weight at least $x$. We relate RPP to the $r$-dependent set problem, derive several NP-hardness results, model RPP as a compact (polynomial in size) Integer Linear Program, and take the first steps of a polyhedral study.
