Network caching asks how to place contents in distributed caches so that future requests are served close to their users. Ganian, Mc Inerney and Tsigkari recently initiated the parameterized-complexity study of the problem and, for the homogeneous unit-size variant (HomNC), isolated an unresolved family of six parameterizations: by the number of caches $C$, the number of users $U$, $U+K$, $C+U$, $C+λ$, and the vertex-cover number $\text{vc}(G)$, where $K$ is the maximum cache capacity and $λ$ is the maximum number of contents requested with nonzero probability by any user. Their interreducibility theorem showed that these six cases stand or fall together under parameterized reductions, and they conjectured the family to be W[1]-hard. We resolve this conjecture in the opposite direction. We prove that HomNC is fixed-parameter tractable parameterized by $C$ alone, and therefore fixed-parameter tractable for all six parameterizations. Our algorithm is based on an exact $n$-fold integer programming formulation that reveals a nontrivial block structure in homogeneous network caching, with the repeated part depending only on $C$. Standard algorithms for $n$-fold integer programming then yield a running time of the form $f(C)\lvert I\rvert^{O(1)}$.

翻译：网络缓存问题关注如何将内容分布放置在分布式缓存中，以使未来请求能尽可能在靠近用户的位置得到服务。Ganian、Mc Inerney 和 Tsigkari 近期开创了该问题的参数化复杂性研究，并针对同构单位容量变体（HomNC）分离出一个包含六个参数化的未解决族：缓存数量 $C$、用户数量 $U$、$U+K$、$C+U$、$C+λ$ 及顶点覆盖数 $\text{vc}(G)$，其中 $K$ 为最大缓存容量，$λ$ 为任意用户以非零概率请求的最大内容数。他们的互归约定理表明，这六种情形在参数化归约下同进退，并推测该族为 W[1]-难。我们反向解决了这一猜想。我们证明 HomNC 在仅以 $C$ 为参数时是固定参数可解的，因此对于所有六种参数化均为固定参数可解。我们的算法基于一种精确的 $n$ 折整数规划建模，该模型揭示了同构网络缓存中非平凡的块结构，其重复部分仅依赖于 $C$。随后，$n$ 折整数规划的标准算法可导出形如 $f(C)\lvert I\rvert^{O(1)}$ 的运行时间。