Homogeneous Network Caching is Fixed-Parameter Tractable Parameterized by the Number of Caches

TL;DR

This paper proves that homogeneous network caching is fixed-parameter tractable when parameterized by the number of caches, resolving a longstanding conjecture and providing an efficient algorithm based on integer programming.

Contribution

The authors demonstrate that the homogeneous network caching problem is fixed-parameter tractable with respect to the number of caches, contrary to previous conjectures.

Findings

Proved fixed-parameter tractability of HomNC parameterized by cache count C.

Developed an exact n-fold integer programming formulation revealing a block structure.

Provided an algorithm with runtime f(C) * |I|^{O(1)} based on standard n-fold integer programming algorithms.

Abstract

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 $U$, $U+K$, $C+U$, $C+\lambda$, and the vertex-cover number $\text{vc}(G)$, where $K$ is the maximum cache capacity and $\lambda$ 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 $f(C)\lvert I\rvert^{O(1)}$.