Parameterized Complexity of Caching in Networks

TL;DR

The paper analyzes the parameterized complexity of Network-Caching under three variants—HomNC, HetNC-U, and HetNC-B—across six structural parameters. It delivers a nuanced landscape of fixed-parameter tractable regimes and hardness results, deriving upper-bound algorithms (FPT and XP) and extensive reductions (e.g., from $NAE$-SAT, Unary Bin Packing, Knapsack) to establish intractability for many parameterizations. Structural graph restrictions (treewidth, planarity, etc.) largely do not yield tractability, and the authors show strong interreducibility among open HomNC cases, indicating a single open frontier. The work also discusses generalizations to broader caching objectives, practical implications for CDN and edge-inference deployments, and potential avenues for ETH-based improvements and heuristics.

Abstract

The fundamental caching problem in networks asks to find an allocation of contents to a network of caches with the aim of maximizing the cache hit rate. Despite the problem's importance to a variety of research areas -- including not only content delivery, but also edge intelligence and inference -- and the extensive body of work on empirical aspects of caching, very little is known about the exact boundaries of tractability for the problem beyond its general NP-hardness. We close this gap by performing a comprehensive complexity-theoretic analysis of the problem through the lens of the parameterized complexity paradigm, which is designed to provide more precise statements regarding algorithmic tractability than classical complexity. Our results include algorithmic lower and upper bounds which together establish the conditions under which the caching problem becomes tractable.

Figures (4)

• Figure 1: Complexity landscapes of HetNC-B (top), HetNC-U (middle), and HomNC (bottom). We (mostly) omit parameterizations including $C+\Delta$, $U+\Delta$, $U+\lambda$, or $S+\lambda$, as the first two, the third, and the last are complexity-theoretically equivalent to $C+U$, $U+S$, and $S$, respectively. Indeed, for any pair of combined parameters claimed to be equivalent, each of them can be bounded by a computable function of the other. To prove the first two equivalences, note that $\Delta\leq C+U$, $U\leq C\cdot \Delta$, and $C \leq U\cdot \Delta$. For the latter two, note that $S\leq U \cdot \lambda$ and $\lambda\leq S$.
• Figure 2: Illustration of a bipartite graph between users and subsets of the caches in our setting.
• Figure 3: Illustration of the subdivided star $G'$ constructed in the proof of Theorem \ref{['thm:vc-red']}, where $ab$, $aq$, and $xy$ are edges in the graph $G$ from the instance of Maximum $k$-Vertex Cover.
• Figure 4: Illustration of the planar graph $G$ constructed from an instance $\phi$ of Planar 3-SAT-E3 in the proof of Theorem \ref{['thm:sat-red']}. Here, $\phi$ contains the clause $C_1$ containing the variables $x_1$, $x_2$, and a third arbitrary one (denoted by a line protruding from the vertex $u_{C_1}$), the clause $C_2$ containing the variables $x_1$, $x_2$, and $x_n$, and the clause $C_m$ containing the variables $x_1$, $x_n$, and a third arbitrary one.

Theorems & Definitions (26)

• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• proof
• Corollary 4