Non-Linear Paging

TL;DR

This work generalizes online paging to non-linear feasibility, replacing fixed cache size with a monotone function $f$ and introducing the width parameter $\ell(f)$ to capture the difficulty of infeasibility. It presents a tight deterministic $\ell$-competitive algorithm for general non-linear paging via a novel LP relaxation, and proves a $o(\log^2(\ell))$ lower bound for randomized algorithms, highlighting inherent limits. For the important special case of supermodular paging, the authors develop a polylogarithmic randomized algorithm with matching lower bounds up to polylog factors, leveraging a submodular-cover LP and randomized rounding; they also connect online supermodular paging to online set cover and discuss offline approximation results. The framework unifies classic paging, weighted paging, and submodular/supermodular variants, with practical implications for caching with overlaps, deduplication, and hypergraph-based data systems, while outlining avenues for stronger LP formulations and potential polylogarithmic randomized algorithms.

Abstract

We formulate and study non-linear paging - a broad model of online paging where the size of subsets of pages is determined by a monotone non-linear set function of the pages. This model captures the well-studied classic weighted paging and generalized paging problems, and also submodular and supermodular paging, studied here for the first time, that have a range of applications from virtual memory to machine learning. Unlike classic paging, the cache threshold parameter $k$ does not yield good competitive ratios for non-linear paging. Instead, we introduce a novel parameter $\ell$ that generalizes the notion of cache size to the non-linear setting. We obtain a tight deterministic $\ell$-competitive algorithm for general non-linear paging and a $o\left(\log^2 (\ell)\right)$-competitive lower bound for randomized algorithms. Our algorithm is based on a new generic LP for the problem that captures both submodular and supermodular paging, in contrast to LPs used for submodular cover settings. We finally focus on the supermodular paging problem, which is a variant of online set cover and online submodular cover, where sets are repeatedly requested to be removed from the cover. We obtain polylogarithmic lower and upper bounds and an offline approximation algorithm.

Figures (3)

• Figure 1: An illustration of a shared memory systems with four pages and four shared memory units ("atoms"). The cache is of size $k = 3$ and contains pages $p_2,p_3$ (in blue) whose shared memory is of size $3$ - the three red atoms. Observe that either fetching $p_1$ or evicting it will not change the number of atoms stored in cache, however, fetching $p_4$ requires evicting $p_2$.
• Figure 2: An illustration of the LP constraints. Each colored shape represents a minimally infeasible set of pages upon the arrival of a page $p_t$. Removing pages $p_1$ and $p_2$ ensures cache feasibility.
• Figure 3: An illustration of the construction for $X = \{1,\ldots,4\}$ and ${\mathcal{S}} = \{S_1,S_2\}$. The considered moment in time takes place after a request for element-pages $1,4$; to serve these requests, the set-page $S_1$ (in red) is evicted from cache (taken to the cover). Observe that $\{2,3,S_1\}$, the pages outside of the cache, form a feasible cover of the elements $\{1,\ldots,4\}$.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Lemma 2.1