Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.03947

Stochastic Caching via Subset Entropy

Ravi Kumar·Roie Levin·Joseph·Naor·Debmalya Panigrahi
View original source ↗Read source document ↗Export metadata ↓

Abstract

A classic approach to beyond worst-case algorithm design is to impose stochastic assumptions on the input. However, a limiting feature of stochastic analyses is that, by the min-max principle, performance on worst-case distributions mirrors that of randomized algorithms on worst-case inputs. In other words, the same shortcoming of worst-case analysis -- its inability to distinguish "easy" and "hard" instances -- reappears as an inability to distinguish "easy" and "hard" distributions. This raises a natural question: Can we characterize "easy" input distributions with useful beyond worst-case bounds? A canonical example is the stochastic caching problem (Aho et al. 1971). When the page requests are drawn i.i.d. from the uniform distribution, the best achievable competitive ratio is $O(\log k)$, matching the performance of the best randomized algorithm on worst-case instances (Fiat et al. 1991). However, when the input distribution has less entropy, intuition suggests that we should be able to do better by exploiting the information provided by the distribution. We formalize this by defining a new information-theoretic parameter called subset entropy which we use to give a fine-grained characterization of the competitive ratio of stochastic caching, including a new analysis for the well-known LRU algorithm on stochastic inputs. While our technical results are for the caching problem, we believe the broader principle -- parameterizing algorithmic performance by an entropy measure of the input -- is of independent interest and might apply to other online/stochastic optimization problems. Indeed, for problems such as (comparison-based) sorting, online matching, load balancing, etc., the hardest stochastic instances involve high-entropy distributions. We hope our work is a step toward a broader theory of fine-grained algorithmic performance for this class of problems.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ravi Kumar, Roie Levin, Joseph, Naor, Debmalya Panigrahi. 2026-07-04. Stochastic Caching via Subset Entropy. https://arxiv.org/abs/2607.03947

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Kruskal: Polynomial-Time Tensor Decomposition under the Lovitz-Petrov Condition

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

cs.DS↗
arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗
arXiv2026

Approximating Prize-Collecting TSP below 1.556

The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and Nägele gave the previously best-known LP-relative $1.599$-approximation. We show that a simpler version of their algorithm, obtained by omitting the splitting-off preprocessing before the tree decomposition, has an LP-relative approximation ratio of $1.555761$. The improvement comes entirely from a new analysis of the parity-correction step: a simple analysis already gives $1.56$, and the stated factor follows from a numerical parameter search with exact verification.

cs.DS↗