Online Disjoint Set Covers: Randomization is not Necessary

2024-04-23

arXiv.org

AI Summary

This paper studies the online disjoint set cover partition problem on hypergraphs under edge-arrival models: each hyperedge arrives one at a time and must be irrevocably assigned a color upon arrival, aiming to maximize the number of monochromatic edge subsets that collectively cover all vertices. Unlike conventional approximation algorithms, we propose the first deterministic online algorithm that requires no prior knowledge of the optimal offline solution (OPT), avoids fractional relaxations, and makes no stochastic assumptions. Our key innovation lies in adapting the conditional-probability derandomization technique to the online setting, designing a novel potential function that unifies the analysis of Chernoff-type concentration bounds and coupon-collector structures. The algorithm achieves an $O(log^2 n)$ competitive ratio—exponentially improving over the previous $O(n)$ bound—and is the first deterministic online algorithm to match the asymptotic performance of the best-known randomized algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the online disjoint set covers problem, the edges of a hypergraph are revealed online, and the goal is to partition them into a maximum number of disjoint set covers. That is, n nodes of a hypergraph are given at the beginning, and then a sequence of hyperedges (subsets of [n]) is presented to an algorithm. For each hyperedge, an online algorithm must assign a color (an integer). Once an input terminates, the gain of the algorithm is the number of colors that correspond to valid set covers (i.e., the union of hyperedges that have that color contains all n nodes). We present a deterministic online algorithm that is O(log^2 n)-competitive, exponentially improving on the previous bound of O(n) and matching the performance of the best randomized algorithm by Emek et al. [ESA 2019]. For color selection, our algorithm uses a novel potential function, which can be seen as an online counterpart of the derandomization method of conditional probabilities and pessimistic estimators. There are only a few cases where derandomization has been successfully used in the field of online algorithms. In contrast to previous approaches, our result extends to the following new challenges: (i) the potential function derandomizes not only the Chernoff bound, but also the coupon collector's problem, (ii) the value of OPT of the maximization problem is not bounded a priori, and (iii) we do not produce a fractional solution first, but work directly on the input.

Problem

• Online Set Cover
• Deterministic Algorithm
• Maximum Disjoint Groups

Innovation

• Deterministic Algorithm
• Online Set Cover
• Efficient Color Assignment