The Online Disjoint Set Cover Problem and its Applications

TL;DR

This paper introduces an online algorithm for the disjoint set cover problem with a competitive ratio of ln n, establishes a lower bound of sqrt(ln n), and discusses practical applications in various online resource allocation scenarios.

Contribution

It presents the first online algorithm with a competitive ratio of ln n for the DSCP and proves a lower bound of sqrt(ln n), advancing understanding of online set cover problems.

Findings

Proposed an online algorithm with competitive ratio ln n.

Established a lower bound of Omega(sqrt(ln n)) for any online algorithm.

Demonstrated applications in crowd-sourcing, sensor networks, and resource allocation.

Abstract

Given a universe $U$ of $n$ elements and a collection of subsets $\mathcal{S}$ of $U$, the maximum disjoint set cover problem (DSCP) is to partition $\mathcal{S}$ into as many set covers as possible, where a set cover is defined as a collection of subsets whose union is $U$. We consider the online DSCP, in which the subsets arrive one by one (possibly in an order chosen by an adversary), and must be irrevocably assigned to some partition on arrival with the objective of minimizing the competitive ratio. The competitive ratio of an online DSCP algorithm $A$ is defined as the maximum ratio of the number of disjoint set covers obtained by the optimal offline algorithm to the number of disjoint set covers obtained by $A$ across all inputs. We propose an online algorithm for solving the DSCP with competitive ratio $\ln n$. We then show a lower bound of $\Omega(\sqrt{\ln n})$ on the competitive ratio for any online DSCP algorithm. The online disjoint set cover problem has wide ranging applications in practice, including the online crowd-sourcing problem, the online coverage lifetime maximization problem in wireless sensor networks, and in online resource allocation problems.