An $O(\log n)$-approximation for the Set Cover Problem with Set Ownership

TL;DR

This paper formulates the network edge validation in distributed Internet measurement as a set cover problem, proves its NP-hardness, and provides an $O(rac{ ext{log} n}{ ext{log} ext{log} n})$ approximation algorithm with matching lower bounds.

Contribution

It introduces a novel formulation of the validation problem as a set cover variant and develops an efficient approximation algorithm with proven bounds.

Findings

Validation problem is NP-hard.

Presented an $O(rac{ ext{log} n}{ ext{log} ext{log} n})$-approximation algorithm.

Proved the approximation ratio is tight under P ≠ NP.

Abstract

In highly distributed Internet measurement systems distributed agents periodically measure the Internet using a tool called {\tt traceroute}, which discovers a path in the network graph. Each agent performs many traceroute measurement to a set of destinations in the network, and thus reveals a portion of the Internet graph as it is seen from the agent locations. In every period we need to check whether previously discovered edges still exist in this period, a process termed {\em validation}. For this end we maintain a database of all the different measurements performed by each agent. Our aim is to be able to {\em validate} the existence of all previously discovered edges in the minimum possible time. In this work we formulate the validation problem as a generalization of the well know set cover problem. We reduce the set cover problem to the validation problem, thus proving that the validation problem is ${\cal NP}$-hard. We present a $O(\log n)$-approximation algorithm to the validation problem, where $n$ in the number of edges that need to be validated. We also show that unless ${\cal P = NP}$ the approximation ratio of the validation problem is $\Omega(\log n)$.