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

##### Date

##### Authors

Gonen, Mira

Shavitt, Yuval

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

##### Description

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 $\Omega(\log n)$.

##### Keywords

Computer Science - Networking and Internet Architecture, Computer Science - Computational Complexity