Network Design with Coverage Costs

TL;DR

This paper addresses network design problems with coverage costs, providing approximation algorithms for two settings: laminar packet sets and intersecting packet sets, using primal-dual methods and novel spanner constructions.

Contribution

It introduces a primal-dual 2-approximation for laminar packet sets and a new group spanner construction achieving O(log g)-approximation for intersecting packet sets.

Findings

Improved approximation ratio for laminar case from logarithmic to 2-approximation.

Development of a novel group spanner with cost close to minimum spanning tree.

Achieved O(log g)-approximation for intersecting packet set network design.

Abstract

We study network design with a cost structure motivated by redundancy in data traffic. We are given a graph, g groups of terminals, and a universe of data packets. Each group of terminals desires a subset of the packets from its respective source. The cost of routing traffic on any edge in the network is proportional to the total size of the distinct packets that the edge carries. Our goal is to find a minimum cost routing. We focus on two settings. In the first, the collection of packet sets desired by source-sink pairs is laminar. For this setting, we present a primal-dual based 2-approximation, improving upon a logarithmic approximation due to Barman and Chawla (2012). In the second setting, packet sets can have non-trivial intersection. We focus on the case where each packet is desired by either a single terminal group or by all of the groups, and the graph is unweighted. For this setting we present an O(log g)-approximation. Our approximation for the second setting is based on a novel spanner-type construction in unweighted graphs that, given a collection of g vertex subsets, finds a subgraph of cost only a constant factor more than the minimum spanning tree of the graph, such that every subset in the collection has a Steiner tree in the subgraph of cost at most O(log g) that of its minimum Steiner tree in the original graph. We call such a subgraph a group spanner.