Hardness of robust network design

Abstract

The authors settle the complexity status of the robust network design problem in undirected graphs. The fact that the flow-cut gap in general graphs can be large, poses some difficulty in establishing a hardness result. Instead, the authors introduce a single-source version of the problem where the flow-cut gap is known to be one. They then show that this restricted problem is coNP-Hard. This version also captures, as special cases, the fractional relaxations of several problems including the spanning tree problem, the Steiner tree problem, and the shortest path problem.

Volume-oriented routing and its modifications

“…More precisely, answering the question whether a given traffic polytope can be dynamically routed through a known capacitated network is a co-N Pcomplete problem [11].…”

“…An interesting observation is that for each D there exists a simple way to obtain an optimal pair of vectors α and β. (11) yield an optimal solution, i.e., there are no cheaper solutions to General Volume-Oriented Routing for any other α and β.…”

“…While stable routing is easy to implement, it can be expensive in terms of cost when compared to an optimal dynamic strategy where routing depends on the current traffic matrix. However, dynamic routing has two drawbacks: it is difficult to implement and is also difficult to compute as shown in [11].…”

Dynamic vs. Oblivious Routing in Network Design

“…The construction in [13] uses meshes (grids), building on work of [4,14]. This construction does not seem to extend to the total cost model however, and we use instead a construction based on expanders, extending and simplifying a connection shown in earlier work [7].…”

“…This answers a question in [6], and is tight up to constant factors. Our proof technique builds on a connection between expander graphs and robust design for single-sink traffic patterns [7]. …”