Disjoint-Path Facility Location: Theory and Practice

Abstract

This paper is a theoretical and experimental study of two related facility location problems that emanated from networking. Suppose we are given a network modeled as a directed graph G = (V, A), together with (not-necessarily-disjoint) subsets C and F of V , where C is a set of customer locations and F is a set of potential facility locations (and typically C ⊆ F). Our goal is to find a minimum sized subset F ′ ⊆ F such that for every customer c ∈ C there are two locations f 1 , f 2 ∈ F ′ such that traffic from c to f 1 and to f 2 is routed on disjoint paths (usually shortest paths) under the network's routing protocols. Although we prove that this problem is impossible to approximate in the worst case even to within a factor of 2 log 1−ǫ n for any ǫ > 0 (assuming no NPcomplete language can be solved in quasipolynomial time), we show that the situation is much better in practice. We propose three algorithms that build solutions and determine lower bounds on the optimum solution, and evaluate them on several large real ISP topologies and on synthetic networks designed to reflect real-world LAN/WAN network structure. Our main algorithms are (1) an algorithm that performs multiple runs of a straightforward randomized greedy heuristic and returns the best result found, (2) a genetic algorithm that uses the greedy algorithm as a subroutine, and (3) a new "Double Hitting Set" algorithm. All three approaches perform surprising well, although, in practice, the most cost-effective approach is the multirun greedy algorithm. This yields results that average

The design space of probing algorithms for network-performance measurement

“…As we do here, Breslau et al [7] provide a theoretical treatment of problems relevant to network monitoring, but they focus on novel facility-location problems. Their work, like ours, includes properties of IP networks in the definition of an optimization problem requiring the choice of paths to cover network resources.…”

“…The output corresponds to an implementation of selecting paths to probe over time. Like the problem definitions in [7]-but unlike routing-agnostic formulations, e.g., the Maximum Edge Disjoint Paths problem [11], in which any set of links can be chosen to form paths between sourcedestination pairs-we assume that the set of possible probing paths is somehow constrained (e.g., by some underlying routing system) and thus forms part of the input to the problem. For an undirected network G = (V, E) and a set of paths P ⊆ 2 E , define a (possibly randomized) algorithm f : N → 2 P that, for each discrete timestep t = 1, 2, .…”

Experiments with LAGRASP heuristic for set k-covering

“…Though many of these applications can be modeled as set covering problems, for reliability purposes they are treated as multicovering. Other variants of the set covering problem and related applications can also be found in [5][6][7].…”

Solving Set Cover with Pairs Problem using Quantum Annealing

“…We will present its formal definition in the Preliminaries section. SCP and its variants arise in a wide variety of contexts including Internet traffic monitoring and content distribution 57 , computational biology 58 59 , and biochemistry 60 . On classical computers, the SCP problem is at least as hard to approximate as SC.…”

“…Specifically, its difficulty on classical computers can be manifested in the results by Breslau et al . 57 , which showed that no polynomial time algorithm can approximately solve Disjoint-Path Facility Location, a special case of SCP, on n objects to within a factor that is for any ε > 0. Due to its complexity, various heuristics 56 and local search algorithms 60 have been proposed.…”