Connections in Networks: A Hybrid Approach Carla P. Gomes, Willem-Jan van Hoeve, Ashish Sabharwal Cornell University CP-AI-OR Conference, May 2008 Paris, France

May 23, 2008Ashish Sabharwal CP-AI-OR '082 Connection Subgraph: Motivation Motivation 1: Resource environment economics  Conservation corridors (a.k.a. movement or wildlife corridors) [Simberloff et al. ’97; Ando et al. ’98; Camm et al. ’02]  Preserve wildlife against land fragmentation  Link zones of biological significance (“reserves”) by purchasing continuous protected land parcels  Limited budget; must maximize environmental benefits/utility Reserve Land parcel

May 23, 2008Ashish Sabharwal CP-AI-OR '083 Connection Subgraph: Motivation Real problem data:  Goal: preserve grizzly bear population in the U.S.A. by creating movement corridors  3637 land parcels (6x6 miles) connecting 3 reserves in Wyoming, Montana, and Idaho  Reserves include, e.g., Yellowstone National Park  Budget: ~ $2B

May 23, 2008Ashish Sabharwal CP-AI-OR '084 Connection Subgraph: Motivation Motivation 2: Social networks  What characterizes the connection between two individuals? The shortest path? Size of the connected component? A “good” connected subgraph? [Faloutsos, McCurley, Tompkins ’04]  If a person is infected with a disease, who else is likely to be?  Which people have unexpected ties to any members of a list of other individuals?  Vertices in graph: people; edges: know each other or not

May 23, 2008Ashish Sabharwal CP-AI-OR '085 The Connection Subgraph Problem Given An undirected graph G = (V,E) Terminal vertices T  V Vertex cost function: c(v); utility function: u(v) Cost bound / budget C; desired utility U Is there a subgraph H of G such that H is connected cost(H)  C; utility(H)  U ? Cost optimization version: given U, minimize cost Utility optimization version: given C, maximize utility

May 23, 2008Ashish Sabharwal CP-AI-OR '086 Previous Results Theoretical NP-hard Cost optimization NP-hard to approximate within a factor of 1.36 Empirical: Typical-case complexity w.r.t. increasing budget fraction Without terminals: pure optimization version, always feasible, still a computational easy-hard-easy pattern With terminals: a) Phase transition: Problem turns from mostly infeasible to mostly feasible at budget fraction ~ 0.13 b) A coinciding computational easy-hard-easy pattern c) Proving optimality can be substantially easier than proving infeasibility in the phase transition region [CP-AI-OR ’07]

May 23, 2008Ashish Sabharwal CP-AI-OR '087 Graph Ensemble for Evaluation Problem evaluated on semi-structured graphs  m x m lattice / grid graph with k terminals Inspired by the conservation corridors problem  Place a terminal each on top-left and bottom-right Maximizes grid use  Place remaining terminals randomly  Assign uniform random costs and utilities from {0, 1, …, 10} m = 4 k = 4

May 23, 2008Ashish Sabharwal CP-AI-OR '088 Pure MIP: feasibility vs. optimization Split instances into feasible and infeasible; plot median runtime For feasible ones : computation involves proving optimality For infeasible ones: computation involves proving infeasibility Infeasible instances take much longer than the feasible ones! [CP-AI-OR ’07]

May 23, 2008Ashish Sabharwal CP-AI-OR '089 The MIP Approach  MIP model based on network flow  Revealed interesting tradeoffs between testing for infeasibility and optimization  Easy-hard-easy phenomena [CP-AI-OR ’07] connection subgraph instance MIP model feasibility + optimization CPLEX solution Problem?  MIP+Cplex really weak at feasibility testing  Poor scaling: couldn’t even get close to handling real data Can we do better?

May 23, 2008Ashish Sabharwal CP-AI-OR '0810 A Hybrid Solution Approach CPLEX connection subgraph instance solution MIP model optimization feasibility compute min-cost Steiner tree ignore utilities greedily extend min-cost solution to fill budget APSP matrix min-cost solution static pruning higher utility feasible solution starting solution 40-60% pruned like knapsack: max u/c

May 23, 2008Ashish Sabharwal CP-AI-OR ' x10 random lattices, 3 reserves ~20x improvement in runtime on feasible instances Infeasible instances solved instantaneously!

May 23, 2008Ashish Sabharwal CP-AI-OR ' x10 random lattices, 3 reserves Peak of hardness still strongly correlated with budget slack

May 23, 2008Ashish Sabharwal CP-AI-OR '0813 Real Data, 40x40km Parcels Gap between optimal and extended-optimal solutions peaks in a critical region right after min-cost

May 23, 2008Ashish Sabharwal CP-AI-OR '0814 Real Data, Best Parcels Grid 25 sq km hexagonal parcels work very well Best found solution (green) very close to MIP upper bound Extended-optimal (blue) often better than best found After 1 month of cpu time: Experiments still running after 3.5 months :-)

May 23, 2008Ashish Sabharwal CP-AI-OR '0815 Summary  MIP+Cplex gives a natural way to model and solve the optimization problem But has difficult in feasibility testing  A hybrid approach with an external feasibility testing algorithm improves performance dramatically on both feasible and infeasible instances Also provides additional information for pruning Makes it possible to scale to real-life data!