Transcript slides
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
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, 2008
Ashish Sabharwal
CP-AI-OR '08
2
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, 2008
Ashish Sabharwal
CP-AI-OR '08
3
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;
May 23, 2008
edges: know each other or not
Ashish Sabharwal
CP-AI-OR '08
4
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, 2008
Ashish Sabharwal
CP-AI-OR '08
5
[CP-AI-OR ’07]
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
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
6
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, 2008
Ashish Sabharwal
CP-AI-OR '08
7
Pure MIP: feasibility vs. optimization
[CP-AI-OR ’07]
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!
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
8
[CP-AI-OR ’07]
The MIP Approach
connection
subgraph
instance
MIP
model
solution
MIP model based on network flow
Revealed interesting tradeoffs
between testing for infeasibility
and optimization
Easy-hard-easy phenomena
Problem?
MIP+Cplex really weak at
feasibility testing
Poor scaling: couldn’t even get
close to handling real data
CPLEX
Can we do better?
feasibility + optimization
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
9
A Hybrid Solution Approach
min-cost solution
connection
subgraph
instance
compute min-cost
Steiner tree
ignore utilities
APSP
matrix
MIP
model
0
3
6
2
8
40-60%
pruned
solution
3
0
7
4
1
6
7
0
5
9
2
4
5
0
1
8
1
9
1
0
greedily extend
min-cost solution
to fill budget
like knapsack: max u/c
static
pruning
higher utility
feasible solution
CPLEX
starting solution
optimization
May 23, 2008
feasibility
Ashish Sabharwal
CP-AI-OR '08
10
10x10 random lattices, 3 reserves
Infeasible instances
solved instantaneously!
~20x improvement
in runtime on
feasible instances
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
11
10x10 random lattices, 3 reserves
Peak of hardness
still strongly
correlated with
budget slack
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
12
Real Data, 40x40km Parcels
Gap between optimal
and extended-optimal
solutions peaks in a
critical region right
after min-cost
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
13
Real Data, Best Parcels Grid
25 sq km hexagonal
parcels work very well
After 1 month of cpu time:
Best found solution (green)
very close to MIP upper bound
Extended-optimal (blue)
often better than best found
Experiments still running after 3.5 months :-)
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
14
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
May 23, 2008
on both feasible and infeasible instances
Also provides additional information for pruning
Makes it possible to scale to real-life data!
Ashish Sabharwal
CP-AI-OR '08
15