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
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Conservation corridors (a.k.a. movement or wildlife corridors)
[Simberloff et al. ’97; Ando et al. ’98; Camm et al. ’02]
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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
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Ashish Sabharwal
CP-AI-OR '08
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Connection Subgraph: Motivation
Real problem data:
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Goal: preserve grizzly bear
population in the U.S.A. by
creating movement corridors
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3637 land parcels (6x6 miles)
connecting 3 reserves in
Wyoming, Montana, and Idaho
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Reserves include, e.g.,
Yellowstone National Park
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Budget: ~ $2B
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
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Connection Subgraph: Motivation
Motivation 2: Social networks
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What characterizes the connection between two individuals?
The shortest path?
Size of the connected component?
A “good” connected subgraph?
[Faloutsos, McCurley, Tompkins ’04]
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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?
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Vertices in graph: people;
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edges: know each other or not
Ashish Sabharwal
CP-AI-OR '08
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The Connection Subgraph Problem
Given
An undirected graph G = (V,E)
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Terminal vertices T  V
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Vertex cost function: c(v); utility function: u(v)
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Cost bound / budget C;
desired utility U
Is there a subgraph H of G such that
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H is connected
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cost(H)  C; utility(H)  U ?
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Cost optimization version : given U, minimize cost
Utility optimization version : given C, maximize utility
May 23, 2008
Ashish Sabharwal
CP-AI-OR '08
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[CP-AI-OR ’07]
Previous Results
Theoretical
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NP-hard
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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
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Ashish Sabharwal
CP-AI-OR '08
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Graph Ensemble for Evaluation
Problem evaluated on semi-structured graphs
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m x m lattice / grid graph with k terminals
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Inspired by the conservation corridors problem
Place a terminal each on top-left and bottom-right
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Maximizes grid use
Place remaining terminals randomly
Assign uniform random costs and utilities
from {0, 1, …, 10}
m=4
k=4
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CP-AI-OR '08
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Pure MIP: feasibility vs. optimization
[CP-AI-OR ’07]
Split instances into feasible and infeasible; plot median runtime
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For feasible ones : computation involves proving optimality
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For infeasible ones: computation involves proving infeasibility
Infeasible instances take much longer than the feasible ones!
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Ashish Sabharwal
CP-AI-OR '08
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[CP-AI-OR ’07]
The MIP Approach
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connection
subgraph
instance
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MIP
model
solution
MIP model based on network flow
Revealed interesting tradeoffs
between testing for infeasibility
and optimization
Easy-hard-easy phenomena
Problem?
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MIP+Cplex really weak at
feasibility testing
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Poor scaling: couldn’t even get
close to handling real data
CPLEX
Can we do better?
feasibility + optimization
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CP-AI-OR '08
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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
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feasibility
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CP-AI-OR '08
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10x10 random lattices, 3 reserves
Infeasible instances
solved instantaneously!
~20x improvement
in runtime on
feasible instances
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CP-AI-OR '08
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10x10 random lattices, 3 reserves
Peak of hardness
still strongly
correlated with
budget slack
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CP-AI-OR '08
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Real Data, 40x40km Parcels
Gap between optimal
and extended-optimal
solutions peaks in a
critical region right
after min-cost
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CP-AI-OR '08
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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 :-)
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Summary
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MIP+Cplex gives a natural way to model and solve the
optimization problem
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But has difficult in feasibility testing
A hybrid approach with an external feasibility testing
algorithm improves performance dramatically
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on both feasible and infeasible instances
Also provides additional information for pruning
Makes it possible to scale to real-life data!
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