minimizing landscape resistance for habitat conservation...
TRANSCRIPT
Minimizing Landscape Resistance for Habitat
Conservation
CPAIOR 2017
Diego de Una, Graeme Gange, Peter Schachte and Peter J. Stuckey
June 6, 2017
University of Melbourne — CIS Department
Table of Contents
1. Motivation
2. Isolation by Resistance
3. Modeling the Problem
4. Solving the Problem
Diego de Una Minimizing Landscape Resistance CPAIOR2017 2
Why did the chicken cross the road?
• Repeatedly shown that isolationleads to extinction.
• Direct perturbation may not lead
to extinction, but isolation does
(Brook et al., 2008).
• Landscape influences genetic
variation (Manel et al., 2003),
(Rosenberg et al., 1997), (Pimm
et al., 1988)
• Action needed to restore to
movement/spread of speciesFragmentation of habitats
Diego de Una Minimizing Landscape Resistance CPAIOR2017 4
Gene Flow Matters
• The Great Prairie Chicken (Westemeier
et al., 1998)
• Threatened Species
• Individuals:
1933 7−→ 25000, 1993 7−→ 50
• Hatching success rate:
1960 7−→ 90%, 1990 7−→ 74%
• Genetic variation declined 30%
(1960-1990)
• Introducing groups from other areas:
Hatching success rate 94%
• Deer Mouse (Schwartz et al., 2005)
• Study compare survival rates:
• Control group 7−→ 88%
• Migrant group introduced 7−→ 98%
Great Prairie Chicken
Deer Mouse
Diego de Una Minimizing Landscape Resistance CPAIOR2017 5
Modeling Movement of Species via Genetic Distance
• Model for genetic distance (a.k.a.
population differentiation, “fixation
index”, FST ).
• Better understanding of movement of
species
• Better action planning to improve survival
• Metrics (R2 “fit the data”):
• IBD: Geographic distance (0.24)
• LCP/LCC: “Shortest” path (0.37)
• IBR: Effective resistance in an electrical
circuit (0.68)Fig. 1: Comparison of models and how
the fit the data (McRae et al., 2007).
Diego de Una Minimizing Landscape Resistance CPAIOR2017 7
Modeling Movement of Species via Genetic Distance
• Model for genetic distance (a.k.a.
population differentiation, “fixation
index”, FST ).
• Better understanding of movement of
species
• Better action planning to improve survival
• Metrics (R2 “fit the data”):
• IBD: Geographic distance (0.24)
• LCP/LCC: “Shortest” path (0.37)
• IBR: Effective resistance in an electrical
circuit (0.68)Fig. 1: Comparison of models and how
the fit the data (McRae et al., 2007).
Diego de Una Minimizing Landscape Resistance CPAIOR2017 8
Optimization Problem
Isolation by Resistance:
• Used by experts e.g. (Amos et al., 2012), (Amos et al., 2014)
• Justified: commute time between nodes ∝ to effective
resistance (Lovasz, 1993), (Doyle et al., 1984)
• Justified: current intensity at a node ∝ probability of random
walker passing through that node
⇒ The more resistance, the more isolation⇒ The more isolation, the least chance of survival
Problem
Given a maximum budget, where to improve the habitat (i.e.
invest) to minimize landscape resistance?
We are trying to decide where to reforest, build an animal bridge,
eradicate invasive species, improve soil, etc
Diego de Una Minimizing Landscape Resistance CPAIOR2017 9
Modeling landscape as an electrical circuit (1)
• Split landscape into “patches”
• Land patches ←→ nodes
• Adjacent patches ←→ resistor
• The value of each resistor is the
“difficulty” of movement between patches
• Eff. res. depends on all the resistors
• Decrease the value of resistors
=⇒ Eff. res. decreases
=⇒ Which resistors matter most?
s
1A
t
Fig. 2: Modeling landscape as a resistor
network (habitats are s and t)
Diego de Una Minimizing Landscape Resistance CPAIOR2017 11
Modeling landscape as an electrical circuit (2)
Fig. 3: Example of landscape: current (left) and conductivity (right)
Definitions
Effective Resistance between s and t : Rst
Conductance = 1/Resistance (G = 1/R)
Intensity at a node n =∑
b∈branches(n) |ib|/2Diego de Una Minimizing Landscape Resistance CPAIOR2017 12
Modeling landscape as an electrical circuit (3)
Fig. 4: Example of landscape: current (left) and conductivity (right)
More than 2 habitats
We define the total resistance as the sum of all-pairs resistances.
Diego de Una Minimizing Landscape Resistance CPAIOR2017 13
Problem Formulation
Given:
• Graph G = (V ,E ).
• Set of pairs of “focal” nodes F ⊂ V 2.
• Conductance function g∅ : E 7→ R for original resistors.
• Conductance function gE : E 7→ R for improved resistors.
• Budget B.
• Cost function c : E 7→ R for the cost of improving a resistor e.
Minimize the Total Effective Resistance, subject to:
•∑
e∈E[
be︸︷︷︸Boolean decisions for investment
∗ c(e)]≤ B
Diego de Una Minimizing Landscape Resistance CPAIOR2017 14
Computing Effective Resistance between s and t
1. Define Laplacian matrix (|V | × |V |):
Li ,j =
−gij if (i , j) ∈ E∑
k∈adj(i) gik if i = j
0 otherwise
2. Remove tth row and column of L.
3. Solve system: Lv = i , where i = (0, ..., 1︸︷︷︸is
, ..., 0)T
4. Effective resistance: vs .
(5. Voltages of all nodes are in v =⇒ We can compute currents.)
Diego de Una Minimizing Landscape Resistance CPAIOR2017 15
Mathematical Problem Formulation
Two habitats:1. Define Laplacian matrix:
Li ,j =
−(be ∗ gE
ij + (1− be) ∗ g∅ij ) if (i , j) ∈ E∑
k∈adj(i)(be ∗ gEik + (1− be) ∗ g∅
ik) if i = j
0 otherwise
2. Remove tth row and column of L.
3. Minimize vs subject to the system:
Lv = i , where i = (0, ..., 1︸︷︷︸is
, ..., 0)T .
Three or more habitats:• Define a system Lv = i for each pair of 〈si , ti 〉 ∈ F .
• Minimize∑
〈si ,ti 〉∈F v〈si ,ti 〉si
Diego de Una Minimizing Landscape Resistance CPAIOR2017 16
Solving as a MIP problem
• IBM CPLEx 12.4
• Use indicator constraints of the form:
be =⇒ ge = gE (e)
¬be =⇒ ge = g∅(e)
Problem:
• Weak relaxation
• Optimality gap above 40% even afte 5 hours for 10×10 grids
• Could not find better model (model is “imposed” limited by
the definition of effective resistance)
Diego de Una Minimizing Landscape Resistance CPAIOR2017 18
Greedy algorithm
1. Define the Laplacians for each pair 〈si , ti 〉 ∈ F using g∅.
2. Solve all the systems (No decisions)
3. Compute currents of all branches:
iab = g∅ab ∗
∑〈si ,ti 〉∈F (v
〈si ,ti 〉b − v
〈si ,ti 〉a )
4. Invest in the branches with highest current (more animals
walking through there!) until exhausting the budget.
• Surprisingly good results
• Total resistance dropped between 40% and 60% (for small
landscapes)!
Diego de Una Minimizing Landscape Resistance CPAIOR2017 19
Local Search
1. Begin with the solution s given by the Greedy algorithm.
2. Repeat:
2.1 Choose some “invested resistors” to disinvest (get some
budget back).
2.2 Choose some “uninvested resistors” to invest (exhaust the
budget).
2.3 Compute objective function.
2.4 Accept new solution snew with probability e−(snew−s)
T (simulated
annealing). Update s consequently.
2.5 Update temperature T : less and less likely to accept a worse
solution.
Diego de Una Minimizing Landscape Resistance CPAIOR2017 20
Choosing where we don’t want to invest anymore
• InvRand: Random resistors.
• InvLC: Resistors with least current through them.
• InvLCP: Probability that favors choosing resistors with low
current:
Pr(e) = 1current(e)
Intuition: places where animals aren’t walking anyway.
Diego de Una Minimizing Landscape Resistance CPAIOR2017 21
Choosing new places to invest
• WilRand: Random resistors
• WilBFS: Choose a node with a probability distribution that
favors high current nodes. Perform BFS around that node
until exhausted budget. ⇒ Investments are near each other.
=⇒ Stronger effect in dropping resistance in that region
• WilHC: Choose resistors with highest current.
• WilHCP: Probability that favors choosing high current
resistors.
Intuition: places where animals are walking and it could be made
easier.
Catch: WilHC + InvLC can get stuck in local minimum =⇒do one iteration of WilHCP.Diego de Una Minimizing Landscape Resistance CPAIOR2017 22
Experimental setting
• 500 artificial instances
• Realistically generated instance:
• Resistance 1Ω to 100Ω
• Beta distribution of resistance: much more likely to have high
resistance value (like in the real world)
• Added oasis with low resistance (e.g. small tree areas)
• Instances with homogeneous and heterogeneous cost functions
• 200 Local Search iterations
Diego de Una Minimizing Landscape Resistance CPAIOR2017 23
Results
Averages of the ratios of updated resistance over original resistanceCost
TypeSize
Budget
RatioHabitats Greedy
InvLC InvLCP
WilBFS WilHC WilHCP WilBFS WilHC WilHCP
Homo 20×20 0.11 2 0.30 0.24 0.25 0.25 0.29 0.26 0.27
Homo 25×25 0.09 2 0.30 0.22 0.23 0.24 0.28 0.24 0.26
Homo 25×25 0.09 3 0.38 0.29 0.30 0.29 0.37 0.31 0.35
Homo 30×30 0.07 2 0.37 0.26 0.27 0.28 0.35 0.29 0.31
Homo 30×30 0.07 3 0.42 0.33 0.33 0.33 0.42 0.35 0.41
Homo 50×50 0.04 2 0.44 0.34 0.35 0.36 0.44 0.37 0.40
Homo 50×50 0.04 3 0.48 0.42 0.40 0.41 0.48 0.43 0.48
Homo 50×50 0.04 4 0.52 0.46 0.44 0.45 0.52 0.47 0.52
Homo 100×100 0.02 2 0.49 0.39 0.38 0.41 0.42 0.39 0.37
Homo 100×100 0.02 3 0.53 0.42 0.43 0.42 0.53 0.42 0.50
Homo 100×100 0.02 4 0.55 0.49 0.47 0.46 0.55 0.49 0.55
Homo 100×100 0.02 5 0.58 0.49 0.46 0.50 0.58 0.48 0.58
Rand 20×20 0.11 2 0.44 0.34 0.34 0.38 0.39 0.35 0.40
Rand 25×25 0.09 2 0.45 0.33 0.32 0.36 0.39 0.33 0.38
Rand 25×25 0.09 3 0.53 0.41 0.39 0.38 0.47 0.40 0.44
Rand 30×30 0.07 2 0.52 0.45 0.42 0.44 0.51 0.45 0.48
Rand 30×30 0.07 3 0.57 0.53 0.51 0.52 0.57 0.52 0.56
Rand 50×50 0.04 2 0.56 0.54 0.51 0.53 0.56 0.53 0.56
Rand 50×50 0.04 3 0.60 0.59 0.55 0.58 0.60 0.58 0.60
Rand 50×50 0.04 4 0.63 0.62 0.60 0.63 0.63 0.62 0.63
Rand 100×100 0.02 2 0.61 0.51 0.53 0.52 0.66 0.61 0.65
Rand 100×100 0.02 3 0.64 0.59 0.57 0.60 0.63 0.60 0.64
Rand 100×100 0.02 4 0.65 0.58 0.60 0.62 0.62 0.61 0.64
Rand 100×100 0.02 5 0.68 0.60 0.59 0.62 0.63 0.62 0.65
Diego de Una Minimizing Landscape Resistance CPAIOR2017 24
Conclusions
• Substantial drop on effective resistance: even with investing in
only 2% of the land we get ≥ 40% less resistance.
• Improvement of more than 10% over greedy algorithm
• Small optimality gap: 5% (tested on 10× 10 girds)
• Fast: less than 5 minutes for 200 iterations
Diego de Una Minimizing Landscape Resistance CPAIOR2017 25
Comparison (Original: 53.78Ω, Final: 23.03Ω; 100/4900)
Diego de Una Minimizing Landscape Resistance CPAIOR2017 27
Proof of NP-hardness
The STP: Given a graph G = (N,E ), a set R ⊆ N, weighting function w on the
edges and positive integer K , is there a subtree of G weight ≤ K containing the set
of nodes in R? Reduction:• The electric circuit is the graph G .
• The cost function is w .
• The original resistance of edges is infinite (i.e. g∅(e) = 0,∀e ∈ E ).
• The resistance upon investment of all edges is 1 (i.e. gE (e) = 1,∀e ∈ E ).
• The budget is K .
• The set of pairs of focal nodes F is the set of all pairs of distinct nodes in R,
which can be built in O(|R|2).Assume we have an algorithm to solve the landscape problem that gives a solution
S . By investing in the selected edges S we obtain a resistance RS ∈ R iff there is
enough budget K , and RS =∞ otherwise:1. If RS =∞, there is no Steiner Tree of cost ≤ K , since we could not connect
focal nodes with 1Ω resistors.
2. If RS = 0, G∗ = G \ invested(S). G∗ is of cost ≤ K and a subgraph of G that
connects all nodes in R pairwise. G∗ contains/is a Steiner Tree.Diego de Una Minimizing Landscape Resistance CPAIOR2017 28
MIP model
Minimize
|F |∑i=1
v〈si ,ti 〉si
s.t.
e=|E |∑e=0
bec(e) ≤ B
∀〈si , ti 〉 ∈ P, ∀x ∈ N\si , ti,∑
y∈adj(x)
p〈si ,ti 〉(x ,y),x −
∑y∈adj(x)\ti
p〈si ,ti 〉(x ,y),y = 0
∀〈si , ti 〉 ∈ P,∑
y∈adj(si )
p〈si ,ti 〉(si ,y),si
−∑
y∈adj(si )\ti
p〈si ,ti 〉(si ,y),y = 1
∀〈si , ti 〉 ∈ P, ∀e = (x , y) ∈ E ,∀z ∈ x , y,
p〈si ,ti 〉e,z = beg
E (e)v〈si ,ti 〉z + (1− be)g∅(e)v
〈si ,ti 〉z
Diego de Una Minimizing Landscape Resistance CPAIOR2017 29
Minimize the sum of voltages at the
source habitats
MIP model
Minimize
|F |∑i=1
v〈si ,ti 〉si
s.t.
e=|E |∑e=0
bec(e) ≤ B
∀〈si , ti 〉 ∈ P, ∀x ∈ N\si , ti,∑
y∈adj(x)
p〈si ,ti 〉(x ,y),x −
∑y∈adj(x)\ti
p〈si ,ti 〉(x ,y),y = 0
∀〈si , ti 〉 ∈ P,∑
y∈adj(si )
p〈si ,ti 〉(si ,y),si
−∑
y∈adj(si )\ti
p〈si ,ti 〉(si ,y),y = 1
∀〈si , ti 〉 ∈ P, ∀e = (x , y) ∈ E ,∀z ∈ x , y,
p〈si ,ti 〉e,z = beg
E (e)v〈si ,ti 〉z + (1− be)g∅(e)v
〈si ,ti 〉z
Diego de Una Minimizing Landscape Resistance CPAIOR2017 30
Budget constraint
MIP model
Minimize
|F |∑i=1
v〈si ,ti 〉si
s.t.
e=|E |∑e=0
bec(e) ≤ B
∀〈si , ti 〉 ∈ P, ∀x ∈ N\si , ti,∑
y∈adj(x)
p〈si ,ti 〉(x ,y),x −
∑y∈adj(x)\ti
p〈si ,ti 〉(x ,y),y = 0
∀〈si , ti 〉 ∈ P,∑
y∈adj(si )
p〈si ,ti 〉(si ,y),si
−∑
y∈adj(si )\ti
p〈si ,ti 〉(si ,y),y = 1
∀〈si , ti 〉 ∈ P, ∀e = (x , y) ∈ E ,∀z ∈ x , y,
p〈si ,ti 〉e,z = beg
E (e)v〈si ,ti 〉z + (1− be)g∅(e)v
〈si ,ti 〉z
Diego de Una Minimizing Landscape Resistance CPAIOR2017 31
All non-focal nodes need
to maintain a flow of
current equal 0
MIP model
Minimize
|F |∑i=1
v〈si ,ti 〉si
s.t.
e=|E |∑e=0
bec(e) ≤ B
∀〈si , ti 〉 ∈ P, ∀x ∈ N\si , ti,∑
y∈adj(x)
p〈si ,ti 〉(x ,y),x −
∑y∈adj(x)\ti
p〈si ,ti 〉(x ,y),y = 0
∀〈si , ti 〉 ∈ P,∑
y∈adj(si )
p〈si ,ti 〉(si ,y),si
−∑
y∈adj(si )\ti
p〈si ,ti 〉(si ,y),y = 1
∀〈si , ti 〉 ∈ P, ∀e = (x , y) ∈ E ,∀z ∈ x , y,
p〈si ,ti 〉e,z = beg
E (e)v〈si ,ti 〉z + (1− be)g∅(e)v
〈si ,ti 〉z
Diego de Una Minimizing Landscape Resistance CPAIOR2017 32
The source has a flow of 1
MIP model
Minimize
|F |∑i=1
v〈si ,ti 〉si
s.t.
e=|E |∑e=0
bec(e) ≤ B
∀〈si , ti 〉 ∈ P, ∀x ∈ N\si , ti,∑
y∈adj(x)
p〈si ,ti 〉(x ,y),x −
∑y∈adj(x)\ti
p〈si ,ti 〉(x ,y),y = 0
∀〈si , ti 〉 ∈ P,∑
y∈adj(si )
p〈si ,ti 〉(si ,y),si
−∑
y∈adj(si )\ti
p〈si ,ti 〉(si ,y),y = 1
∀〈si , ti 〉 ∈ P, ∀e = (x , y) ∈ E , ∀z ∈ x , y,
p〈si ,ti 〉e,z = beg
E (e)v〈si ,ti 〉z + (1− be)g∅(e)v
〈si ,ti 〉z
Diego de Una Minimizing Landscape Resistance CPAIOR2017 33The value of each product depends on the investments