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Preprint submitted on 17 Feb 2022

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Optimizing the ecological connectivity of landscapeswith generalized flow models and preprocessing

François Hamonic, Cécile Albert, Basile Couëtoux, yann Vaxès

To cite this version:François Hamonic, Cécile Albert, Basile Couëtoux, yann Vaxès. Optimizing the ecological connectivityof landscapes with generalized flow models and preprocessing. 2022. �hal-03578088�

Optimizing the ecological connectivity of landscapes with generalizedflow models and preprocessing I

Francois Hamonica,b, Cecile Albertb, Basile Couetouxa, Yann Vaxesa

aAix-Marseille Univ, CNRS, Universite de Toulon, LIS, Marseille, FrancebAix-Marseille Univ, CNRS, Univ Avignon, IRD, IMBE, Marseille, France

Abstract

In this paper we consider the problem of optimizing the ecological connectivity of a land-

scape under a budget constraint by improving habitat areas and ecological corridors be-

tween them. We consider a formulation of this problem in terms of graphs in which vertices

represent the habitat areas and arcs represent a probability of connection between two ar-

eas that depend on the quality of the respective corridor. We propose a new generalized

flow model that improves existing models for this problem and an efficient preprocessing

algorithm that reduces the size of the graphs on which generalized flows is computed.

Reported numerical experiments highlight the benefice of these two contributions on com-

putation times and show that larger problems can be solved using them. Our experiments

also show that several variants of greedy algorithms perform relatively well on practical

instances while they return arbitrary bad solutions in the worst case.

Keywords: Environment and climate change, landscape connectivity, generalized

network flow, robust shortest path, linear programming, distances in graphs.

1. Introduction

Habitat loss is a major cause of the rapid decline of biodiversity [5]. Further than

reducing the available resources, it also increases the discontinuities among small habitat

areas called patches, this is a phenomenon known under the name habitat fragmentation [9].

While habitat loss tends to reduce the size of populations of animals and plants, which may

be deleterious to their survival in the long term, habitat fragmentation also makes it harder

for organisms to move around in landscapes. This constrains their access to resources and

reduces gene flow among populations, which can also be deleterious for their persistence

IThe research on this paper was supported by Region SUD, Natural Solutions and the ANR projectDISTANCIA (ANR-17-CE40-0015).

Email addresses: francois.hamonic@lis-lab.fr (Francois Hamonic), cecile.albert@imbe.fr(Cecile Albert), basile.couetoux@lis-lab.fr (Basile Couetoux), yann.vaxes@lis-lab.fr (Yann Vaxes)

Preprint submitted to Networks February 17, 2022

in the long term. Landscape connectivity, defined as the degree to which the landscape

facilitates the movement of organisms between habitat patches [19], then becomes of major

importance for biodiversity and its conservation. Accounting for landscape connectivity

in restoration or conservation plans thus appears as a key solution to maximize the return

on investment of the scarce financial support devoted to biodiversity conservation. For

this purpose, graph-theoretical approaches are useful in modelling habitat connectivity

[20]. In this paper, we consider the landscape connectivity that is distinct from the graph

theoretical notion of connectivity defined in terms of minimum cut.

A landscape can be viewed as a directed graph in which vertices represent the habitat

patches and each arc represents a way for individuals to travel from one patch to another.

Each vertex has a weight that represents the quality of the patch – patch area is often

used as a surrogate for quality – and each arc has a weight that represents the difficulty

for an individual to make the corresponding travel – often approximated by the border to

border distance between the patches. Interestingly, this approach can be used for a variety

of ecological systems like terrestrial (patches of forests in an agricultural area, networks

of lakes or wetlands), riverine (patches are segments of river than can be separated by

human constructions like dams that prevent fishes’ movement) or marine (patches can be

reefs that are connected by flows of larvae transported by currents). With this formalism,

ecologists have developed many connectivity indicators [11, 12, 17] that aim to quantify

the quality of a landscape with respect to the connections between its habitat patches.

Among the proposed indicators, the probability of connection – PC – [17] and its derivative

the equivalent connected area – ECA – [16] seem to perform well; they have received

encouraging empirical support [3, 14, 18]. Definitions and relationship between these

two indicators are described in Section 2. In this paper, our purpose is not to discuss

the merit of ECA with respect to other landscape connectivity indicators but to study the

corresponding combinatorial optimization problem from an algorithmic point of view. One

key question for which landscape connectivity indicators have been used in the last years is

to identify which elements of the landscape (habitat patches, corridors) should be preserved

from destruction or restored in order to maintain a well-connected network of habitat

under a given budget constraint [2]. This means, identifying the set of vertices or arcs that

optimally maintain a good level of connectivity. Many mathematical programming models

have been introduced in the literature to help decision-makers to protect biodiversity, for

a deep and wide review of these models, we refer to the survey paper [4] and the references

2

therein.

Here, we will focus on this problem by measuring ecological connectivity with ECA.

The budget-constrained ECA optimization problem consists in finding the best combina-

tion of graph elements (arcs or vertices) among a set Φ of elements that could be restored

in order to maximize the ECA in the landscape within a limited budget. Interestingly, this

problem addresses both restoration and conservation cases. The restoration case starts

from the current landscape and aims to restore a set of elements among the different

feasible options. The conservation case starts with a landscape in which we search for

the elements to protect among the threatened ones, and before the unprotected ones get

degraded. The conservation problem and the restoration problem are equivalent from an

algorithmic point of view. Indeed, an instance of the conservation problem can be viewed

as an instance of the restoration problem in which the initial landscape is the landscape in

which all threatened elements are degraded and the set Φ of elements than can be restored

in the instance of the restoration problem is the set of elements that can be protected in

the instance of the conservation problem. Until now, ecologists have mostly tackled this

problem by ranking each conservation or restoration option by its independent contribu-

tion to ECA, i.e. the amount by which ECA varies if the option is purchased alone. Such

an approach overlooks the cumulative effects of the decisions made like unnecessary redun-

dancies or potential synergistic effects, e.g. consecutive arcs. This could lead to solutions

that are more expensive or less beneficial to ECA than an optimal solution. Some studies

have tried to overcome this limitation by considering tuples of options [13, 15]. In [15],

the authors show that the brute force approach rapidly becomes impractical for landscape

with more than 20 patches of habitat. However, few studies explore the search for an op-

timal solution, most of which being developed for river dendritic networks. For instance,

in [21], a polynomial time approximation scheme is described for the problem where the

landscape is represented by a tree graph. Recently a mixed integer formulation that is

applicable to our problem has been introduced [22]. This formulation is however expen-

sive to solve. It uses O(|V |2|A|) flow variables and O(|V |3) constraints while O(|Φ|+ |V |2)

integer variables are required to approximate the non-linear objective function. The au-

thors of [22] show that solving exactly this formulation does not scale to landscapes of few

hundreds of patches. Their main contribution was to propose a XOR sampling method

for approximating it. Since it is possible to use the XOR sampling method with our exact

and more compact MIP model, it is better to view our contributions as a complement

3

to those of [22]. Therefore, we did not perform numerical experiments to compare our

methodology against those developed in [22].

Our model is based on a generalized flow formulation (see for instance [1]) instead of

a standard network flow formulation, which has two main advantages. On one side, our

formulation has a linear objective function. We do not need to use a piecewise constant

approximation nor to introduce additional binary variables to handle non-linearity. On the

other side, the new formulation is much more compact because it aggregates into a single

generalized flow the contribution to the connectivity of several source/sink pairs having

the same source. This reduces significantly the number of variables and constraints of the

model. It uses O(|V ||A|) flow variables with O(|V |2) constraints, and |Φ| integer variables

instead of O(|V |2|A|) flow variables with O(|V |3) constraints, and O(|Φ| + |V |2) integer

variables in the model of [22]. Recall that Φ is the set of elements that are threatened or

could be restored. Another contribution of this paper is to propose a preprocessing step

that speeds-up the resolution of the problem by reducing the size of the directed graph

on which a generalized flow to a particular sink t have to be computed. The idea is to

identify, for each sink t, the set of arcs (u, v) that are always on a shortest path from u

to t for all the 2|Φ| possible restoration plans. Such arcs, called t-strong arcs, are always

carrying flow and can be contracted in the generalized flow formulation to t. Conversely,

an arc (u, v) that is never on a shortest path from u to t, called t-useless, can be removed

from the generalized flow formulation to t because it never carries any flow. Once t-strong

arcs have been contracted and t-useless arcs have been removed in each generalized flow

formulation to a particular sink t, it remains to solve the problem on a much smaller

instance containing only the arcs that are not t-strong nor t-useless. Obviously it would

be prohibitive in terms of computation times to solve a shortest path problem for each of

the 2|Φ| possible restoration plans in order to determine whether a given arc is t-strong

(resp. t-useless) or not. In Section 4, we design and analyze a very efficient algorithm, that

computes directly the set of all sinks t such that a given arc is t-strong. This algorithm

takes implicitly into consideration all 2|Φ| possible restoration plans but has the same

complexity as the Dijkstra shortest path algorithm. It improves previous result for the

problem of computing t-strong or t-useless arcs [6]. We anticipate that this algorithmic

result is of independent interest and could be useful in many other contexts, in particular

when shortest path computations are embedded in robust optimization models.

In Section 2 we give more details about the landscape modelization and describe for-

4

mally the problem. Section 3 is devoted to the description of our mixed integer formula-

tion of the problem. In Section 4 we first explain why the removal of t-useless arcs and

the contraction of t-strong arcs does not modify the objective function of any restora-

tion/conservation plan. Then, we design and analyze a O(|A| + |V | log |V |) algorithm

that computes the set of all vertices t for which a given arc (u, v) is t-strong or the set

of all vertices t for which (u, v) is t-useless. Section 5 presents experimental cases for

which we compare our optimization approach with simple greedy algorithms in terms of

running times and quality of the solutions found. These experiments also shows that the

preprocessing is very effective on some real instances.

2. Budget Constrained ECA Optimization problem

Our optimization problem takes as input a landscape represented by a directed graph

G = (V,A). Each patch s ∈ V has a weight ws that represents its size. Each ordered

pair of patches (s, t) such that an individual can move from patch s to patch t without

crossing another patch is joined by an arc of G. For each arc (u, v), we suppose that an

estimate of the probability π(u,v) that an individual successfully moves from u to v. This

value will be determined by the landscape and by the species studied. The input of our

optimization problem also includes a list of arcs that can be improved to make them easier

or less dangerous to cross and the costs of these improvements. As objective function and

measure of the quality of the landscape, we use the Equivalent Connected Area defined

by the following formula [16]:

ECA(G,w, l) =

√∑s∈V

∑t∈V

(wswtΠst)

where Πst is the probability of connection from the patch s to the patch t. This probability

is defined as the probability of the most probable (reliable) path from s to t where the

probability of a path is the product of the probabilities of its arcs. The probability of an

arc a is often defined as πa := exp(−αla) where α is a parameter that depends on the

dispersal ability of the species and la is the euclidean distance between the centers of the

two patches joined by a. This dispersal model has been used in many applications. Its

ecological justification, discussed for instance in [7, 8], is out of the scope of this paper. By

monotonicity of the exponential function, a path P is a most reliable path between s and

t if and only if it is a shortest path with respect to arc lengths l and Πst = exp(−αd(s, t))

where d(s, t) is the length of a shortest st-path with respect to l. The equivalent connected

5

area of a landscape is the area of a single patch equivalent to it. This equivalence comes

from the indicator PC on which ECA is based [16]. Indeed, suppose that the landscape is

a rectangle of area W containing all patches. We consider a stochastic process consisting

of choosing two points p and q uniformly at random in the rectangle. The indicator PC

is the expected value of a random variable equal to 0 if either p or q does not belong to

a patch and Πst if p belongs to s and q belongs to t (recall that Πst = 1 if s = t). Since

the probability that p belongs to a patch u is wu/W, and the events p ∈ s and q ∈ t are

independent, by linearity of expectation, PC can be expressed as follows:

PC(G,w, l) =

∑s,t∈V wswtΠst

W2=ECA(G,w, l)2

W2

where W, ws and wt are respectively the area of the rectangle, patch s and patch t. The

equivalent connected area of a landscape is the area of a single patch whose PC value is

equal the PC value of the original landscape. If the area of the patches and the landscape

are normalized to makeW equal to 1 then PC is the square of ECA. Therefore, optimizing

PC and optimizing ECA are equivalent problems. ECA is often considered by researchers

interested in landscape connectivity because it represents an area, a more concrete quantity

than the expected value of a random variable.

An instance I = (V,A,Φ, w, l, l−, c, B) of the Budget Constrained ECA Optimization

Problems – BC-ECA-Opt – is defined by a graph G = (V,A) with a weight wu for each

vertex u ∈ V and a length la for each arc a ∈ A, a subset Φ ⊆ A of arcs that can be

improved, for each arc a ∈ Φ, the cost ca of reducing its length to l−a and the total budget

B for improving the arcs. An optimal solution of the instance I is a subset of arcs S ⊆ Φ

of total cost c(S) :=∑

a∈S ca at most B that maximizes ECA(G, l′, w) where l′a = l−a if

a ∈ S and la otherwise. For the sake of clarity, we first consider only arcs improvements,

but, as explained in Section 3.1, our model can be extended to the case where it is also

possible to improve the landscape by increasing the weight of the vertices. In this case, a

subset of vertices W is given, as well as the cost cu of increasing the weight of u to w+u for

each vertex u ∈ W, and we want to find a subset of arcs S ⊆ Φ and a subset of vertices

T ⊆ W of total cost c(S) + c(T ) ≤ B that maximizes ECA(G, l′, w′) where l′a = l−a if

a ∈ S and la otherwise and w′u = w+u if u ∈ T and wu otherwise.

6

Remark 1. Our optimization problem is stated in terms of arc length reduction but it

would be equivalent to state it in terms of arc probability augmentation.

Remark 2. In our model, it is possible to add a new connection between two existing

patchs u and v. It suffices to place (u, v) in the set of improvable arcs Φ with an initial

length l(u,v) =∞ (i.e. π(u,v) = 0) and a finite improved length l−(u,v) <∞.

3. An improved MIP formulation for BC-ECA-Opt

We first show how to compute f t =∑

s∈V ws ·Πst as the maximum quantity of gener-

alized flow that can be sent to t across the network if, for every patch s ∈ V , ws units of

flow are available at s and appropriate multipliers are chosen for each arc of the network.

Recall that a generalized flow differs from a standard flow by the fact that each arc a has

a multiplier πa such that the quantity of flow leaving arc a is equal to the quantity of flow

φa entering in a multiplied by πa. In our case πa = exp(−αla). For each vertex u ∈ V, let

δoutu be the set of arcs leaving u and δin

u the set of arcs entering u. As explained below,

the linear program (P) has f t as optimum value.

(P)

max z

s.t.∑

a∈δoutu

φa −∑b∈Γ−u

πb · φb ≤ wu u ∈ V \ {t} (A1)

∑a∈δoutt

φa −∑b∈δint

πb · φb = wt − z (A2)

φa ≥ 0 a ∈ A (A3)

Constraints (A1) require that the total quantity of flow leaving u is at most wu plus the

total quantify of flow entering u, i.e. the quantity of flow available in vertex u. Constraint

(A2) requires that z is equal to the total quantity of flow entering t plus wt minus the total

quantity of flow leaving t. Finally, constraints (A3) state that each arc carries a positive

quantity of flow from its source to its sink.

Lemma 1. Any optimal solution of (P) is obtained by sending, for each every vertex

s ∈ V − {t}, ws units of flow from s to t along most reliable st-paths.

Proof. First notice that the quantity of flow arriving at t when ws units flow are sent

from s to t along an st-path P is ws times the probability of path P. Let φ′ be an optimal

solution of (P) maximizing the quantity of flow routed along a path which is not a shortest

7

path. Suppose by contradiction that φ′ send ε > 0 units of flow along an st-path P ′ whose

probability is smaller than the probability of a most reliable st-path P. Let φ be the flow

obtained from φ′ by decreasing the flow sent on path P ′ by ε and increasing the flow sent

on path P by ε. Since the probability of P is larger than the probability of P ′, φ send

more flow to t than φ′, a contradiction.

Corollary 1. The optimal value of (P) is f t =∑

s∈V ws ·Πst.

Proof. Since the objective is to maximize z, no flow leaves t in any optimal solution of

(P), i.e.∑

a∈δoutt

φa = 0. Constraint (A2) ensures that z is the quantity of flow received

by t plus wt = wt · Πtt. By Lemma 1, there exits an optimal solution φ such that every

vertex s distinct from t send ws units of flow on a most reliable st-path. Hence, for every

vertex s distinct from t, the flow received by t from s is ws ·Πst and thus the value of z is∑s∈V ws ·Πst.

With this linear programming definition of f t, we can now present our MIP program

of BC-ECA-Opt. For each arc a = (u, v) ∈ Φ, we add another arc a′ = (u, v) of length

la′ = l−a that can be viewed as an improved copy of arc a. This improved copy can be

used only if the improvement of arc a is purchased. For each arc a ∈ Φ, that could be

improved, the variable xa is equal to one if the improvement of arc a is purchased and

zero otherwise. We denote by Ψ the set of improved copies of arcs in Φ. In the following

MIP program, δoutu and δin

u are defined with respect to the set of arcs A′ = A ∪Ψ.

BC-ECA-Opt

max∑t∈V

wt · f t

s.t.∑

a∈δoutu

φta −∑b∈δinu

πb · φtb ≤ wu t ∈ V , u ∈ V \ {t} (B1)

∑a∈δoutt

φta −∑b∈δint

πb · φtb = wt − ft t ∈ V (B2)

φta′ ≤ xa ·Ma t ∈ V , a ∈ Φ (B3)∑a∈F

ca · xa ≤ B (B4)

xa ∈ {0, 1} a ∈ Φ (B5)

φta ≥ 0 t ∈ V , a ∈ Φ (B6)

Constraints of (B1) and (B2) are simply constraints of (A1) and (A2) for all possible

target vertex t. For each arc a ∈ Φ, the big-M constraints (B3) ensure that if the im-

8

provement of arc a is not purchased, i.e. xa = 0, then the flow on arc a′ is null. Ma is

an upper bound of the flow value on arc a. We could simply take Ma =∑

u∈V wu for all

a ∈ Φ but smaller estimations are important to guaranty a good linear relaxation and thus

a faster resolution of the MIP program. Constraint (B4) ensures that the total cost of the

improvements is at most B. This MIP program has O(|V |(|A| + |Φ|)) flow variables, |Φ|

binary variables and O(|V |2 + |V ||Φ|) constraints.

3.1. Extension to patch improvements

To extend our model to the version with patch improvements where it is possible to

increase the weight of a vertex u from wu to w+u at cost cu, we process all the occurrences

of wu as follows. Let yu be a binary variable equal to 1 if the vertex u is improved and 0

otherwise. We add a term yucu in the budget constraint for every vertex u in the set W

of vertices that can be improved. When wu appears as an additive constant, we simply

replace it by wu+yu(w+u −wu). Note that wu only appears as a coefficient in the objective

function in the form wuft. In this case, to avoid a quadratic term, we use a standard

McCormick linearization [10]. We replace the product wuft by wuft + (w+u −wu)f ′t where

f ′t is a new variable that is equal to ft if yu = 1 and 0 otherwise. To achieve this values of

f ′t we add the constraints f ′t ≤ ft and f ′t ≤ yuM where M is larger than any values of ft.

As it is a maximization program and f ′t appears with a positive coefficient in the objective

function, the first constraint guaranties that f ′t = ft if yu = 1 in any optimal solution and

the second constraint guaranties that f ′t = 0 if yu = 0.

4. Preprocessing

The size of the mixed integer programming formulation of BC-ECA-Opt given in Sec-

tion 3 grows quadratically with the size of the graph that represents the landscape. In

this section, we describe a preprocessing step that reduces the size of this graph. For that,

we introduce a notion of strongness of an arc with respect to a target vertex t ∈ V . We

will call a solution x ∈ {0, 1}Φ of BC-ECA-Opt a scenario and say that the distances are

computed under scenario x when the length of every a ∈ A is l−a if xa = 1 and la otherwise.

We denote dx(s, t) the distance between s and t when the arc lengths are set according

to the scenario x. An arc (u, v) is said to be t-strong if, for every scenario x ∈ {0, 1}Φ,

(u, v) belongs to a shortest path from u to t, i.e. dx(u, t) = dx(u, v) + dx(v, t) for every

x ∈ {0, 1}Φ. An arc (u, v) is said to be t-useless if, for every scenario x ∈ {0, 1}Φ, arc

9

(u, v) does not belong to any shortest ut-path when arc lengths are set according to the

scenario x. We denote by S(t) the set of arcs (u, v) such that (u, v) is t-strong and by

W (t) the set of arcs (u, v) such that (u, v) is t-useless.

Of course, the naive algorithm that tests whether (u, v) belongs to a shortest ut-path

for every scenario x ∈ {0, 1}Φ is very inefficient. In Section 4.2, we describe and analyse

an algorithm that computes the set S(t) and W (t) for every sink t in a much more efficient

way. But first, we explain how the knowledge of these sets can be used to define a smaller

equivalent instance of the problem.

4.1. Operations to reduce the size of the graph

Removal of an arc from W (t). Let (u, v) ∈ W (t), since for all scenarios x ∈ {0, 1}Φ,

(u, v) does not belong to any shortest path from u to t, its removal does not affect the

distance from any vertex to t. It is clear, from the definition of ECA, that the removal of

(u, v) does not affect the contribution of t to ECA. Let fxt (G) :=∑

s∈V wswtΠxst be the

contribution to ECA of all the pairs having sink t in the graph G when the probability of

connection Πxst is computed under the scenario x.

Lemma 2. Let (u, v) ∈W (t) and let G′ be a graph obtained from G by removing arc (u, v)

then, for all scenario x ∈ {0, 1}Φ, fxt (G) = fxt (G′).

Proof. For every scenario x ∈ {0, 1}Φ, (u, v) does not belong to any shortest path from u

to t. Therefore, the removal of (u, v) cannot affect the probability of connection Πut for

any vertex t and fxt (G) = fxt (G′).

Contraction of an arc in S(t). Now, assume that (u, v) /∈ Φ, (u, v) ∈ S(t). The

contraction of (u, v) consists in replacing every arc (w, u) ∈ δinu by an arc (w, v) of length

l′wv = lwu + luv and by removing the vertex u and all its outgoing arcs. The weight of

u in G is moved to the weight of v in G′. Namely, the weight of v in the new graph is

w′v = wv+wu exp(−αluv). Let G′ be the graph obtained from G by contracting (u, v). The

next lemma establishes that the contribution of t to ECA in G is equal to its contribution

in G′.

Lemma 3. Let (u, v) ∈ S(t) and let G′ be the graph obtained from G by contracting

arc (u, v) and modifying accordingly the weight of wv. For every scenario x ∈ {0, 1}Φ,

fxt (G) = fxt (G′).

10

w

u v t...

lwuluv

wu wv

Γ−u

(a)

w

v t...

lwu + luv

w′v

Γ−u

(b)

Figure 1: (a) represents a graph G before the contraction of an arc (u, v). (b) represents the graph G′

obtained from G by contracting arc (u, v). The weight w′v of v in G′ is equal to wv + wu exp(−αluv).

Proof. Let s be a vertex of G and x be a scenario in {0, 1}Φ. We denote fxst(G) the

contribution to ECA of the pair st with scenario x ∈ {0, 1}Φ in G. If s /∈ {u, v}, it is easy

to check that, for any x ∈ {0, 1}Φ the length of the shortest path from s to t in G is equal

to the length of the shortest path from s to t in G′. Moreover, the weight of s is the same

in G and in G′. Therefore, by the definition of ECA, fxst(G) = fxst(G′), for any x ∈ {0, 1}Φ.

On the other hand, the contributions of the pairs ut and vt in G sum to the contribution

of v in G′. Indeed,

fxut(G) + fxvt(G) = wu exp(−αdx(u, t)) + wv exp(−αdx(v, t))

= wu exp(−α(luv + dx(v, t))) + wv exp(−αdx(v, t))

= (wv + wu exp(−αluv)) exp(−αdx(v, t))

= w′(v) exp(−αdx(v, t))

= fxvt(G′)

We conclude that, for any x ∈ {0, 1}Φ, fxt (G) =∑

s∈V fxst(G) = fxut(G) + fxvt(G) +∑

s∈V−{u,v} fxst(G) = fxvt(G

′) +∑

s∈V−{u,v} fxst(G

′) = fxt (G′).

Graph reduction. We obtain Gt by deleting every arcs of W (t) and contracting every

arcs of S(t). By induction the contribution of t is preserved. The experiments of Section

5 show that, when the number of arcs that can be protected is much smaller than the

total number of arcs, replacing G by Gt in the construction of the mixed integer program

reduces significantly the size of the MIP formulation and the running times.

The problem of identifying S(t) and W (t) has been studied as a preprocessing step

for other combinatorial optimization problems having as input a graph whose arc lengths

may change [6]. In [6], given an arc (u, v) and a vertex t, the authors identify a sufficient

(but not necessary) condition for (u, v) ∈ S(t) and use it to design a O(|A| + |V | log |V |)

11

algorithm that can, in most cases, identify if (u, v) ∈ S(t). The authors also propose a

O(|A|+ |V | log |V |) algorithm to test whether a given arc (u, v) belongs to W (s) or not.

Given an arc (u, v) of G, we will show how to adapt the Dijkstra’s algorithm to compute

in O(|A|+|V | log |V |) the set of all vertices t such that (u, v) ∈ S(t) or the set of all vertices

t such that (u, v) ∈ W (t). This improves the results of [6] by providing a necessary and

sufficient condition for (u, v) to be t-strong, i.e. we compute the entire set S(t) while

the algorithm of [6] computes a subset of S(t). It also reduces the time complexity for

computing S(t) and W (t) for all t by a factor |V | as we only need to run the algorithms

on each arc instead of each pair of arc and vertex.

4.2. Computing S(t) for all t

Given a scenario x ∈ {0, 1}Φ, the fiber Fx(u, v) of arc (u, v) is the set of vertex t such

that (u, v) belongs to a shortest path from u to t when arc lengths are set according to x,

i.e.

Fx(u, v) = {t ∈ V : dx(u, t) = lx(u, v) + dx(v, t)}

Let F (u, v) be the intersection of the fibers of (u, v) over all possible scenarios x ∈

{0, 1}Φ, i.e. F (u, v) :=⋂x Fx(u, v). By definition, an arc (u, v) is t-strong if t belongs to

Fx(u, v) for every scenario x i.e.

S(t) = {(u, v) : t ∈ F (u, v)}

In order to compute every S(t), we first compute F (u, v) for every arc (u, v) and then

transpose the representation to get S(t) for every vertex t ∈ V . Let y ∈ {0, 1}Φ be the

following scenario :

ywt =

0 w ∈ F (u, v) or (w, t) = (u, v)

1 w /∈ F (u, v)(1)

Lemma 4. The intersection F (u, v) of the fibers of (u, v) over all possible scenarios is the

fiber of (u, v) under the scenario y, i.e. F (u, v) = Fy(u, v).

Proof. Since the inclusion F (u, v) ⊆ Fy(u, v) is obvious, it suffices to prove that Fy(u, v) ⊆

Fx(u, v) for any scenario x ∈ {0, 1}Φ. By way of contradiction, suppose that x is a scenario

such that Fy(u, v)\Fx(u, v) contains a vertex t at minimum distance from u in the scenario

y. Let P be a shortest ut-path containing (u, v) under the scenario y. For every vertex

12

w of P, dy(u,w) ≤ dy(u, t) and w ∈ Fy(u, v). Hence, by the choice of t, w belongs to

Fx(u, v) for every scenario x. Therefore w belongs to F (u, v) and the length of every

arc of P is set to its upper bound in the scenario y, i.e. ly(P ) = l(P ). Now, let Q

be a shortest ut-path in the scenario x. If the path Q contains a vertex z ∈ F (u, v)

then dx(u, t) = dx(u, z) + dx(z, t) = lx(u, v) + dx(v, z) + dx(z, t) = lx(u, v) + dx(v, t), a

contradiction with t /∈ Fx(u, v). Therefore, the vertices of Q do not belong to F (u, v) and

thus their lengths are set to their lower bound in y, i.e. l(Q) = l−(Q). We deduce that

lx(P ) ≤ ly(P ) ≤ ly(Q) ≤ lx(Q) a contradiction with t /∈ Fx(u, v).

We describe a O(|A| + |V | log |V |) time algorithm that, given an arc (u, v), computes

simultaneously the scenario y defined by (1) and the fiber Fy(u, v) of arc (u, v) with respect

to this scenario. The algorithm is an adaptation of the Dijkstra’s shortest path algorithm

that assigns colors to vertices. We prove that it colors a vertex w in blue if w belongs to

Fy(u, v) and in red otherwise. At each step, we consider a subset of vertices S ⊆ V whose

colors have been already computed. Before the first iteration, S = {u} and the length

of every arc leaving u is set to its lower bound except the length of (u, v) which is set to

its upper bound according to scenario y. At each step, since the color of every vertex of

S is known, the length, under scenario y, of every arc (w, t) with w ∈ S is also known.

Therefore, it is possible to find a vertex t ∈ V − S at minimum distance from u, under

scenario y, in the subgraph G[S ∪ {t}] induced by S ∪ {t}. Following Dijkstra’s algorithm

analysis, we know that the distance under scenario y from u to t in G[S ∪ {t}] is in fact

the distance between u and t in G. This allows us to determine if there exists a shortest

path from u to t in the scenario y passing via (u, v) and to color the vertex t accordingly.

We end-up the iteration with a new vertex t whose color is known and that can be added

to S before starting the next iteration. The algorithm terminates when S = V.

For every vertex w ∈ V − S, the estimated distance d(w) is the length of a uw-path

under the scenario y in the subgraph G[S ∪{w}]. An arc a is blue if a = uv or if its origin

is blue, the other arcs are red, i.e. a is blue if ya = 0 and red if ya = 1. The estimated

color γ(w) of w is blue if there exists a blue uw-path of length d(w) in G[S ∪{w}] and red

otherwise. The correctness of Algorithm 1 follows from the following Lemma.

Lemma 5. For every vertex t ∈ S, γ(t) is blue if and only if t ∈ Fy(u, v).

Proof. We proceed by induction on the number of vertices of S. When S = {u}, the

property is verified. Now, suppose the property true before the insertion in S of the

13

Algorithm 1: Computes the set S of vertices t such that (u, v) is t-strong

Input : G = (V,A, l, l−), (u, v) ∈ A

Output: {t ∈ V : (u, v) is t-strong}

foreach (u,w) ∈ δoutu \ {(u, v)} dod(w)← l−uw

γ(w)← red

d(v)← luv; γ(v)← blue

S ← {u}; γ(u)← red

while S 6= V do

Pick t ∈ V − S with smallest d(t) breaking tie by choosing a vertex t such that

γ(t) = blue if it exists

if γ(t) is blue then

foreach (t, w) ∈ δoutt such that d(w) ≥ d(t) + ltw do

d(w)← d(t) + ltw

γ(w)← blue

else

foreach (t, w) ∈ δoutt such that d(w) > d(t) + l−tw do

d(w)← d(t) + l−tw

γ(w)← red

S ← S ∪ {t}return {t ∈ V : γ(t) = blue}

vertex t such that d(t) is minimum. By induction hypothesis, the lengths of arcs having

their source in S are set according to y. Therefore, following Dijkstra’s algorithm analysis,

we deduce that d(t) is the length of the shortest path from u to t in the graph G under

scenario y. If t has been colored blue then there exists a blue vertex w ∈ S such that

d(t) = d(w) + lwt. By induction hypothesis w ∈ Fy(u, v) and there exists a shortest ut-

path under scenario y containing (u, v), i.e. t ∈ Fy(u, v). Now, suppose that t has been

colored in red. By contradiction, assume there exists a shortest ut-path under scenario

y that contains (u, v). This path cannot contain a vertex outside S except t because

otherwise, since arc lengths are non-negative, its length according to y would be greater

than d(t) by the choice of t. Therefore, the predecessor w of t in this path belongs to S.

Since w belongs to a shortest ut-path passing via (u, v), by induction, it was colored blue.

But in this case, there exists a blue vertex w such that d(w) + l(w, t) = d(t), and t was

colored blue as well, a contradiction.

14

We are now ready to state the main result of this section.

Proposition 1. Given a graph G = (V,A), two arc-length functions l− and l such that 0 ≤

l−a ≤ la for every arc a ∈ A, and an arc (u, v), Algorithm 1 computes in O(|A|+|V | log |V |)

the set of vertex t such that (u, v) is t-strong.

Proof. The correctness of Algorithm 1 is given by Lemma 4 and Lemma 5. Analogously

to the Dijkstra’s algorithm, it can be implemented to run in O(|A| + |V | log |V |) using a

Fibonacci heap as priority queue.

4.3. Computing W (t) for all t

The algorithm that computes, for every arc (u, v), the set of vertices t such that

(u, v) is t-useless is obtained from Algorithm 1 by applying two small changes. Before

describing these changes, we explain how the two problems are related. For that, we first

introduced a strengthening of the notion of strongness. We will say that an arc (u, v)

is strictly t-strong if it belongs to all shortest ut-paths for every scenario x ∈ {0, 1}Φ.

Recall that t-strongness requires only the existence of a shortest ut-path passing via (u, v)

for every scenario x ∈ {0, 1}Φ. Algorithm 1 can be easily adapted to compute for every

arc (u, v) the set of vertex t such that (u, v) is strictly t-strong. It suffices to change

the way, the algorithms breaks tie between a red and a blue path and the choice of t

in case of tie. Namely, in the first internal loop, the condition for coloring w in blue

becomes d(w) > d(t) + ltw while the condition for coloring w in red in the second internal

loop becomes d(w) ≥ d(t) + l−tw. Moreover, when we choose t such that d(t) is minimal,

we break tie by choosing a red vertex if it exists. Clearly, these small changes exclude

the existence of a red path of length d(w) between u and a blue vertex w. Therefore,

(u, v) belongs to every path of length d(w) and (u, v) is strictly w-strong whenever w is

blue. We call the resulting algorithm the strict version of Algorithm 1. The next step

is to extend the notion of strict strongness to a subset of arcs having the same source.

For any vertex u ∈ V, a subset Γ ⊆ Γ+u of arcs is strictly t-strong if, for every scenario

x ∈ {0, 1}Φ, all shortest ut-paths intersect Γ. By definition, an arc (u, v) is t-useless if and

only if Γ+u \ {(u, v)} is strictly t-strong. Indeed, every ut-path avoiding (u, v) intersects

Γ+u \{(u, v)} and, conversely, (u, v) belongs to every ut-path avoiding Γ+

u \{(u, v)}. Hence,

computing the set of vertex t such that (u, v) is t-useless amounts to compute the set

of vertex t such that Γ+u \ {(u, v)} is strictly t-strong. An algorithm that computes this

15

set of vertices can be obtained from the strict version of Algorithm 1 by modifying only

the initialization step: arc (u, v) is colored in red and its length is set to l−uv while arcs

of δoutu \ {(u, v)} are colored in blue and their lengths are set to their upper bound. A

correctness proof very similar to the one of Algorithm 1 (and that we will not repeat)

shows that a vertex is colored blue by Algorithm 2 if and only if (u, v) is t-useless. Since

the two algorithms have clearly the same time complexity, we conclude this section with

the following result.

Algorithm 2: Computes the set of vertex t such that (u, v) is t-useless

Input : G = (V,A, l, l−), (u, v) ∈ A

Output: {t ∈ V : (u, v) is t-useless}

d(v)← l−uv; γ(v)← red

foreach (u,w) ∈ δoutu \ {(u, v)} dod(w)← luw

γ(w)← blue

S ← {u}; γ(u)← blue

while S 6= V do

Pick t ∈ V − S with smallest d(t) breaking tie by choosing a vertex t such that

γ(t) = red if it exists

if γ(t) is blue then

foreach (t, w) ∈ δoutt such that d(w) > d(t) + ltw do

d(w)← d(t) + ltw

γ(w)← blue

else

foreach (t, w) ∈ δoutt such that d(w) ≥ d(t) + l−tw do

d(w)← d(t) + l−tw

γ(w)← red

S ← S ∪ {t}return {t ∈ V : γ(t) = blue}

Proposition 2. Given a graph G = (V,A), two arc-length functions l− and l such that 0 ≤

l−a ≤ la for every arc a ∈ A, and an arc (u, v), Algorithm 2 computes in O(|A|+|V | log |V |)

the set of vertex t such that (u, v) is t-useless.

16

5. Numerical experiments

In this section, we report our computational experiments in order to demonstrate the

added benefit of our MIP formulation and preprocessing step. The numerical experiments

were performed on a desktop computer equipped with an Intel(R) Core(TM) i7-8700k 4.8

gigahertz and a memory of 16 gigabytes. Our model was implemented in Gurobi 9.0.1

with default parameters and the other algorithms were coded in multithreaded C++ using

the LEMON Graph Library.

5.1. Quality of the solutions

In the lack of an efficient method to compute an optimal solution of BC-ECA-Opt,

ecologists often use greedy algorithms to compute sub-optimal solutions. In this subsec-

tion we will compare an optimal solution obtained with our MIP model to the solutions

obtained with different greedy algorithms. The incremental greedy algorithm (IG) starts

from the graph with no improved element, at each step i, the algorithm selects the ele-

ment e with the greatest ratio δie/ce until no more element fits in the budget. Here, δie

denotes the difference between the value of ECA with and without the improvement of

the element e at the step i and ce is the cost of improving the element e (that can be

either an arc or a vertex). The decremental greedy (DG) starts from the graph with all

improvements performed and iteratively removes the improvement of the element e with

the smallest ratio δie/ce. DG finishes with incremental steps to ensure there is no free bud-

get left. These algorithms perform at most |Φ| steps and at each step i need to compute

δie for each element e. It is easy to implement the IG algorithm in O(|V |3 + |Φ|2 · |V |2)

by using an all pair shortest path algorithm to compute in O(|V |3) the initial distance

matrix and then by performing |Φ| steps in which the computation of δie for each arc

e ∈ Φ takes O(|V |2). Indeed, when we decrease the length of an edge we can update the

distance matrix in O(|V |2). For the implementation of DG, when we increase the length

of an edge we cannot update the distance matrix as easily as for the IG algorithm. We

can recompute in |V |3 the whole distance matrix for computing each δie and the algorithm

runs in O(|Φ|2 · |V |3) (we did not try to improve the implementation). These complexities

are already too large for the practical instances handled by ecologists which can have few

thousands of patches. Most of the studies using the PC or ECA indicators use simpler

algorithms that we call static increasing (SI) and static decreasing (SD). These algorithms

are simpler variants of the greedy algorithms which do not recompute the ratio δe/ce of

17

each element e at each step and thus are faster but do not account for cumulative effects

nor redundancies.

In the next section, we provide instances on which IG and DG performs poorly com-

pared to an optimal solution. On these instances, it is easy to check that the solutions

returned by algorithms SI and SD are not better than the solutions returned by algorithms

IG and DG.

5.1.1. Limits of the greedy algorithms

In the instances presented below, all arcs have a probability 1 if improved and 0

otherwise and have unitary costs. Recall that a spider is a tree (sorry for ecologists)

consisting of several paths glued together on a central vertex.

Case A: The graph is a spider with 2k branches: k long branches with two edges, an

intermediate node of weight 0 and a leaf node of weight 1, and k short branches consisting

of a single edge with a leaf of very small weight ε > 0, see Fig.1 (a). All branches are

connected to a central node of weight 1. IG performs poorly on this instance. Indeed,

IG is tricked into purchasing short branches with very small ECA improvement because

purchasing an edge of a long branch alone does not increase ECA at all. An optimal

solution results in larger value of ECA by improving pairs of arcs of long branches.

In this case, IG does not perform well while DG finds an optimal solution computed by

the MIP solver except when the budget is 1. In this case, the reverse occurs: DG performs

badly while IG is optimal. Indeed, DG realizes that the budget is not sufficient to improve

two arcs of a long branch only after removing the improvements of all short branches.

Case B: The graph is obtained from a star with k + 1 branches by replacing one branch

by a path of length k. The central node and all leaves except the leaf of the path have

weight 1. The leaf of the path has weight 1+ε. The internal nodes of the path have weight

0. DG performs poorly on this instance because it removes one by one the branches of

the star for which δe/ce = 1 before removing an edge of the path for which δe/ce = 1 + ε.

When the budget is at least 2, an optimal solution removes all the edges of the path before

removing an edge of another branch.

Case C: The graph is a spider with k+ 1 branches. All branches except one are paths of

length 2 with an internal node of weight 0 and a leaf of weight 1. The last branch is a path

18

8

7

6

5

4

3

2

10

(a)

0 2 4 6 8 10 12budget

0.0

0.2

0.4

0.6

0.8

1.0

opt

rati

o

incremental greedydecremental greedyoptimum (MIP)

(b)

Figure 2: An instance on which Incremental Greedy fails. (a) the graph of case A with k = 4, (b) ratio of

the increase in ECA between the solutions returned by IG and DG and an optimal solution for several

budgets.

����

65

4

3 2

10

(a)

0 2 4 6 8 10budget

0.0

0.2

0.4

0.6

0.8

1.0

opt

rati

o

incremental greedydecremental greedyoptimum (MIP)

(b)

Figure 3: An instance on which Decremental Greedy fails. (a) the graph of case B with k = 5, (b) ratio

of the increase in ECA between the solutions returned by IG and DG and an optimal solution for several

budgets.

of length 2k with internal nodes of weight ε > 0 and a leaf of weight 1 + ε. All branches

intersect in a central node of weight 1. In this case, both Incremental and Decremental

Greedy fail. On one hand, IG selects the edges of the path of length 2k one by one and

does not realize that by taking two edges of a short branch it could improve much more

ECA. On the other hand, DG removes first the edges of the short branch because the

weight of leaf of a long branche is 1 + ε while the weight of the leaf of a short branch is 1.

Hence, DG and IG return the same law quality solution.

19

151413121110987

65

4

3 2

10

(a)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0budget

0.0

0.2

0.4

0.6

0.8

1.0

opt

rati

o

incremental greedydecremental greedyoptimum (MIP)

(b)

Figure 4: An instance on which both Incremental and Decremental Greedy algorithms fail. (a) the graph

of case C with k = 5, (b) ratio of the increase in ECA between the solutions returned by IG and DG and

an optimal solution for several budgets.

Case C illustrates the fact that the greedy algorithms IG and DG do not provide any

constant approximation guarantee (even for trees), i.e. for any constant 0 < α < 1 there

exists an instance of BC-ECA-Opt such that ALG < αOPT where ALG is the value of

ECA for the best solution among those returned by IG and DG and OPT is the value of

ECA for an optimal solution.

5.1.2. In Practice

In this subsection we investigate on the performances of the greedy algorithms we

discuss in the previous subsection on a real instance. We perform these numerical experi-

ments on a model of the landscape around Montreal from [2]. The landscape model of [2]

is a Gabriel graph – a subset of the Delaunay triangulation – of 8733 patches. For each

patch, we have an estimation of its area that could be lost by 2050. In order to obtain

20

several examples with a reasonable computing time, we extract 72 subgraphs of sizes be-

tween 150 and 400 nodes from this large instance. Then, we consider that protecting a

patch corresponds to buying the portion of it that would otherwise be lost and has a cost

proportional to the purchased area. To model the protection of a patch u, we set its de-

fault weight, wu, to the area it would have in 2050 if unprotected and its improved weight,

w+u , to its current area. If a patch is threatened of total loss, i.e. all its area would be lost

by 2050, we prevent it to contribute to shortests paths if it is not purchased as follows.

We add a node uout, replace each outgoing arc (u, v) by an arc (uout, v) and connect u and

uout with an arc (u, uout) of default probability 0 and enhanced probability 1. In this way,

if a patch threatened of total loss is not protected then it does not contribute to ECA by

its quality nor by its use as stepping stone for other pairs of nodes.

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Figure 5: Quebec landscape modelization from [2]

Contrastingly to what we have seen in the previous section, greedy algorithms perform

surprisingly well in the experiments we perform on this real case study. As shown in Fig

6, greedy algorithms lead on average to gains that reach 95% of the one obtained with

optimal resolution with respect to the gain expected with a random solution.

5.2. Scalability and benefits of the preprocessing

In this subsection we address the scalability of our approach and the added benefits of

our preprocessing step. For this purpose we build a plausible instance from which we can

easily vary the number of changing elements and integer variables. We take the case of

the city of Marseille which is surrounded by 3 big forest massifs and contains 54 parks of

21

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0budget in %

0.95

0.96

0.97

0.98

0.99

1.00av

erag

e op

tim

um r

atio

incremental greedydecremental greedyoptimum (MIP)

Figure 6: Average value of returned solutions normalized between the average random solution and an

optimal one

#wastelandMIP preprocessed MIP greedy

#var #const time #var #const time time

20 345775 18410 12246 10716 7197 316 766

50 717901 36410 142306 34613 21485 1964 7217

80 1306597 59810 1914478 76281 42039 13141 29479

110 - - - 132225 66759 62268 87948

140 - - - 207999 96600 200558 208179

170 - - - 308355 132701 1701684 416751

Table 1: Comparison of the MIP, the preprocessed MIP and the greedy (algorithm DG) according to the

number of variables (#var), the number of constraints (#const) and the time (in ms) it takes to execute

variable sizes. The idea is to consider a certain number of wastelands throughout the city

that can be protected to preserve the stepping stones between massifs and parks. The base

graph is composed of 96 nodes of which 42 models the frontier of massifs, 43 represents

small parks and 11 big parks. The graph is complete and each arc is weighted with the

border to border distance between the two patches it connects. We model each wasteland

as a patch which is totally threatened (see Subsection 5.1.2). A non purchased wasteland

has then a null quality and does not contribute to any shortest path. We perform our

experiments by adding various number of wastelands, randomly generated across the city,

and looking for variations of the MIP formulation size and computation time.

We see in Table 1 that the computation time of the MIP without preprocessing in-

22

creases very quickly. It takes more than 30 minutes on average for instances with 80+

wastelands whereas the preprocessed one can be solved in less than 30 minutes with up to

170 wastelands. This is due to the preprocessing step that significantly reduces the time

required to solve the linear relaxation by reducing the number of variables, constraints

and non-zero entries of the mixed integer program. The preprocessed MIP is faster than

the greedy algorithm for instances with at most 140 wastelands (the preprocessing step

was not used for greedy algorithms).

Finally, we run our preprocessing on 400 instances randomly generated from the Que-

bec’s graph (Fig. 5) in order to study the impact of the preprocessing on the MIP for-

mulation size. For building these instances, we take 20 connected subgraphs of 500 nodes

and for each graph we create 20 instances by randomly picking a percentage of arcs whose

length could be improved by half. Figure 7 shows the percentage of reduction of the num-

ber of constraints, variables and non-zero entries of the MIP formulation with respect to

the percentage of arcs that could be restored. The preprocessing removes almost all the

elements of the MIP formulation when the number of restorable arcs arrives close to zero.

When 20% of the arcs could be restored, the preprocessing removes more than 75% of the

model’s variables and non-zero entries and around 60% of the constraints.

0 20 40 60 80 100percentage of restorable arcs

0

20

40

60

80

100

perc

enta

ge o

f ele

men

tsre

mov

ed b

y th

e pr

epro

cess

ing constraintsvariablesnon-zero entries

Figure 7: Average percentage of constraints, variables and non-zero entries that the preprocessing

removes with respect to the percentage of arcs that could be restored

6. Conclusion

In this paper we introduce a new MIP formulation for BC-ECA-Opt and we show that

this formulation allows solving optimally instances with around 150 habitat patches while

23

previous formulations [22] are limited to instances of about 30 patches. In order to make

our formulation scale to even larger instances, we present a preprocessing step that reduces

significantly the size of the graphs on which a generalized flow has to be computed. We

show that this preprocessing step allows to greatly reduce the MIP formulation size and

that its benefits increase when the proportion of arcs whose lengths can change decrease.

Thus allowing us to treat instances of up to 300 patches.

This new method allows us to solve optimally larger instances of BC-ECA-Opt and to

compare optimum solutions with the solutions returned by greedy algorithms. Interest-

ingly, we found that greedy algorithms perform well in practice despite the arbitrary bad

cases we spotted. Our next goals will be to experiment our approach on other practical

instances of the problem arising from different ecological contexts. On the theoretical side,

we would like to investigate the problem from an approximation algorithm point of view.

For instance, is it possible to find reasonable assumptions under which greedy algorithms

are guaranteed to return a solution whose ECA value is at least a constant fraction of the

optimal ECA ? If these assumptions are fulfilled by the real instances that we considered,

this would explain our experimental observations. Moreover, since a polynomial time ap-

proximation scheme has been given in the case of trees [21], it could also be interesting

to know for which larger classes of graphs constant factor approximation algorithms for

this problem exist. Another interesting question is to determine whether good solutions

could be obtained by decomposing geographically the problem, by solving independently a

subproblem for each region and then by reassembling the solutions. In this case, a notion

of fairness could help to allocate the budget among the regions so that each region can

enhance its own internal connectivity keeping a part of the budget to enhance the connec-

tivity between the regions. Since ECA is based on the equivalence between a landscape

and a patch, such a multilevel optimization approach looks promising.

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