A Multi-period Vertex Cover Problem and Application to

Fuel Management

Marc Demange and Cerasela Tanasescu

School of Science, RMIT University, GPO Box 2476, Melbourne, Vic., Australia

Keywords:

Multi-period Vertex Cover, Wildﬁre, Fuel Management, Planar Graphs, Polynomial Approximation, Approx-

imation Preserving Reductions.

Abstract:

We consider a generalisation of MIN WEIGHTED VERTEX COVER motivated by a problem in wildﬁre pre-

vention. The problem is deﬁned for a ﬁxed number of time periods and we have to choose, at each period,

some vertices to be deleted such that we never have two adjacent remaining vertices. The speciﬁcity is that

whenever a vertex is deleted it reappears after a given number of periods. Consequently we may need to delete

a single vertex several times. The objective is to minimise the total weight (cost) of deleted vertices. The

considered application motivates the case of planar graphs. While similar problems have been mainly solved

using mixed integer linear models (MIP) we investigate a graph approach that allows to take into account the

structure of the underlying graph. We use a reduction to the usual MIN WEIGHTED VERTEX COVER to devise

efﬁcient approximation algorithms and to raise some polynomial classes.

1 INTRODUCTION

Fuel management problems is one of the main tech-

niques used for reducing the impact of wildﬁres (Boer

et al., 2015). The main idea consists in dividing a

landscape into a number of cells representing candi-

date locations for fuel treatment like harvesting (EU,

North America) or burning (USA, Australia). Treat-

ments are generally applied on multiple periods (sev-

eral years) and key decisions are to determine which

cells should be treated during each time period (i.e.,

each year).

Spatially explicit multi-period fuel treatment

scheduling is a very complex problem (Chung, 2015).

A particularity to take into account is the transient

effect since vegetation begins to re-grow after it has

been treated. The risk of ﬁre spread from one cell

to another one becomes important if the ages of the

vegetation in the two adjacent cells achieve a speciﬁc

threshold and does not signiﬁcantly increase for older

vegetation (Boer et al., 2015).

Most of the known optimisation solutions for

this problem involve a Mixed Integer Linear Pro-

gramming approach (MIP) for locating fuel treat-

ments (see, e.g., (Hof et al., 2000; Kim et al., 2009;

Minas et al., 2014; Rachmawati et al., 2015; Wei

et al., 2008)) with various objectives. For further de-

tails on this area, the reader is referred to (Ager et al.,

2010; Chung, 2015; Minas et al., 2012).

Due to computational constraints, many of the

modelling efforts have considered very small land-

scapes in a single period problem (see e.g., (Hof et al.,

2000; Wei et al., 2008)). Multi-year spatial fuel treat-

ment planning has been considered in (Kim et al.,

2009; Minas et al., 2014). In the latter paper, the un-

derlying adjacency graph structure is explicitly used

in the model and computational results are proposed

on small grid-like partition of the map. However, to

our knowledge, the adjacency graph of cells has never

been used for algorithmic purpose.

MIP has the advantage of ﬂexibility and of easy

representation of diverse constraints and objectives as

well as the possibility to use efﬁcient solvers. How-

ever, as an almost universal model, a well known

drawback is that it makes it difﬁcult to take into ac-

count the true nature of the problem and the speci-

ﬁcity of the instances appearing in applications. In

particular, the MIP approaches that have been devel-

oped so far for fuel management problems do not

use any speciﬁc property of the underlying matrix

and thus, induce exponential time algorithms that be-

come very quickly intractable (Chung, 2015). As a

consequence, their use is limited to small instances

and is justiﬁed only for problems that are hard, even

for the considered classes of instances appearing in

practice. Therefore, preliminary theoretical complex-

Demange, M. and Tanasescu, C.

A Multi-period Vertex Cover Problem and Application to Fuel Management.

DOI: 10.5220/0005708900510057

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 51-57

ISBN: 978-989-758-171-7

ity considerations are important to justify these ap-

proaches. The last drawback of the current models

is that they make it very hard to detect and use links

with other problems studied in other areas of appli-

cations. The notion of approximation preserving re-

ductions gives a suitable theoretical framework for es-

tablishing such links and using them in practice. The

reader is referred to (Ausiello et al., 1999) for basic

deﬁnitions in polynomial approximation theory.

In this work we propose to use a graph model as a

complementary approach for some fuel management

problems. We consider the example of the model de-

veloped in (Minas et al., 2014) and give ﬁrst results in

this direction that motivate such though process and

illustrate how it can be used for better understand-

ing the problem, studying its complexity

, designing

efﬁcient algorithms in some speciﬁc graph instances

and identifying some links with extensively studied

problems. As a preliminary work, this paper remains

essentially theoretical and considers a simpliﬁed ver-

sion of the practical problem. However, our example

illustrates how such though process can raise some

new research questions as well as particular kind of

solutions that are not naturally produced by a usual

MIP model. We strongly believe that such approach

is interesting, as a complement of MIP solutions, and

we hope it will foster further works in this direction.

2 THE MODEL

In this study we will mainly follow the model pro-

posed in (Minas et al., 2014). We assume that the

landscape is already partitioned into zones (called

cells), each having a speciﬁc predominant fuel pat-

tern. Each cell is associated with a discrete time step

function representing the age of the fuel that is in-

creased by one each year if the cell is untreated and

reset to zero if it is treated; however, up to some

threshold the age of the cell is considered as old and

does not need to be incremented since the vegeta-

tion’s behaviour is essentially the same up to the old-

vegetation threshold. From this threshold the risk of

ﬁre spread becomes high and a critical situation oc-

curs when two such cells are adjacent (i.e. share a

boundary line). This age function is characterised by

two parameters supposed to be known:

1. the initial fuel age;

2. the old-vegetation threshold.

The model is formulated with the following nota-

tions:

Thus, eventually motivating some previous MIP ap-

proaches.

• I = {1, ...,n} will denote the set of all cells in the

landscape and we will denote each cell with index

i ∈ I;

• V = {1, ...,m} is a ﬁnite set of vegetation types

and v

∈ V, i = 1,. .. ,n is the predominant vegeta-

tion type of cell i ∈ I;

• Each vegetation type v ∈ V is associated with

an old-vegetation threshold o

: the vegetation

will be called old if its old-vegetation threshold

is reached; a cell with old vegetation is called old;

• T is the number of time periods in the planning

horizon and t = 1,.. .,T is the time parameter;

• a

i,0

, i ∈ I is the initial fuel age of cell i and

i,t

, t = 1,. ..,T + 1 will denote the age of cell

i at the beginning of time period t where t = T +1

corresponds to the end of the last time period;

• For everycell i ∈ I, c

(t,a

i,t

),t = 1...,T is the cost

for treating the cell i at time t if its age vegetation

is a

i,t

The risk of ﬁre spread from one cell to another one

is represented by a graph G

= (I,E) with the cells as

vertex set and E as edges. Two cells i, j ∈ I are con-

nected if there is a risk of ﬁre spread from one to the

other when both cell’s vegetations are old (a

i,t

= o

and a

j,t

= o

). In this work we consider G

as non-

directed and we will mainly consider planar G

s cor-

responding to the case where the edge set E corre-

sponds to the usual spatial adjacency of cells.

The aim is to decide, for each time period t, which

set C

⊂ I to treat. If a cell i is treated during the time

period t then its vegetation age is set to 0 at the next

period while is is just increased by one - up to the

old-vegetation threshold - if it is not selected:

∀t = 1,. .., T,∀i = 1,...,n,

i,t+1



min(a

i,t

+ 1,o

) if i /∈ C

0 if i ∈ C

The following combinatorial problem, called min-

imum Zero Risk Fuel Treatment Scheduling(ZFTS),

is a simpliﬁed abstraction of the real problem to be

solved in practice. It provides a theoretical frame-

work that allows us to understand the combinatorial

structure of the problem.

ZFTS’s objective is to minimise the total cost over

the time period [1,T] in order to never have two cells

with old vegetation that are connected in G

. In the

conclusion we will also mention the version where we

are given some a budget b

for each period t and the

objective is to minimise over the time period [1, T]

the total number of connections between two old veg-

etation cells such that the total budget used during

each time period t does not exceed b

. This problem,

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

called minimum Budget constrained Fuel Treatment

Scheduling (BFTS) is exactly the one studied in (Mi-

nas et al., 2014).

2.1 Some Graph Theory Related

Notations

For all graph notions not deﬁned here, the reader is

referred to (Diestel, 2012) and for all complexity no-

tions he is referred to (Garey and Johnson, 1979).

All graphs considered in this work are simple

undirected graphs. Given a graph G = (I, E), I de-

notes the set of vertices and E the set of edges. Given

′

⊂ I, we denote by G[I

′

] the subgraph of G induced

by I

′

. A vertex cover is a set of vertices such that ev-

ery edge has at least one extremity in this set. The

problem of deciding, for any graph G and integer K

whether G contains a vertex cover of size at most

K is a well-known NP-complete problem (Garey and

Johnson, 1979). MIN VERTEX COVER is the problem

of determining in any graph instance G a vertex cover

of minimum size; since its decision version is NP-

complete in general graphs, it is itself NP-hard. In the

weighted version, non negative weight ω

is associ-

ated to vertex i ∈ I and the weight of a subset I

′

⊂ I is

ω(I

′

) =

∑

i∈I

′ ω

. MIN WEIGHTED VERTEX COVER

is to determine, for any weighted graph a vertex cover

of minimum weight.

A graph G = (V,E) is called planar if it can be

embedded in the two dimension plane without cross

edges. In particular, given a set of connected ar-

eas in the plane, the adjacency graph with vertex set

these areas and two areas linked by an edge if they

share a common bounder-line of positive length is

planar. A basic example of planar graph is the case

of grid graph. A grid graph of size n × m is deﬁned

as follows: its vertex set is {1,. .., n} × {1,. ..,m}

with edges between (i, j) and (i

′

, j

′

) if and only if

|i−i

′

|+| j− j

′

| = 1. Grid graphs are planar and bipar-

tite, which means that their vertex set can be divided

into two stable sets (set of vertices pairwise not linked

by an edge).

3 A VERTEX COVER APPROACH

FOR ZFTS

We will say that a treatment cost c(t,a) is not increas-

ing if ∀(t,a) ∈ N

,c(t + 1,a+ 1) ≤ c(t,a).

Let us ﬁrst note that, for any instance of ZFTS

on a graph G

, if there is only one single period, then

the problem is to ﬁnd a minimum vertex cover of the

graph induced by the vertices associated with old veg-

etation cells: the vertices in a vertex cover correspond

to the cells to be treated. So, we immediately get the

following result:

Remark 1. ZFTS is at least as hard as MIN VER-

TEX COVER and in particular it is NP-hard in planar

graphs (Garey and Johnson, 1979).

This result holds even for the case with only one

vegetation type and all costs are equal to one. It moti-

vates us working in two different research directions:

either ﬁnding polynomial classes of instances or de-

vising good approximations for NP-hard cases.

For a minimisation combinatorial problem, a

polynomial-time algorithm (Ausiello et al., 1999) is

said to guarantee the approximation ratio ρ if, for each

instance H of optimal value β

it computes a feasible

solution of value λ

such that λ

≤ ρβ

. Moreover,

a family of polynomial algorithms indexed by ε > 0

and guaranteeing the ratio 1+ ε is called a polynomial

approximation scheme (PTAS).

Proposition 1. For any instance of ZFTS with not in-

creasing treatment costs, for every feasible solution S,

there is a feasible solution

S, with an objective value

not greater than S for which every treatment occurs

on old vegetation cells.

Proof. (sketch) We iteratively transform S as follows.

Consider the ﬁrst time t a cell i ∈ I is treated in S

with a vegetation age a

verifying a

< o

. If T − t <

− a

, then we just withdraw the related treatment,

else we delay it until t + o

− a

. All other treatments

are kept unchanged. We denote by S

the transformed

solution. We can easily verify that S

is still feasible

(consider the two cases, dates up to t + o

− a

and

later dates).

We then deﬁne a measure m of a solution σ as

the sum

∑

i∈I

, where d

is either the ﬁrst date when

i is treated with an age less than its old-vegetation

threshold in σ or T + 1 if such a date does not ex-

ist. m(σ) is an integer that ranges between 0 and

(T + 1)|I| and moreover m(σ) = (T + 1)|I| if and

only if all treatments in solution σ occur on old veg-

etation cells. It is straightforward to verify that if

m(S) < (T + 1)|I|, then m(S

) > m(S). As a conse-

quence, by repeating the previous process, we itera-

tively build instances S

,... until obtaining an in-

stance S

such that m(S

) = (T +1)|I|.

S = S

satisﬁes

the required constraints.

In what follows, we show how to transform any

algorithm for MIN WEIGHTED VERTEX COVER into

an algorithm for ZFTS with some performance guar-

anties (approximation preserving reduction). Propo-

sition 2 corresponds to short time periods and multi-

vegetation while Proposition 3 deals with the single

A Multi-period Vertex Cover Problem and Application to Fuel Management

vegetation case and any time period. Both results lead

to polynomial cases. Finally, our main result is Propo-

sition 4 that leads to interesting asymptotical results

for long time periods and multi-vegetation.

Proposition 2. Let C be a hereditary class of graphs

for which MIN WEIGHTED VERTEX COVER can be

approximated in polynomial time within the ratio ρ.

The problem ZFTS on the class C with T ≤ o

for all

vegetation types v and with constant treatment costs

can be approximated in polynomial time within the

same approximation ratio ρ.

Proof. We consider an instance satisfying the hy-

potheses and denote by G

the related graph with cells

set I. We then associate each vertex (cell) i with the

weight c

corresponding to the treatment cost.

Using Proposition 1 we can restrict ourselves to

feasible solutions for which only cells with old veg-

etation are treated. Moreover, since T ≤ o

, when-

ever a cell is treated during the period, it will never

achieve its old-vegetation threshold and will not be

treated again. We call nice a solution for which each

cell is treated no more than once and every treated

cell is treated at the date it achieves its old-vegetation

threshold. The above arguments show that there is an

optimal solution that is nice.

Let I

′

denote the set of cells i with an initial age

satisfying a

≥ o

− T, then the set of nice optimal

solutions is in one-to-one correspondence with vertex

covers of the graph G

′

] and moreover, the cost of

such solutions is exactly the weight of the related ver-

tex cover. Consider indeed an edge xy of G

′

], since

x,y ∈ I

′

they both achieve their old-vegetation thresh-

old during the period and consequently one of them

needs to be treated at least once. Conversely, for any

vertex cover of G

′

] with weight W, treating each of

its vertices when it achieves its old-vegetation thresh-

old constitutes a nice solution with a cost equal to W.

This concludes the proof.

In (Baker, 1994), a polynomial time approxima-

tion scheme for Min Vertex Cover in planar graphs is

devised and the dynamic programming approach also

holds for the weighted case. Another interesting par-

ticular case is the bipartite case since a landscape with

square cells, leading to a subgrid as the ﬁre spread

graph, is considered in different papers (see e.g. (Mi-

nas et al., 2014)). In this case the weighted vertex

cover is known to be polynomially solvable. We de-

duce the following corollary.

Corollary 1. The problem ZFTS with T ≤ o

for all

vegetation types v and with constant treatment costs

has:

1. a polynomial time algorithm in bipartite graphs

and in particular in sub-grids.

2. a polynomial time approximation scheme in pla-

nar graphs.

For real applications however, it is natural to

consider long time horizons with, possibly, a re-

optimisation with a smaller periodicity. We then pro-

pose the following strategy built from a weighted ver-

tex cover V

′

for some weight system.

At each time period one treats all the cells in

′

which have achieved their old-vegetation

threshold together with at least one adjacent

cell.

Such strategy is very easy to be implemented once

′

is determined and moreover it has the property to

be periodic, which means that, once a cell i is treated

it will be treated again every o

years. We ﬁrst con-

sider the single vegetation case and then turn to the

multi-vegetation case.

Proposition 3. Let C be a hereditary class of graphs

for which MIN WEIGHTED VERTEX COVER can be

approximated in polynomial time within the ratio ρ.

The problem ZFTS on the class C with a single vege-

tation type of old-vegetation threshold o and with con-

stant treatment costs can be approximated in polyno-

mial time within:

(1) the same approximation ratio ρ if T ≡ 0(modo).

(2) the approximation ration



⌊

⌋



ρ else.

Moreover, these ratios can be guaranteed by a pe-

riodic solution of period o: the same treatment pro-

gram is proposed during each time interval [ko,(k +

1)o),k ∈ N.

Proof. As in the proof of Proposition 2, we consider

an instance satisfying the hypotheses and denote by

the related graph with cells set I. We then associate

each vertex (cell) i with the weight c

corresponding

to the treatment cost. Using similar arguments we get

that for any edge xy, either x or y should be treated

during any time period of length o. As a consequence,

the optimal value β satisﬁes:

β ≥





(1)

where τ

denotes the minimum weight of a vertex

cover of G

As approximated solution, we then consider a ver-

tex cover I

′

⊂ I computed by the considered vertex

cover approximation algorithm. Denoting by τ

′

the

cost of I

′

, we have τ

′

≤ ρτ

. The solution we take

for the whole instance just consists in treating dur-

ing each time period [ko, (k+ 1)o),k ∈ N vertices in

′

at the time they achieve their old-vegetation thresh-

old. During the last period





o;T



, one does not

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

treat vertices that do not achieve their old-vegetation

threshold. Whenever a vertex is treated, at time t, it

will be treated again at time t + o and not before; this

shows that the solution is o-periodic.

Since treatment costs are not time-dependent, the

total cost λ

′

of this solution satisﬁes:

1. if T ≡ 0(modo), λ

′

and as a consequence,

using the relation 1 we immediately get λ

′

≤ ρβ.

2. else, we have λ

′

≤





+ 1



′

≤





+ 1



ρτ

and using the relation 1 we get λ

′

≤



⌊

⌋



ρβ

This completes the proof.

Note that the strategy we have used is less restric-

tive than the previous quoted strategy since we treat a

vertex in the pre-determined solution as soon as it be-

comes old, even if it does not have an old neighbour.

However the more precise strategy that would be used

in practice does not allow to improve the worst case

approximation ratio.

Corollary 2. The problem ZFTS with a single vege-

tation type of old-vegetationthreshold o and with con-

stant treatment costs has:

1. a periodic polynomial time algorithm (resp.,

asymptotically optimal) in bipartite graphs if T ≡

0(modo) (resp., if T ≫ 0).

2. a periodic polynomial time approximation scheme

(resp., asymptotic PTAS) in planar graphs if T ≡

0(modo) (resp. if T ≫ 0).

We now present our main result.

Proposition 4. Let C be a hereditary class of graphs

for which MIN WEIGHTED VERTEX COVER can be

approximated in polynomial time within the ratio ρ.

The problem ZFTS on the class C with constant treat-

ment costs, multi-vegetation type (we denote respec-

tively by o

and a

the old-vegetation threshold and the

initial vegetation age in cell i) can be approximated in

polynomial time within

T + a

T − ℓ

ρ ≤ (1+

T − o

)ρ

on a time period T ≥ ℓ, where a = max

i∈I

and ℓ =

max

i∈I

− a

Proof. (sketch) Consider an instance of ZFTS with

= (I,E) the associated graph. We denote by c

the

treatment cost of vertex (cell) i and deﬁne the follow-

ing weight system on I: ∀i ∈ V, ω

. We denote by

) the minimum weight of a vertex cover in G

Let I

′

⊂ I be a ρ-approximated weighted vertex cover

for this weight system:

ω(I

′

) =

∑

i∈I

′

≤ ρτ

) (2)

We denote by S

′

the ZFTS solution constructed from

′

and by S

∗

an optimal solution. According to Propo-

sition 1 we assume that only old vegetation cells are

treated in this optimal solution. The cost of S

′

and

∗

are denoted by c(S

′

) and c(S

∗

), respectively. We

have c(S

∗

) ≤ c(S

′

By deﬁnition of the solution S

′

a vertex i ∈ I

′

treated at most

T+a

times and consequently, using

Relation 2:

c(S

′

) ≤

∑

i∈I

′



T + a



≤ (T + a)ρτ

) (3)

For each time period t = 1, ...,T we deﬁne the set

= {i ∈ I, a

+ t − 1 ≥ o

} that can be seen as the

set of cells that would have an old vegetation at the

beginning of period t if no treatment was applied at

all. Sets I

,t = 1,... ,T, are nested (∀t < T,I

⊆ I

t+1

)

and for t ≥ ℓ + 1, we have I

= I. For t = 1,. .., T, we

denote by G

= G

Consider now the optimal solution S

∗

. We asso-

ciate to S

∗

as subset U of U = {(i,t),t = 1,.. .,T, i ∈

} as follows: whenever a vertex i ∈ I is treated dur-

ing the periodt, we add in U the vertices (i,t +k), k =

0,. .., min{o

− 1,T − t}, and assign to each one the

weight

. Roughly speaking this corresponds to

spread the treatment cost over the whole treatment

and regrowing period. For each period t we deﬁne

= {i,(i,t) ∈ U}. We have:

c(V

∗

) ≥

∑

t=1

ω(U

) (4)

Since the sets I

,t = 1, ... ,T, are nested and only

vertices in I

are treated during period t, we have

∀t = 1,. ..,T,U

⊂ I

. Now, it is easy to see that

for every t = 1,...,T, U

is a vertex cover of G

Since the weights of vertices in U

correspond to the

weights in G

, we deduce from Relation 4:

c(V

∗

) ≥

∑

i=1

) ≥ (T − ℓ)τ

) (5)

Relations 3 and 5 immediately conclude the proof:

c(V

′

) ≤

T + a

T − ℓ

c(V

∗

) (6)

We emphasise some interesting particular cases:

A Multi-period Vertex Cover Problem and Application to Fuel Management

Remark 2. Note that the result still holds if we re-

place a by a

′

= max

i∈V

′

Then, if a

′

= 0, meaning that the computed vertex

cover only includes cells with new vegetation, the re-

lated ratio will be

T − ℓ

ρ ≤

T − o

with o = max

i∈I

If, on the contrary, all cells are old at the beginning

of the process (ℓ = 0), the ratio will be

T + a

ρ =

T + o

Remark 3. For large T, in particular if T ≥

, ε > 0,

we get the ratio (1+

2ε

1−ε

)ρ

Corollary 3. The problem ZFTS with constant treat-

ment costs, multi-vegetation type has:

1. a periodic asymptotical optimal polynomial time

algorithm in bipartite graphs and in chordal

graphs if T ≫ 0.

2. a periodic asymptotic polynomial time approxi-

mation scheme in planar graphs if T ≫ 0.

4 DISCUSSION AND FUTURE

DIRECTIONS

In this work we have studied the problem ZFTS that

can be seen as a multi-period vertex cover problem:

how to remove a set of vertices of minimum weight

so as to eliminate all edges with the particularity that

a removed vertex reappears after some time periods

(regrowing process). We have shown how to build

efﬁcient solutions by using efﬁcient algorithms for

MIN WEIGHTED VERTEX COVER. This example il-

lustrates the potential advantage of such a graph ap-

proach:

1. It is suitable to understand links with well known

problems - MIN WEIGHTED VERTEX COVER in

our case - that has been extensively studied. The

proposed reduction allows even to transform good

heuristics for the latter into heuristics for the for-

mer. Since the reduction preserves good approxi-

mated values we can expect the resulting heuristic

to be good.

2. It is suitable to derive some properties and algo-

rithms in speciﬁc graph classes from the known

properties of MIN WEIGHTED VERTEX COVER

in these classes.

3. It also allows to better understand the hardness of

the problem in some particular cases using their

structural properties.

The continuation of this work includes numerical tests

on real instances and the analysis of the solutions with

ﬁnal users in Victoria State.

Our results emphasised the notion of periodic so-

lutions that has some advantages. Once the original

vertex cover is computed, the treatment schedule can

be computed in linear time and has a better behaviour

for long time periods contrary to other methods. It is

also very simple to implement in practice. This raises

mainly two research questions. The ﬁrst one is to de-

termine the best periodic solution and we conjecture

that our approach is asymptotically optimal. On the

contrary if periodicity is not desirable since the same

cells are always treated, it motivates an opposite no-

tion for solutions that spread uniformly the treatments

over all cells for long time periods.

The problem ZFTS is a very simpliﬁed abstrac-

tion of the real problem. We plan to generalise the

results obtained here to some generalisations of ver-

tex cover that are more relevant for fuel management.

The notion of fragmenting a graph (Edwards and Farr,

2001) is particularly interesting in this context. It cor-

responds to remove as few vertices as possible so that

the components of the resulting graph do not have

more than k vertices, for a ﬁxed k. The case k = 1

corresponds to the notion of vertex cover.

A second research direction is to investigate the

problem with ﬁxed budgets at each time period

(BFTS). Both complexity results and approximation

results similar to the ones presented here would be

interesting. So far, using a general though process

described in (Demange and Ekim, 2013), we have al-

ready proved it is NP-hard even in grid graphs, with

unitary treatment costs, single vegetation type and

T < o. This hardness result emphasises an important

difference with the problem addressed in this paper

and, in some way, justiﬁes a posteriori the MIP ap-

proach proposed in (Minas et al., 2014). Our proof

however still does not prove whether the same prob-

lem remains hard with a constant number of periods

and constant old-vegetation threshold. Such require-

ments are natural from the application point of view.

We conjecture that this case is still hard in grids.

ACKNOWLEDGEMENTS

Cerasela Tanasescu was supported by the Grant Prob-

ability of Fire Ignition and Escalation, Schedule 7 -

Bushﬁre and Natural Hazard CRC, for the Depart-

ment of Environment, Land, Water and Planning of

Victoria State, Australia. This support is greatly ac-

knowledged.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

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A Multi-period Vertex Cover Problem and Application to Fuel Management