Integer Linear Programming Models for the Problem of Covering a Polygonal Region by Rectangles Текст научной статьи по специальности «Математика»

Integer Linear Programming Models for the Problem of Covering a Polygonal Region by Rectangles

G. Scheithauer, Yu. Stoyan and T. Romanova

Abstract— The aim of the paper is to develop integer linear programming (ILP) models for the problem of covering a polygonal region by rectangles. We formulate a Beasley-type model in which the number of variables depends on the size parameters. Another ILP model is proposed which has O(n2 max{m,n}) variables where m is the number of edges of the target set and n is the number of given rectangles. In particular we consider the case where the polygonal region is convex. Extensions are also discussed where we allow the polygonal region to be a union of a finite number of convex subsets.

Index terms — Covering, Integer Linear Programming, Mathematical Modelling, Optimization

Introduction and Problem Formulation

In this paper the problem of covering a polygonal target set Q by a finite number of given rectangles is considered. The main aim is to formulate integer linear programming or optimization models. Rotation of rectangles is not allowed. In particular, the target set is assumed to be an arbitrary convex polygon. Since covering with axes-parallel rectangles is considered the target set can also be assumed to be an orthogonally convex rectilinear polygon.

The decision version of problem CPR (Covering a Polygonal set with Rectangles) asks whether there exists a covering or not. It is known to be NP-complete since the decision version of the Bin Packing Problem ([6]) can be reduced to the CPR problem (cf. [4]). Notice, in difference to e.g. [4] where a finite number of points has to be covered, we consider the covering of an infinite point set. Therefore, the verification that a certain configuration of the rectangles forms a cover of the target set cannot be done by inspecting a finite number of points, another technique is needed.

A solution approach based on a so-called Г-function is proposed in [9]. The Г-function of a certain configuration of all given rectangles attains a non-negative value if and only if this configuration forms a feasible covering of the target set. Based on an enumeration scheme, instances for which no cover exists can

require a lot of computational effort to prove that circumstance.

In [8] one-dimensional bar relaxations for the CPR problem are proposed to be used as necessary conditions for existence of coverings.

Covering problems are of interest in many fields of application. There are many relations between covering and packing or cutting problems. For an annotated survey on Cutting and Packing we refer to [5]. Covering problems arise naturally in a variety of applications. For a comprehensive overview we refer to [4].

As an example, query optimization in spatial databases is a source of covering problems. In this setting a query may correspond to a geometric region and be phrased in a generic form using geometric parameters. Given a set of existing geometric, parametrized, query regions and a set of points or regions, we might want to ask if there are values of the parameters that allow the query regions to cover the set of points or even regions. Another field of application (also mentioned in [4]) is shape recognition for robotics, graphics or image processing applications. In these cases it is sometimes useful to represent a shape as a collection of parts. However, given a collection of parts and a shape, it can be difficult to determine if the shape can be described by that collection of parts. If the goal is to obtain an outer approximation of the shape using the parts, then this can be posed as a covering problem.

Besides the decision problem whether a covering exists or not, related problems can be of interest. For instance, one can ask for a minimum number of (identical) rectangles needed to form a cover for the target region. Or, if there exist several covers one can look for a best cover where best means that some objective function is regarded.

In computational geometry, decomposition of a polygon is of high interest involving partitioning and covering problems. For details and further literature we refer to [10, 1, 7].

In the approach proposed in [9] to solve the problem of covering a compact polygonal region with a finite family of rectangles, the choice of a suitable starting configuration is in particular an essential aspect.

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Therefore, and since the CPR problem is hard to solve, enumeration algorithms like branch-and-bound have to be used in general in an exact solution approach. Another possibility to attack the CPR problem consists in formulating integer linear programming models and to solve them.

The aim of this contribution is to develop models and to discuss advantages and disadvantages with respect to the construction of exact solution approaches. Without loss of generality, we assume that all input-data are integers.

The paper is organized as follows. In the rest of this section we give the input-parameter and state the problem considered in the paper. In the next section a Beasley-type model of the covering problem is discussed. Then, in section 3, a basic model is developed. In section 4, we present a corresponding ILP model. Some extensions and alternative formulations are discussed in section 5. Finally, some conclusions follow.

Within this paper we consider the following optimization problem. We assume that target set H is a convex polygon. Consequently, we can use the representation

^ = {(x,y): gj(x,g) < 0, j € In}

= conv{(Xj, Yj) : j € In},

In = {l,...,m}.

All the linear functions gj, j € In are assumed to be necessary for the representation of П. Hence, the number of vertexes (corners) (Xj,Yj) of H coincides with the (minimum) number of functions gj. In order to cover the given target set H the following rectangles are available:

Ri = {(x, y) : -a < x < a, -bi < y < bi}, i € In = {1,. . .,n}

where 2ai is the length and 2bi the width of rectangle Ri. We assign to each rectangle its (positive) value ci, i € In, e. g. ci = aibi, and we denote the placement parameters (translation vectors) by ui = (xi,yi), i € In. Hence,

Ri(xi,yi) = {(x,y) : xi - ai < x < xi + ai, yi - bi < y < yi + bi}

represents the translated rectangle Ri. Then the problem CPR under consideration is:

Find a subset I* of In of the rectangles and corresponding placement parameters ui, i €

I * such that Uiei* Ri(ui) forms a cover of H and its total valuation Y}iei* c is minimal.

We always assume that problem CPR has a solution. This can be done by adding some sufficiently large rectangle R0 (i.e. the minimum H enclosing rectangle) with c0 > Y}iei ci. Consequently, if the optimal value of the CPR problem is not smaller than c0 then the original problem has no covering.

A Beasley-type Model:

ILP Model 1

Beasley [3, 2] proposed an integer linear programming (ILP) model with 0/1-variables for the twodimensional rectangle packing problem. There, the 0/1-variables xipq are used to describe the placement of the reference point (lower left corner) of rectangle Ri at position (p, q).

For an ILP model of the CPR problem we can also use these xipq-variables. Let

Xe := maxi£i^ Xi, Xw := miniei^ Xi,

Yn := maxiei^ Yi, Ys := miniEin Yi

denote the extremal coordinates of П. To visualize different directions we use N for the north-direction, E, S and W for east, south and west direction, respectively. Clearly, directions N, E, S and W can be understood as top, right, bottom or left direction, respectively. Without loss of generality, we may assume XW = 0 and YS = 0. Consequently, target set H is completely contained within a rectangle R0 of dimensions L0 = XE - XW and Wo = YN - YS. Since all input-data are assumed to be integral we can restrict the coordinates of all allocation point to be integral too. Let Li = {0,..., Lo-li}, Wi = {0,..., Wo -wi}, i € In. If rectangle Ri is placed with its lower left corner at point (p, q) with p € Li and q € Wi then it covers the point set

Ri (p, q) = {(x, y): p < x<x + li,

< y < q + wi}

where li = 2ai and wi = 2bi, i € In. Notice, here the upper and right boundary are assumed not to be in Ri (p, q) Let denote the minimal orthogonally convex rectilinear polygon with only integral corner points enveloping П. Since we are looking for a covering of H with only rectangles we have, the convex target set is covered if is covered. Notice, using an appropriate scale the approximation of by is tight enough to obtain an exact solution. We assume that is a closed point set. According to definition (1) not all integer lattice points in have to be covered namely those lying on the NE or SE boundary,

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accept the leftmost point lying on the SE boundary. Let denote the integer lattice points in which have to be covered by the rectangles.

Then the optimization problem can be formulated as follows:

Beasley-type model: ILP model 1

^2 ^2 ci xipq ^ min (2)

ie1n (p,q)en

subject to

s t

E E E xipq E 1, v (s,t) e Q, (3)

i£ln p=s(i) q=t(i)

^ ^ ^ ^ xipq < 1? V i e /n , (4)

p^Li q£Wi

xipq e {0, 1}, p e Li, q e Wi, * e in (5) where s(i) = max{0, s — li + 1}, t(i) = max{0,t —

Wi + 1}.

This Beasley-type model has some drawbacks. First of all, the number of 0/1-variables depends on the dimensions of the target set. Moreover, the choice of an appropriate scale can further increase this number. Furthermore, the continuous (or linear programming, LP) relaxation (restrictions xipq e {0,1} are replaced by xipq e [0,1]) is weak. It yields feasibility if the total area of the rectangles is not smaller than the area of Ll. This makes it difficult to solve the (integer) Beasley-type model.

Notice, the number of 0/1-variables can be somewhat reduced regarding the shape of Ll instead of the enveloping rectangle L x W. Another opportunity results from the principle of normalized patterns or raster points.

Basic Model

In the following, an attempt is made to model the CPR problem using a polynomial number of variables and restrictions.

Notations

The following sets of directions will be used:

D := {N,E,S,W},

Dt

{N, S}, if t e{E,W}, {E, W}, if t e{N,S}.

t e D.

Since the polygonal region Q is assumed to be convex the following denotations and abbreviations are meaningful. We identify Q defining functions visible from different directions as follows:

TNW .

1n :

{j € In : 5-j2~ < 0 dx

5jh_

Sy

> 0},

mNw := |,

%E : !./ ^ h : J; <*• %>0},

. I tNE I mNE := |1fi t

TSW 1П :"

Sg

Sy

Sgj

Ь e In : wf < o, -2 < 0}

Sx

Sy

msw := |/SW |,

Sgj

Sgj

/.v : !./ ^ /о; ■ <». ;;. 0}

Sy

mSE := |/nE |,

Moreover, let

mrs = msr Vr e Ds, s e D.

In order to characterize the relation between two rectangles we define the constants

aij

bij

1, ai > a j ,

0, ai < aj,

1, bi > bj,

0, bi < bj,

i,j e tn, * = j. (6)

These constants are used to combine different cases, and therefore, to shorten the description.

For a given 0/1-vector a = (ai,..., an) and a given vector u = (u1,..., un) e 1R2n of placement parameters ui e 1R2, i e /n let

P(u, a) := У Ri(ui),

i:ai= 1

H(u, a) := 1R2 \ int(P(u, a)).

Polygonal set P(u, a) represents the point set covered by the chosen rectangles whereas H(u, a) denotes the closure of the complement of P(u, a). For technical purposes let

dt

4, if t e{N,W}, 2, if t e {S, E},

t e D.

(7)

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Placement Parameters

As already introduced above, we denote the placement parameters (translation vectors) to be found

by ui (xi , yi)7 i G J-n • Hence,

Ri(xi,Vi) = {(x,y) :

Xi - ai < x < xi + ai,yi - bi < y < yi + bi}

represents the region covered by the translated rectangle Ri.

If more then one rectangle is used to cover Q not all of these restrictions have to be fulfilled depending on the relative positions of the rectangles. Here we assume that no coverings are of interest where some covering rectangle Ri(ui) is completely contained within another covering rectangle Rj (uj). Such a configuration cannot be optimal because of assumption ci > 0, i G In.

Relative Position Variables

Selection Variables

If rectangles Ri and Rj are both used to cover Q, i. e. if ai = aj = 1, 0/1-variables

In order to indicate those rectangles Ri, i G In which are used to cover the polygonal region Q we define 0/1-variables ai according to

ai

1, Ri (ui) is used for the cover,

0, Ri (ui) is not used for the cover.

Let Ri be given in the form

Ri(xi,yi) = {(x,y): /T(x,y) > ^ r G D} i G Я,

where

fiN(x,y) := yi + bi - y, fiE(x, y) := xi + ai - x,

fiS(x,y) := y + bi - yi,

fW(x, y) := x + ai - xi.

In case that Q fits within a single rectangle Ri(ui) and in related situations the following conditions on the translation vector ui = (xi, yi) are probably nontrivial since they form the boundary of H(u, a):

ttj and , i,j G In i = A r G {1,..., 5}

are introduced to characterize the relative position of the two rectangles to each other. In doing so we consider five different situations in horizontal and ialso n vertical direction (cf. Fig. ??). Consequently, using both rectangles Ri and Rj that means we have

r =1

r = 3

r=5

r = 2 r = 4

i

Figure 1: labelfig-1 Different relative horizontal positions

5

r = 1

5

1 and Фij

r = 1

1.

(9)

fiN(xi, yi) := yi + bi - Yn > 0,

fiE(xi, yi) := xi + ai - Xe > 0,

(xi, yi) := Ys + bi - yi > 0,

fW (xi, yi) := Xw + ai - xi > 0.

These inequalities should hold only if rectangle Ri is used. Therefore we modify them by adding some term depending on ai:

Я(xi,yi) + M(1 - ai) > 0, (8)

r G D, i G In, У f

where M is a sufficient large number, e.g. M = max{Lo,Wo}. For every i G In these four restrictions (in a somewhat modified manner) have to be added to an ILP model.

With other words, equations (9) should be fulfilled if and only if both rectangles Ri and Rj are used to cover Q in order to get a unique description of the different configurations/interactions of the two rectangles which are as follows.

We define = 1 if and only if Rj (uj) is completely left to Ri (ui) which leads to the inequality

xj + aj < xi - ai + M (1 - ф1)

(non-trivial if ф1 = 1)

where M is a sufficient large number, e. g. M = Xe + ai + aj. The case when Rj (uj) is completely right to Ri(ui) is characterized by ф5 = 1. If int(Ri(ui)) П Rj (uj) = 0 then we have three subcases, namely r = 2: xj - aj < xi - ai < xj + aj,

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r = 3: either Xj — aj < Xi — ai < x3 + ai < Xj + aj or xi — ai < Xj — aj < Xj + aj < xi + ai, r = 4: Xj — aj < Xi + ai < Xj + aj.

Obviously, only one of two cases with r = 3 can occur according to the definition of aij and bj in formula (6). Altogether, we have the following restrictions (describing the relative horizontal position of R3 and Rj) which are non-trivial if some ф-variable has value one:

(Аф) :=

Xi — ai — Xj — aj + M(1 — ф1) > 0, (Аф) :=

Xj + aj — Xi + ai + M(1 — 4>lj) > 0, (Аф) :=

xi — ai — Xj + aj + M(1 — ф2) > 0, h4j(x, ф) := (aij — aji)(xj — aj — xH + ai)

+M(1 — фА) > 0,

h5j(x, ф) := (aij — aji)(xi + ai — Xj — aj) +M(1 — фЗ ) > 0,

hj (Аф) :=

Xj + aj — Xi — ai + M(1 — ф4) > 0,

hlj(x,ф) :=

Xi + ai — Xj + aj + M(1 — ф4 ) > 0, (Аф) :=

Xj — aj — Xi — ai + M(1 — ф5 — > 0.

(10)

In a similar way the ^-variables are defined to characterize the relative vertical position of the two rectangles:

(У,^) :=

yi — bi — yj — bj + M(1 — ^ij) > 0,

(y,^) :=

yj + bj — yi + bi + M(1 — ^i2j) > 0, (y,^) :=

yi — bi — yj + bj + M(1 — ^2j) > 0,

(y, ^ := (bij — bji)(yj — bj — yi + bi) _ +M(1 — ) > 0,

hij (y, ^ := (bij — bji)(yi + bi — yj — bj) _ +M(1 — ) > 0,

hj (y,^) :=

yj + bj — yi — bi + M(1 — ) > 0,

(y,^) :=

yi + bi — yj + bj + M(1 — ) > 0,

(y,^) :=

yj — bj — yi — bi + M(1 — ^i5j) > 0.

Because of definition we have

фа = ф-г , V*j = vjr,

r e {1,...,5}, у j e /„, * = j.

(11)

Notice, every combination of ф- and ^-values which fulfills conditions (9) defines a subset of R4 of possible placement parameters u3 = (xi,yi) and Uj =

(xj ,yj).

Overlap Characterizing Variables

If two rectangles R3 and Rj are used to cover U then two essential different situations have to be considered: is the intersection of the two rectangles empty or not. For that reason 0/1-variables eij can be introduced which gets value one if and only if the intersection of Ri(ui) and Rj (uj) is non-empty.

It is obvious, the в-variables are dependent on the ф-and ^-variables. It holds

44

Aj =J2 фА ^j , * j e ^ * = j.

r=2 r=2

Note that although in cases r = 1 or r = 5 some ” touching” is allowed, these situations cannot lead to a real overlap.

Inner Corners

Every pair R3 and Rj of used rectangles with nonempty intersection (i.e. eij = 1) determines some points, called inner corners which probably define a cone usable in the representation of H(u, a)

In order to obtain a description of H(u, a) we consider all eight possibilities of inner corners which can arise. By means of 0/1-variables we identify in dependence of the ф- and ^-variables these inner corners which are formed with respect to this configuration. Using further 0/1-variables we characterize those inner corners which form cones for the description of H(u, a).

In total, there are eight different types of inner corners. We define

Ej s e D, t e Ds, ij e /„, * = j,

as follows:

EjW := (xj —aj^i+Ah

Eij := (xj —aj, yi —bi), := (Xi+ai,yj+bj), EjW := (xi—ai,yj+bj),

Eij : (xj + aj, уЗ+ЬА

Eij : (xj +aj, yi bi),

Eij : (xi +ai, yj —bj),

Eij := (xi — a^ yj —bj).

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In order to identify those inner corners which are induced by Rj, Rj, and the ф- and ф-variables we define 0/1-variables e|j as follows.

1.

3.

2.

The variable ejW which corresponds to point ENW = (xj — aj, yj + bj) has to be one if and only if

aj = aj = 1, EjW G Rj(ui) П Rj(uj).

In this case, the set

{(x, y) : x < xj — aj, y > yj + bj}

forms a cone in North-West direction. There are several situations where point ENjW forms an inner corner, namely:

1 ф% = 1 1

2. ф! = 1, ф2 = 1, bjj = 1,

3. фу,' = 1, aij = 1 Фу = 1,

4. ф3 = 1, aij = ^ ф3- = 1, bjj = 1.

These four situations are depicted in Ffigure 2.

Figure 3: Interaction between ENjW and target region Q

1. Either Q C {(x, y) : y < yj + bj}, or

2. Q C {(x,y) : x > xj — aj}, or

3. gl(ENNW) > 0 for at least one l G I^W.

This can be modelled using the e-variables as follows::

max {yj + bj YN, xj + aj,

max{gi(EjW) : l G I^W}} (12)

+M(1 — ejW) > 0.

For every pair of rectangles and every kind of potential inner corner such an inequality has to be considered, i. e. approximately 4n2 restrictions. But this inequality should be redundant if the point ENjW is covered by a third rectangle.

2. j

i

CO j 4.

j

i i

1. j

i

Figure 2: Situations where inner corner ENjW can arise

Linear equations or inequalities are needed which ensure that eNjW becomes one in exactly these four cases. It holds for i = j:

Similar conditions hold for the other types of inner corners.

If only two rectangles are needed to cover Q, i. e. 'Thiel aj = 2 and pjj = 1, then the resulting two or four inner corners determine cones usable in the description of H(u, a).

If more than two rectangles are needed to cover Q then some of the resulting inner corners can be covered by another third rectangle, and hence, are not useful for the description of H(u, a).

ejW =(ф4 + ajj ф% )(ф4' + bjjф3 ^

j = (ф2,' + ajj ф% )(Фу + bjjф3), efjW =(ф4? + ajjф3 )(ф2 + bjj^3j),

j =(ф2? + ajj ф%)(ф2' + bjj ф%). Furthermore, for i = j we have

Active Inner Corners

In case of more than two rectangles used to cover Q it may happen that some inner corner, e.g. ENjW, is covered by a third rectangle Rk so that ENjW does not form a part of the complement of the union of covering rectangles as drawn in Fig. 4. In order to charac-

rs

e

j

sr

e

j

r G D, s G Dr.

If we suppose that ejW = 1 and that ENW g H (u, a) then at least one of the following inequalities, illustrated in Fig. 3,

must be fulfilled for the convex region Q to ensure non-overlapping:

j

k

i

Figure 4: Inner corner ENjW is not active

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terize such situation we introduce a 0/1-variable pjW which has to be one if and only if EvVW is not covered by another rectangle. More precisely, pjW = 1 should hold true if and only if there does not exist any k £ In \ {i,j} with

restriction (14) can be formulated as

A jo Я (Xi, yi) + £ t= j grta(E%)

+xij,mrs+i f?(xj ,yj) > 0 r £ D, s £ Dr, i,j £ In, i = j,

(16)

_ ,o where qt s are the functions corresponding to ЯЯ

1. Ф% + akj 4>\j = 1, and yt n

2 Фгк + bfci^fc = 1 Summation

Hence, we define

pNjW = 1 ^

£*=i,j (V + «fcj V- )(V + bki V) = °-

In case of pjW = 1 the bounding constraints in North-direction for Ri and in West-direction for Rj (according to (1)) must be removed. This can be achieved as follows:

fv(xi,yi) + M(1 - a.i)

+ M£j:j= j > 0, (13)

fjV(xj,yj) + M(1 - aj)

+ M £j:j= j > °.

If for all relations between the а-, ф-, ф-, p-, mad A- variables linear inequalities or equalities can be found then an ILP model for the problem of covering a convex polygon by rectangles can be obtained. As an intermediate result we have the following formulation of the CPR problem:

Compute placement parameters xi, yi, i £ In and 0/1-variables ai, i £ In, фР, фР p £ {1,..., 5}, pr?, and A jt, r £ D, s £ Dr, t £ {0,..., mrS + 1}, i, j £ In, i = j such that

Ci ai ^ min, (17)

iein subject to

Instead of the two removed restrictions, another condition has to become relevant which guaranties that H is either in one of the two half-planes determined by ff or fjW or that EjW £ int(Q):

max{fvv(xi, yi),fjW(xj, yj), max{gi(EjW) : l £ IfW}}

+M(1 - pjW) > 0.

Similar restrictions have to be introduced for the other types of active inner corners:

5 5

I>t = а r = 1 iaj, ХФА: r = 1 = ai a j, VV j (18)

Л1 Я я -ST 0, t £ {1, . . . , 8}, Vi = j, (19)

IV 0, t £ {1, . . . , 8}, Vi = j, (20)

V = (Ф j + aij Ф3 )(£ijr + bji Ф3), (21)

r £ D, s £ Dr, i = j.

prs psr i = j , r £ D, s £ D r. (22)

prS < £rS ij ij Vr £ D, s £ Dr, i = j, (23)

max{fi (xi ,Уi), fj (xj ,yj), max{g;(Ej): l £ яя}}

+M(1 - pj > 0, r £ D, s £ Dr, i, j £ In, i = j.

pj = 4s ^ £k=i,j Vj +«fcj ф|^- )(фЯ + bfci£ik) = 0, Vr £ D, s £ Dr, i = j,

Я (xi, yi) + M(1 - ai)

(24)

Constraint Selection Variables

In case of an active inner corner EJj, i.e. with pj = 1, 0/1-variables Aj), t = 0,..., mrs + 1, are needed to identify a single constraint which ensures the nonoverlapping similar to (12). Because of

mrs + 1

XI Ajt = pry Vr,s,i,j. (15)

t=0

+m£ j=v:seDr prr > 0, (25)

r £ D, i=j

v^mrs | + 1 a rs 2^t=0 Aijt = pj, Vr, s, i,j. (26)

A jo Я (xi, yi) + £V Aijt grs(EijS)

+ ArjS,mrs + 1fI(xj ,yj ) > 0, (27)

r £ D, s £ Dr, i, j £ In, i = j.

In the following we are going to develop a corresponding ILP model.

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ILP Model 2

In order to get an ILP formulation with polynomial number of variables and constraints, all conditions in basic model (17) - (27) have to be formulated as linear restrictions.

fa < E ФГу, к, < E к*,

r=2 r=2

л, > E ф,+E *<i -1-

Restricting the Placement Parameters

Relations Between e-, ф- and ф-Variables

In order to ensure that Ri does not overlap D if Ri is not used, i. e. if a = 0, we define the following inequalities:

ai < xi < (XE + 2ai)ai ai?

bi < yi < (YN + 2bi)ai bi7 i G In -

(28)

Relations Between а-, ф- and ф-Variables

Lower bounds for ф- and ^-variables:

о < ФГ,, о < Ф1,,

Vi,j G In, i = j r = 1,---, 5-

ФГ, = фб-г, Ф, = Ф|Г,

r G {1, - - -, 5} i,j G In, i = j-Realization of the logical AND:

E^=1 фг, < ai, E^=i фг, < ,

E^ = 1 ф, > ai + - 1, i,j G In, г = j-

5

Er=1 ФГ, < “G Er=1 ФГ, < ,

r

Naturally, we have

о < esI, Vs G D, t G Ds, i,j G In, i = j, (33)

Realization of the logical AND:

4S < ф, + a, ф%, ers < ф, + ^i, ,

r G D, s G Dr, i = j,

£rS > ф, + ai,ф3 + ф, + bJi^ii - 1,

(29)

r G D, s G Dr, i = j, Because of definition:

4S = eSr Vi,j,r,s-

(34)

(35)

(36)

Symmetry conditions (source of reducing the number of variables):

(30)

(31)

5 (32)

Er = 1 ф, > ai + a, - 1, i, j G In, i = j-

Moreover, inequalities (19) and (20) have to be fulfilled.

Relations Between в-, ф- and ф-

Variables

According to basic model (17) - (27) where no в-variables are used the following inequalities are not needed. On the other hand, if it is intended to exploit в-variables then these inequalities yield the relations between в-, ф- and ф-variables.

Realization of the logical AND:

0 < вг,, 'Vi,j G In, i = j,

Relations Between e- and p-Variables

By definition we have

0 < p, < , Vr G D, s G Dr, i = j, (37)

In order to get an ILP formulation for (24), i. e. for

PrS = £rS ^ pi, bi, W

Efe=i,, (фк, + ак,фк, )(фТЙ + bkiфф ) = 0,

Vr G D, s G Dr, i = j, we introduce 0/1-variables р^к by

Рфк = 1 ^ (фк, + ак, фк, )(Фг£ + bki ф3д, ) = 1, Vr G D, s G Dr, i = j, k G In \ {i, j}-

This can be modelled as follows:

0 < Рфк < фк, + ак, фк,,

рфк < + 6Ьф3к,

Р, > фк, + ак, фк, + ф^k( + ^i ф3к - 1,

PrS <

фк, + Ф&

Vr G D, s G Dr, i = j, k G In \{i,j}-Now we have the condition

P, = 1 ^ E P, =0

k=i,,

(38)

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which is modelled as

ers __ nrs < prs

ij rij — A^k=i,j rijk'>

(n - 2)(ej - pj >£k=i,j prjk, (39)

Vr G D, s G Dr, i = j.

In the last condition it is assumed that at least two rectangles are available to build a cover, i.e. n > 2.

Moreover, we have

prjs, i = j G In, r G D, s G Dr,

O(n2m) constraint selection variables Aij, i = j G

In, r G D, s G Dr, t G Iq , and

O(n3) variables prjsk, i = j = k G In, r G D, s G Dr. Hence, the total number of variables is proportional to n2 • max{n, m}.

The number of constraints has the same order of magnitude.

pij

pj, Vr,s,i,j.

(40)

and, last but not least,

fir(xi,yi) + M(1 - Oj)

+M Ej=i EseDr pj > 0,

r G D, i = j)

(41)

mrs | + 1

E\rs Aijt

t=0

PijV Vr, s,i,j.

(42)

A

rs

ij0

iT(xi ,yi) + E

mst

t=1

Aijt 9™(Щ£)

+Aij,mrs +1 fj (xj ,yj ) > 0,

r G D, s G Dr, i, j G In, i = j.

(43)

Feasibility

In this model, objective function (17) and restrictions (28) - (43), it is possible that no covering exists. For computational purposes it may be better to have the existence of feasible solutions in the optimization problem.

There are several possibilities to guarantee feasible solutions, e.g. by adding artificial rectangles with sufficient high costs, or by weakening some of the restrictions similar to [9].

Extensions

Here we propose directions of further research.

ILP Model 3: Non-convex Region П

If Q is the union of a finite number of convex polygons, i. e.

Q = |^J Qq where Qq is convex for all q, (44)

q

then, in analogy to [9], for every subset Qq a complete set of A-variables has to be introduced. The placement parameters xi and yi and the а-, ф- and ф-variables are maintained.

An Alternative ILP Model

Another way of modelling is as follows. Given the ф- and ф-values we can obtain the e-values. For a certain subset Qq we derive the p-values regarding only these rectangles which are used to cover Qq. For every Qq a set of a-, p- and A-variables is needed.

The Linear Model

Besides the linear objective function (17) and the linear restrictions (19), (20), (22), (23), (25) - (27) of the basic model now we have to add all linear constraints (28) - (43) to get a linear formulation of the CPR problem with continuous and binary variables. In this formulation we have 2n continuous variables xi, yi, i G In, n rectangle selection variables ai, i G In,

10n(n — 1) relative position variables ф- and ф-, i = j G r G {1,..., 5}

8n(n — 1) inner corner identification variables ej i = j G In, r G D, s G Dr,

8n(n - 1) active inner corner identification variables

Restricting the Placement Parameters

For every q G Q 0/1-variables aiq are defined. Then, a rectangle Ri, i G In, is used to cover Q if for at least one q G Q aiq = 1 holds true. In order to ensure that Ri does not overlap Q = UneQ Qq if R is not used for any subset of Q, i. e. if EqeQ °-щ = 0, we define the following inequalities:

ai < xi < (XE + 2ai) ^0qGQ aiq

-bi < yi < (Yn + 2bi) EqeQ aiq - bi, (45)

i G In.

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Conclusions

Two ILP models have been developed for problem CPR. In the first one, the number of variables and constraints is dependent on the size of the target region; in the second model it is polynomially bounded but even large. Investigations to reduce this number as well as numerical experiments are needed. Moreover, alternative formulations can possibly help.

References

[1] V. S. Anil Kumar and H. Ramesh. Covering rectlinear polygons with axis-parallel rectangles. SIAM J. Comput. vol. 32, no. 6, 2003, pp. 1509-1541.

[2] J. E. Beasley. Bounds for two-dimensional cutting. J. Oper. Res. Soc., vol. 36, no. 1, pp. 71-74, 1985.

[3] J. E. Beasley. An exact two-dimensional non-guillotine cutting tree search procedure. Oper. Res., vol. 33, no. 1, pp. 49-64, 1985.

[4] K. Daniels and R. Inkulu. An incremental algorithm for translational polygon covering. Technical Report 2001-001, University of Massachusetts at Lowell Computer Science, 2001.

[5] H. Dyckhoff, G. Scheithauer, and J. Terno. Cutting and packing. In M. Dell?Amico, F. Maffioli, and S. Martello, editors, Annotated Bibliographies in Combinatorial Optimization, chapter 22, pp. 393-412. John Wiley & Sons, Chichester, 1997.

[6] M. R. Garey and D. S. Johnson. Computers and Intractability — A Guide to the Theory of NP- Completeness. Freeman, San Francisco, 1979.

[7] A. Lingas, A. Wasylewicz, and P. Zylinski. Note on covering monotone orthogonal polygons with star-shaped polygons. Information Processing Letters vol. 104, no. 6, pp. 220-227, 2007.

[8] G. Scheithauer. One-dimensional relaxations for the problem of covering a polygonal region by rectangles. Preprint MATH-NM-04-2009, Technische Universit”at Dresden, 2009.

[9] G. Scheithauer, Yu. Stoyan, T. Romanova, and A. Krivulya. Covering a polygonal region by rectangles. Computational Optimization and Applications, Springer, online: http://www.citeulike.org/user/ima/article/4575123

[10] San-Yuan Wu and Sartaj Sahni. Covering rectlinear polygons by rectangles. Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on. 01/05/1990; 9(4):377-388.

G. Scheithauer 1983 Doctor rer. nat. at Technische Universit”at Dresden, Germany, since 1983 member of the scientific staff at the Institute for Numerical Mathematics, Department of Mathematics, TU Dresden. Selected papers: 1. Zuschnitt- und Packungsoptimierung, Vieweg + Teubner, 2008 (in German, 338 pages) 2. Go-mory Cuts from a Position-Indexed Formulation of 1D

Stock Cutting. In Bortfeldt et al (eds), Intelligent Decision Support, Gabler Edition Wiss. 2008, pp. 1-14 (with G. Belov, ,C. Alves, J.M.V. de Carvalho) 3. LP-based bounds for the Container and Multi-Container Loading Problem. Int. Trans. Opl. Res., 6 (1999) 199-213. Address: Institute for Numerical Mathematics, Department of Mathematics, Technische Universit”at Dresden, 01062 Dresden, Germany,

Field of specialisation: Operational Research, Mathematical modeling, Optimization, Cutting, Packing, Covering Member of ESICUP

Yu. Stoyan 1966 Ph. D./Candidat in Physical-Mathematical Sciences, Institute of Cybernetics of National Academy of Sciences of Ukraine, Kiev, 1970 Doctor of Technical Science, Aviation Engineering Institute in Moscow, 1972 Professor for Computational Mathematics, the Kharkov National University of Radioelectronics, 1985 Corresponding-member of National Academy of Sciences of Ukraine, since 1972 Head of Department of Mathematical Modeling at the Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine.

Selected papers: 1. Packing of convex polytopes into a parallelepiped, Optimization, vol. 54, 2005, pp. 215-235.(with N. Gil, G. Scheithauer, A. Pankratov, I. Magdalina) 2. Packing of Various Radii Solid Spheres into a Parallelepiped. Central Europen Journal of Operational Research, 2003, Vol. 11, Issue 4, pp.389-407. (with G. Yaskov, G. Scheithauer) 3. Packing cylinders and rectangular parallelepipeds with distances between them, Eu-rop. J. Oper. Research 197, 2008, 446-455.(with A. Chugay).

Address: Institute for Problems in Machinery, National Ukrainian Academy of Sciences, 2/10 Pozharsky St., Kharkov, 61046, Ukraine

Field of specialisation: Computational Geometry, Operational Research, Mathematical modeling, Optimization, Cutting, Packing, Covering Member of ESICUP

T. Romanova 1990 Ph. D./Candidat in Physical-Mathematical Sciences, Institute of Cybernetics of National Academy of Sciences of Ukraine, Kiev, 2003 Doctor of Technical Science, Institute of Cybernetics of National Academy of Sciences of Ukraine, Kiev, 2005 Professor for Computational Mathematics, the Kharkov National University of Radioelectronics, since 2003 Senior staff scientist of Department of Mathematical Modeling at the Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine.

Selected papers: 1. Phi-functions for complex 2D-

objects//4OR Quarterly Journal of the Belgian, French and Italian Operations Research Societies. vol. 2, no 1, 2004 pp. 69 - 84. (with G. Scheithauer, N. Gil, Yu. Stoyan) 2. Construction of a Phi-function for two convex polytopes, Applicationes Mathematicae, 2002, Vol.29, No 2, pp. 199 - 218. (with Yu. Stoyan, J. Terno, M. Gil, G. Scheithauer) 3. Phi-function for 2D primary ob-jects//Studia Informatica Universalis, 2002, Vol. 2, No 1, pp. 1-32. (with Yu. Stoyan, J. Terno, G. Scheithauer, N. Gil)

Address: Institute for Problems in Machinery, National Ukrainian Academy of Sciences, 2/10 Pozharsky St., Kharkov, 61046, Ukraine

Field of specialisation: Computational Geometry, Operational Research, Mathematical modeling, Optimization, Cutting, Packing, Covering Member of ESICUP

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