Constant Approximation Algorithms for Rectangle Stabbing and Related Problems

DOI: 10.1007/s00224-005-1273-8

Theory Comput. Systems 40, 187–204 (2007) Theory ofComputing

Systems© 2005 Springer Science+Business Media, Inc.

Constant Approximation Algorithms for Rectangle Stabbingand Related Problems∗

Guang Xu and Jinhui Xu

Department of Computer Science and Engineering, University at Buffalo,Buffalo, NY 14260, USA{guangxu,jinhui}@cse.buffalo.edu

Abstract. In this paper we present constant approximation algorithms for twoNP-hard rectangle stabbing problems, called the weighted rectangle stabbing (WRS)problem and the rectangle stabbing with rejecting cost (RSRC) problem. In the WRSproblem a set of axis-aligned rectangles is given, with each rectangle associated witha positive weight, and a set of weighted horizontal and/or vertical stabbing linesis sought so that each rectangle is intersected by at least one stabbing line with aweight (called cost) no less than that of the rectangle and the total cost (or weight) ofall stabbing lines is minimized. In the RSRC problem each rectangle is associatedwith an additional positive rejecting cost and is required to be either stabbed bya stabbing line or rejected by paying its rejecting cost. For the WRS problem,we present a polynomial time 2e-approximation algorithm, where e is the naturallogarithmic base. Our algorithm is based on a number of interesting techniques suchas rounding, randomization, and lower bounding. For the RSRC problem, we give a3e-approximation algorithm by using a simple but powerful LP rounding techniqueto identify those to-be-rejected rectangles. Our techniques are quite general andcan be easily applied to several related problems, such as the stochastic rectanglestabbing problem and polygon stabbing problem from fixed directions. Algorithmsobtained by our techniques are relatively simple and can be easily implemented forpractical purpose.

∗ This research was supported in part by an IBM faculty partnership award, and an IRCAF award fromSUNY Buffalo.

188 G. Xu and J. Xu

1. Introduction

Determining optimal stabbing of a set of geometric objects in Euclidean space is afundamental problem in computational geometry and combinatorial optimizations. Manyvariants of this problem with different stabbing objectives have been studied previouslyand a number of important results have been obtained [1], [2], [5], [6], [9]–[11], [13]–[18], [20], [21], [25], [27]. Among all these stabbing problems, an interesting variantis the rectangle stabbing problem (or more precisely the cardinality rectangle stabbingproblem). In this problem we are given a set R of axis-aligned rectangles, and are requiredto determine a minimum set L of horizontal and/or vertical lines so that each rectangleis stabbed by at least one line. The rectangle stabbing problem is a very useful model andfinds applications in several areas. In [17] Hassin and Megiddo have showed that somemilitary operations (such as bombing) and some medical applications in radiotherapy[17], [26] can be modeled as rectangle stabbing problems. In [16] Gaur et al. have showedthat the rectangle stabbing problem can be used to solve the load balancing problem inparallel computing for image processing (e.g., convolution) and numerical computation(e.g., finite element methods).

In this paper we consider two interesting generalizations of the rectangle stabbingproblem called the weighted rectangle stabbing (WRS) problem and the rectangle stab-bing with rejecting cost (RSRC) problem and some related problems. In the WRS problemwe are given a set R of axis-aligned rectangles and a set W of positive numbers witheach rectangle ri ∈ R associated with a weight wi ∈ W . The objectives is to find a set Lof vertical and/or horizontal lines and to determine a cost cl for each line l ∈ L so thateach rectangle ri ∈ R is intersected by at least one line l ∈ L with cost cl ≥ wi and thetotal cost

∑l∈L cl is minimized. Note that each stabbing line in L may stab more than

one rectangle and similarly each rectangle in R may be intersected by more than oneline (see Figure 1(a) for an example where the optimal solution contains two verticalstabbing lines, each associated with a cost n). In the RSRC problem we are given a set Rof axis-aligned rectangles and two sets W and P of positive numbers with each rectangle

2 2

1 1c=1

c=2

n−1 n−1

n

c=n−1

c=n

c=n c=n

1c=1

nw1 = 35p1 = 30

R1

R2 w2 = 40

p2 = 38

R3

c1 = 40l1

l2 c2 = 30 R4

R5

R6

R7

R8

l3c3 = 40

p3 = 20

w3 = 30

w4 = 30p4 = 20

w5 = 35p5 = 30

p6 = 38w6 = 40

p7 = 10w7 = 30

p8 = 10w8 = 30

(a) (b)

Fig. 1. (a) An example of the WSR problem. (b) An example of the RSRC problem.

Constant Approximation Algorithms for Rectangle Stabbing and Related Problems 189

ri ∈ R associated with a weight wi ∈ W and a rejecting cost pi ∈ P . The objective isto determine a set O ⊆ R of rectangles, a set L of horizontal and/or vertical stabbinglines, and a cost cl for each line l ∈ L so that each rectangle ri ∈ R\O is intersectedby at least one line l ∈ L with a cost cl ≥ wi and the total cost of

∑l∈L cl +

∑ri∈O pi

is minimized. The rectangles in O are omitted from being stabbed and are viewed asoutliers (see Figure 1(b) for an example).

The WRS and RSRC problems we considered are motivated by applications inradiation therapy. In radiation therapy a key problem is to plan a set of radiation beamsor needles to stab (or penetrate) a set of abnormal regions (e.g., tumors or lesions) so thateach region receives a certain amount of radiation and the total amount of radiation isminimized [4], [9]–[12], [22]. The radiation beams (or needles) are often generated (orinserted) from fixed directions, and each region (or tumor) may demand a different levelof radiation. Thus the treatment planning problem of such an application can be modeledas a weighted polygon or polyhedron stabbing problem from fixed directions which is anatural generalization of the WRS problem. (We will show that the techniques for theWRS problem can be naturally extended to this problem.) Some abnormal regions mightbe close to or surrounded by critical organs. Radiation beams to such regions could causea great deal of damage. Thus, one way to achieve a minimum-damage treatment plan isto associate each region with a rejecting cost so that it can be either stabbed or rejected.Thus the treatment planning problem of such an application can be modeled as an RSRCproblem (with more stabbing directions).

The rectangle stabbing problems considered in this paper are in general quite difficultto solve. It is known that even the simplest version, i.e., the cardinality stabbing problem,has been shown to be NP-hard [17]. Existing algorithms for the stabbing problem includea 3-approximation for the (axis-parallel and equal-length) segment stabbing problem[17] and a 2-approximation for the cardinality rectangle stabbing problem [16]. Bothalgorithms do not consider weights for the segments and rectangles. To the best of ourknowledge, we are not aware of any approximation result for the rectangle stabbingproblems considered in this paper.

A major difficulty of the WRS problem comes from the existence of different weightsassociated with the rectangles. The success of the 2-approximation for the cardinalityrectangle stabbing problem relies on a key fact that all rectangles have a unit weightso that the corresponding integer program can be reduced to subproblems with totalunimodularity property. Unfortunately in our WRS problem such a nice property nolonger exists. To achieve a good approximation algorithm for the WRS problem, we firstdevelop a grouping technique which groups rectangles in such a way that with a smallpenalty on the performance ratio the total unimodularity can be recovered locally foreach group. We then present an interesting lower bounding technique to show that thesolution obtained locally by using the total unimodularity property actually achieves agood approximation ratio for the whole problem. We finally show that the quality ofthe obtained solution can be further improved by using an interesting randomizationtechnique. For the RSRC problem, we propose a powerful LP rounding technique firstto identify those rectangles which should be rejected (as outliers), and then to reducethe problem to a weighted rectangle stabbing problem. Our techniques for the WRS andRSRC problems are quite general and can be easily applied to several related problems,such as the weighted polygon stabbing problem from k fixed directions, the polygon

190 G. Xu and J. Xu

stabbing problem from k fixed directions with rejecting cost, and the stochastic rectanglestabbing problem.

Below is a list of our results:

• For the WRS problem, we present a deterministic 8-approximation algorithmand a randomized 2e-approximation algorithm, where e is the natural logarithmicbase.• For the RSRC problem, we present a 3e-approximation algorithm, and a 3-

approximation algorithm for the case that all rectangles have unit weight.• For the polygon stabbing problem from k fixed directions, we give a k-approximat-

ion for the unweighted case, a ke-approximation for the weighted case, and a(k + 1)-approximation for the unweighted case with rejecting costs.• For the stochastic rectangle stabbing problem, we give a 4-approximation algo-

rithm.

The rest of the paper is organized as follows. In Section 2 we present our algorithmsfor the WRS problem. In Section 3 we give our algorithms for the RSRC problem. InSection 4 we show that our techniques for the WRS and RSRC problems can be extendedto several related problems.

2. The Weighted Rectangle Stabbing Problem

Let R = {r1, r2, . . . , rn} be the given set of rectangles and let W = {w1, w2, . . . , wn} bethe given set of non-negative weights, with each ri ∈ R associated with a weightwi ∈ W .For simplicity, we use the following notations in this section. We use OPT(R,W ) to standfor the cost of an optimal solution of an instance of the WRS problem with rectangle setR and weights W . If all the weights are 1 (i.e., a cardinality rectangle stabbing problem),we denote its optimal cost by OPT(R).

2.1. Algorithms and Analysis

To solve the WRS problem, one might try to use a similar approach for solving thecardinality rectangle stabbing (or CRS for short) problem [16]. As shown in [16], itis possible first to partition the optimization task of the CRS problem into two integerlinear programs (one for each stabbing direction) by suffering a quality loss which isno more than twice of the optimal solution, and then to solve optimally the two integerlinear programs in polynomial time by making use of the total unimodularity propertyof the coefficient matrices of the two integer linear programs ensured by the unit weightassociated with each rectangle. Therefore the total quality loss only comes from thepartition step and can be bounded by a constant ratio of 2. In our WRS problem, however,since different rectangles may have different weights, the integer linear program for eachdirection is in general not totally unimodular and could be very challenging to obtain anoptimal or near optimal solution, making such an approach quite difficult to bound thequality of the solution.

To better make use of the 2-approximation of the CRS problem, our main idea is to tryto partition the problem into a set of subproblems with each being a CRS problem so that


we can obtain a 2-approximation for each subproblem and then combine the solutions tosubproblems to generate a quality guaranteed solution for the WRS problem. However,a direct implementation of such an approach could result in very poor solutions. Suchan example is shown in Figure 1(a) where all the rectangles can be partitioned into ngroups with each group consisting of all the rectangles with the same weight. The totalcost of the optimal solutions (shown by the horizontal dashed lines) to the n subproblemsinduced by the n groups is n(n+ 1)/2+ 1, while the cost of the optimal solution is only2n. The big difference between the two costs seemingly suggests that such an approachcould not lead to any constant approximation for the problem. Surprisingly, we showbelow that a simple rounding procedure could overcome the difficulty.

Thus the main steps of our approach are the following. First we round up the weight ofeach rectangle to one of a small set of different weights. Then we partition the rectanglesinto groups with each containing all the rectangles with the same rounded weight, andsolve the CRS problem for each group. Finally, we combine all the solutions to form asolution to the WRS problem.

2.1.1. An 8-Approximation Algorithm. To reduce the number of different weights, wefirst modify the problem instance by rounding up the weight wi of ri ∈ R to the closestpower of 2. That is, if 2t−1 < wi ≤ 2t , then the modified weight w′i becomes 2t forsome integer t . Let W ′ denote the set of rounded weights {w′1, w′2, . . . , w′n}. Then thefollowing lemma can be easily proved.

Lemma 1. Any feasible solution to the WRS problem of R and W ′ is a feasible solutionto the WRS problem of R and W . Furthermore, OPT(R,W ′) ≤ 2OPT(R,W ).

Proof. Since the two problems share the same set of rectangles and the rounded weightsare no less than their original weights, a feasible solution to the WRS problem of R andW ′ is clearly a feasible solution to the WRS problem of R and W . For the second partof the lemma, let L be the set of stabbing lines of OPT(R,W ) and let L ′ be the sameset of lines as L but with the cost cl of each line l ∈ L being doubled. Since the roundedweight w′i of each rectangle ri is at most twice that of wi , L ′ is a feasible solution to theWRS problem of R and W ′. Thus, OPT(R,W ′) ≤∑l∈L ′ 2cl = 2OPT(R,W ).

Once the weights are rounded, we partition R into a number of subsets R0, R1, . . . ,

RT with each subset Rt containing all rectangles of rounded weight 2t . For each subsetRt , we apply the 2-approximation algorithm for the CRS problem to obtain a set Lt ofvertical or horizontal stabbing lines. We associate each stabbing line l ∈ Lt generatedby the approximation algorithm with a cost 2t , and return the union L =⋃T

t=0 Lt as thesolution to the WRS problem. Since each rectangle in R has been stabbed by at least oneline in L , clearly L is a feasible solution to the WRS problem.

To estimate the quality of such a solution L , we first need to provide a lower boundfor the optimal solution of the WRS problem so that a connection between the optimalsolution and the above solution can be established. The following lemma shows that theoptimal solution to the WRS problem can be well approximated by the optimal solutionsto the set of subproblems of R0, R1, . . . , RT . Our proof of this lemma is based on an

192 G. Xu and J. Xu

interesting lower bounding technique which generates multiple copies of each stabbingline in an optimal solution to the WRS problem.

Lemma 2. OPT(R,W ′) ≥ 12

∑Tt=0 2t OPT(Rt ).

Proof. Note that the cost of a stabbing line in an optimal solution must be equal tothe weight of some rectangle. Let LWRS

OPT (R,W ′) = {l1, . . . , lk} be the set of stabbinglines of an optimal solution to the WRS problem of R and W ′, and let CWRS

OPT (R,W ′) ={2c1 , . . . , 2ck } be the corresponding costs of LWRS

OPT (R,W ′) with lj associated with cost

2cj . For each line lj ∈ LWRSOPT , we make cj copies of line lj , l0

j , l1j , . . . , l

cj−1j , and associate

each lrj , 0 ≤ r ≤ cj − 1, with a cost 2r . Let LWRS(R,W ′) = {l1, . . . , lm} be the

union of LWRSOPT (R,W ′) and the copies of all stabbing lines in LWRS

OPT (R,W ′), and letCWRS(R,W ′) = {2c1 , . . . , 2cm } be the corresponding costs.

To lower bound the optimal solution OPT(R,W ′), we first bound the total cost ofLWRS(R,W ′). Since

∑i=cj−1i=0 2i = 2cj −1 ≤ 2cj , the total costs added to OPT(R,W ′) is

no more than OPT(R,W ′). Thus, the total costs in CWRS(R,W ′) is no more than twicethat of OPT(R,W ′).

Next we show that the stabbing lines in LWRS(R,W ′) form a feasible solution toeach of the T subproblems R1, R2, . . . , RT . For each Rt , since OPT(R,W ′) is a feasiblesolution to the WRS problem, each rectangle r ∈ Rt must be stabbed by some line l inLWRS

OPT (R,W ′) associated with a cost 2c ≥ 2t . Since we have made c copies of l with costs2c−1, 2c−2, . . . , 2, 1, respectively, in LWRS(R,W ′)we can find a line, say l ′, with cost 2t

to stab r ∈ Rt . Thus for each Rt , there exists a subset of stabbing lines in LWRS(R,W ′) ofcosts 2t which stabs every rectangle in Rt . This means that we can partition LWRS(R,W ′)into T subsets with each being a feasible solution to some subproblem associated withRt . Hence the total costs of CWRS(R,W ′) is at least

∑Tt=0 2t OPT(Rt ), and the lemma

follows.

With the above lemmas and a β-approximation algorithm for the CRS problem, wecan design a 4β-approximation algorithm, which has the following main steps, for theWRS problem:

1. For each ri ∈ R, let w′i = 2�logwi2 �.

2. Let Rt = {ri ∈ R : w′i = 2t } be the set of rectangles with the same roundedweight 2t .

3. For each nonempty Rt , apply theβ-approximation algorithm for the CRS problemto obtain a set of stabbing lines Lt .

4. Assign each stabbing line li ∈ Lt a cost 2t and return the union of all Lt ’s withthe assigned costs as the solution to the WRS problem.

Theorem 1. There is a polynomial time 8-approximation algorithm for the WRSproblem.

Proof. By Lemmas 1 and 2, we know that each of the rounding and lower boundingsteps introduces a factor of 2 to the performance ratio, and the approximation algorithm


for the set of CRS problems contributes another factor of β. Thus, the obtained solution isa 4β-approximation algorithm. Since a 2-approximation algorithm for the CRS problemis given in [16], the theorem follows.

Redundant stabbing lines may exist in the above solution. For example, a line withhigher cost may also stab a number of rectangles with lower weights, making the stabbinglines for those lower weighted rectangles redundant. To avoid such a problem, we canslightly modify the above algorithm to improve the quality of our solution further. Theidea is that when we solve the T + 1 CRS problems, we solve them in the decreasingorder of their corresponding rounded weights. That is, we first solve the CRS problemof RT , then that of RT−1, and so forth. Once we have obtained a solution for the CRSproblem of Rt , 1 ≤ t ≤ T , we remove from

⋃t−1j=0 Rj all rectangles that have been

stabbed by the stabbing lines of Rt .

2.1.2. An Improved 2e-Approximation Algorithm. In the previous algorithm we roundthe weight of each rectangle to the closest power of 2 to obtain an 8-approximationof the WRS problem. One might wonder whether it is possible to achieve a betterapproximation ratio by rounding each weight to the closest power of some other number,say α. Unfortunately, the answer to this question is no if we still use the same roundingand lower bounding scheme. The reasons are the following.

First, we analyze how the rounding number α affects the approximation ratio.Consider an optimal solution OPT(R,W ) to the WRS problem with stabbing linesLWRS

OPT = {l1, . . . , lk} and costs CWRSOPT = {c1, . . . , ck}, where ci is the cost associated with

li . Recall that each ci has to be equal to the weight wk of some rectangle rk since it isthe cost of a stabbing line in an optimal solution. Since in our algorithm wk is roundedto the closest power of α, say α jk ≥ wk , the cost for stabbing rk by our algorithm istherefore α jk . This increases the stabbing cost of rk by a factor of α ji /ci , which couldbe arbitrarily close to α since ci (or wk) could be only slightly larger than α ji−1. Hence,choosing α as the rounding number introduces a factor of α to the performance ratio.

Second, we consider the factor introduced by our scheme for bounding the qualityof our solution. In our approach the quality of our solution is bounded by copyingeach stabbing line li ∈ LWRS

OPT of cost ci into ji + 1 lines of costs α ji , α ji−1, . . . , 1,respectively, where α ji is the rounded number of ci . The total cost of all the copies is∑k= ji

k=0 αk ≤ α ji

∑∞k=0(1/α)

k = α ji (1/(1− 1/α)), and hence our solution has a cost ofno more than α ji (1/(1 − 1/α)). This means that copying contributes another factor of1/(1−1/α) to the performance ratio. Thus choosingα as the rounding number introducesin total a factor of α(1/(1−1/α)) to the performance ratio, which achieves its minimumat α = 2.

From the above analysis, we know that in order to obtain a better performance ratio,we have to reduce the factors introduced by either the rounding, the lower bounding, orboth. Next we show how to reduce the factor introduced by rounding by extending aninteresting randomization technique in [3], [7], [23], and [28] to our algorithm.

For each weight w = αlogwα , rounding changes it into α�logwα �. Clearly, this roundednumber could be arbitrarily close toα×w and thus introduces a factor ofα. To reduce thisfactor we can imagine that we somehow know in advance the value of α�logwα �/α. Thus,we can reduce this factor by allowing a fractional number y ∈ [0, 1) in the exponent

194 G. Xu and J. Xu

of the rounded number. That is, we round w to the smallest number αyαkw ≥ w forsome integer kw ≥ 0. Let w = α fw+iw , where iw is a non-negative integer and fwis a fractional number in [0, 1). Since fw + iw ≤ y + kw, when fw ≤ y we havekw = iw and αyαk

w/αfw+iw = αy− fw , and when fw > y we have kw = iw + 1 and

αyαkw/α

fw+iw = α1+y− fw . Thus by choosing a suitable y, we can reduce from α thefactor introduced by rounding.

To achieve the best performance ratio, each weightwwould expect to choose such a ythat minimizes min{αy− fw , α1+y− fw }. However, different weights may have contradictingexpectations. To achieve an overall best performance, we use randomization to avoid thisproblem. Below are the main steps of our improved approximation algorithm, where αis a parameter to be determined:

1. Uniformly randomly choose y from [0, 1).2. Round the weight wi of each ri ∈ R to αy+ki , where ki is the smallest integer

such that wi ≤ αy+ki .3. Let Rt = {ri ∈ R|ki = t} be the set of rectangles whose weights have been

rounded to the same number αy+t . Apply the 2-approximation algorithm for theminimum CRS problem to each Rt .

4. For each t , assign cost αy+t to the set of stabbing lines obtained from the solutionof the CRS problem of Rt .

5. Return the union of the lines with assigned costs as the solution to the WRSproblem.

Lemma 3. There exists an α so that the expected cost of the solution returned by theabove randomized algorithm is no more than 2e times the optimal cost.

Proof. Consider an optimal solution OPT(R,W ) of the WRS problem. LetLWRS

OPT (R,W ) = {l1, . . . , lm} be the set of stabbing lines in an optimal solution and letCWRS

OPT (R,W ) = {c1, . . . , cm} be their corresponding costs. Rewrite ci as α fi+�logciα �. Let

ki be the smallest integer such that y + ki ≥ fi + �logciα �. From the above analysis, we

have

αy+ki

ci={αy− fi , if y ≥ fi ,

α1+y− fi , if y < fi .(1)

It is not hard to see that the line set L = {l1, . . . , lm} with corresponding cost setC = {αy+k1 , . . . , αy+km } forms a feasible solution to the rounded instance of R and W ′.For each line li ∈ L with cost αy+ki , we make ki additional copies and associate themwith costs αy, αy+1, . . . , αy+ki−1, respectively.

Let Si be the total cost of all copies of li . We have

Si =ki∑

p=0

αy+p ≤ αy+ki

∞∑p=0

(1

α

)p

= αy+kiα

α − 1.

Let S =∑mi=1 Si be the total cost of all stabbing lines (including the copied lines). Since

our algorithm returns a solution with cost no more than 2S, the expected cost of our


algorithm is no more than 2E(S). By the linearity of expectation, E(S) =∑mi=1 E(Si ),

where

E(Si ) ≤ E

(αy+ki

α

α − 1

)= α

α − 1E(αy+ki ).

Since y is chosen uniformly randomly from [0, 1), plugging (1) into the above equation,we have

E(Si ) ≤ α

α − 1

(∫ fi

0ciα

1+y− fi dy +∫ 1

fi

ciαy− fi dy

)= α

lnαci .

Thus,

E(S) ≤∑

i

α

lnαci = α

lnα

∑i

ci = α

lnαOPT(R,W ).

Setting α = e, we have E(S) ≤ e × OPT(R,W ) and the cost of our algorithm is nomore than 2e × OPT(R,W ), where e is the natural logarithmic base.

Theorem 2. There is a polynomial time 2e randomized approximation algorithm forthe WRS problem, where e is the natural logarithmic base.

Proof. It follows from Lemmas 1 and 3, and the above discussion.

3. The Rectangle Stabbing Problem with Rejecting Costs

Let R = {r1, r2, . . . , rn} be the given set of rectangles in the RSRC problem, let W ={w1, w2, . . . , wn} be the given set of non-negative weights, and let P = {p1, p2, . . . , pn}be the given set of rejecting costs with each ri ∈ R associated with a weight wi ∈ Wand a rejecting cost pi ∈ P .

In some way we can view those omitted rectangles as outliers. We notice that theidea of using rejecting costs to determine outliers has been used in other problems suchas the clustering problem in [8]. However, due to the different nature of our problem, itis not clear to us how to apply the primal–dual method developed in [8], [19], and [29]to solve our problem.

To solve the RSRC problem, we first consider a simpler version (called the un-weighted RSRC problem) of the RSRC problem in which all the rectangles have unitweight, and then use the technique for the unweighted case to derive a solution to theRSRC problem.

3.1. Algorithm for the Unweighted RSRC Problem

To solve the unweighted RSRC problem, our main idea is first to formulate this prob-lem as an integer program, then to solve the relaxed linear program, and finally to use

196 G. Xu and J. Xu

an interesting rounding technique to obtain a constant approximation for the integerprogram.

In [16] it is shown that we can compute a set H of horizontal lines and a set Vof vertical lines in O(n) time such that we can assume without loss of generality thatstabbing lines are chosen from H ∪ V since any stabbing line can be glided to a line inH ∪ V without changing the stabbing condition of all rectangles.

To simplify our discussion, we assume that all lines in V = {v1, v2, . . .} and H ={h1, h2, . . .} are sorted in the increasing order of their x and y coordinates, respectively.LetHk be the set of horizontal lines in H stabbing rectangle rk , and let Vk be the set ofvertical lines in V stabbing rectangle rk . Below is the integer program of the unweightedRSRC problem, denoted as IP-URSRC, using H and V as the candidates for stabbinglines:

Minimize∑vi∈V

xi +∑hj∈H

yj +k=n∑k=1

pk zk (2)

subject to∑vi∈Vk

xi +∑

hj∈Hk

yj + zk ≥ 1, ∀k ∈ {1, . . . , n}, (3)

xi ∈ {0, 1}, ∀vi ∈ V, (4)

yj ∈ {0, 1}, ∀hj ∈ H, (5)

zk ∈ {0, 1}, ∀k ∈ {1, . . . , n}. (6)

In the above integer program, for each vertical (or horizontal) line vi (or hj ), weuse a 0–1 variable xi (or yj ) to indicate whether it is selected as a stabbing line of theunweighted RSRC problem. For each rectangle rk ∈ R, we use another 0–1 variablezk to indicate whether rk will be rejected or not. Constraint (3) ensures that if rk is notrejected, then it must be stabbed by some line in H ∪ V .

Let LP-URSRC denote the linear relaxation of the integer program IP-URSRC withconstraints (4), (5), and (6) replaced by xi ≥ 0, yj ≥ 0, and zk ≥ 0, respectively. Let(x∗, y∗, z∗) andw∗ be an optimal solution and the optimal value of the objective functionof the linear program LP-URSRC, respectively. Obviously, w∗ ≥ w∗.

We define three subsets RV, RH, Rrej of R as below. If more than one inequalitieshold for a rectangle, we break the tie arbitrarily.

1. RV = {rk ∈ R|∑vi∈Vkx∗i ≥ 1

3 }, i.e., the set of rectangles that should be stabbedby vertical lines.

2. RH = {rk ∈ R|∑hj∈Hky∗j ≥ 1

3 }, i.e., the set of rectangles that should be stabbedby horizontal lines.

3. Rrej = {rk ∈ R|z∗k ≥ 13 }, the set of rectangles that should be rejected.

Lemma 4. RH ∪ RV ∪ Rrej = R.

Proof. For each rk ∈ R, we have∑

vi∈Vkx∗i +

∑hj∈Hk

y∗j +z∗k ≥ 1 since (x∗, y∗, z∗) is afeasible solution to the relaxed linear program LP-URSRC. Hence, one of the followingthree inequalities must be true: (1)

∑vi∈Vk

x∗i ≥ 13 , (2)

∑hj∈Hk

y∗j ≥ 13 , (3) z∗k ≥ 1

3 .Therefore, RH, RV, and Rrej form a partition of R.


The above lemma is very useful to our algorithm since it provides a way to determinewhether a rectangle should be rejected, and, if not, in which direction it should be stabbed.Our algorithm rejects all rectangles of Rrej, stabs rectangles of RV by the vertical linesof V , and stabs rectangles of RH by the horizontal lines H . The remaining task is to finda low cost solution for RV and RH.

We first formulate the stabbing problems of RH and RV into two separate integerprograms using H and V as the candidates for the stabbing lines, and then try to obtainintegral solutions to the two integer programs. Finally, we make a connection betweenthe relaxed linear program of the unweighted RSRC problem and the integer programsof RH and RV to ensure the quality of the solution.

The integer program formulation for the problem of stabbing rectangles in RV bylines in V , denoted as IP-VERTICAL, is as follows:

Minimize∑vi∈V

xi (7)


xi ≥ 1, ∀rk ∈ RV, (8)

xi ∈ {0, 1}, ∀vi ∈ V . (9)

Similarly, we have the integer program formulation for the problem of stabbingrectangles in RH by lines in H , denoted as IP-HORIZONTAL:

Minimize∑hj∈H

yj (10)

subject to∑

hj∈Hk

yj ≥ 1, ∀rk ∈ RH, (11)

yj ∈ {0, 1}, ∀hj ∈ H. (12)

Relaxing constraints (9) and (12) to xi ≥ 0 and yj ≥ 0, respectively, we obtain twolinear programs, denoted as LP-VERTICAL and LP-HORIZONTAL. Letw∗V andw∗H de-note the optimal cost of the linear programs LP-VERTICAL and LP-HORIZONTAL, re-spectively, and letw∗V andw∗H denote the optimal cost of integer programs IP-VERTICLEand IP-HORIZONTAL, respectively. Let wrej =

∑rk∈Rrej

pk be the cost incurred by re-jecting rectangles in Rrej. The following lemma establishes a relation between w∗H, w∗V,wrej, and w∗.

Lemma 5. w∗V + w∗H + wrej ≤ 3w∗.

Proof. We first estimate the cost of wrej. By the definition of Rrej, we have

wrej =∑

rk∈Rrej

pk ≤ 3×∑

rk∈Rrej

z∗k pk ≤ 3×k=n∑k=1

z∗k pk .

To estimate w∗V, we observe that 3x∗ is a feasible solution to the linear program LP-VERTICAL, since rk ∈ RV implies

∑vi∈Vk

x∗i ≥ 13 . Thus, w∗V ≤ 3

∑vi∈V x∗i , since w∗V

198 G. Xu and J. Xu

is the optimal value of the linear program of RV. Similarly, we have w∗H ≤ 3∑

hj∈H y∗j .Combining the three inequalities, we have

w∗V + w∗H + wrej ≤ 3∑vi∈V

x∗i + 3∑hj∈H

y∗j + 3 ∗k=n∑k=1

z∗k pk = 3w∗.

The following lemma follows from Lemma 1 in [16]. We sketch its proof here forcompleteness.

Lemma 6. w∗H = w∗H and w∗V = w∗V.

Proof. Since all horizontal lines in H are sorted by their y coordinates, all lines inHk appear continuously in the sorted sequence of H . Thus the rows of the coefficientmatrix of the integer linear program for RH have the interval property, and the coefficientmatrix is totally unimodular [24]. This means that an optimal solution to the linearprogram relaxation is also an integral optimal solution to the integer linear program.Thus w∗H = w∗H. Similarly, w∗V = w∗V.

Lemma 6 ensures that the solutions to the linear programs of LP-HORIZONTAL andLP-VERTICAL are all integral. Thus we can use the lines determined by the linear pro-grams LP-HORIZONTAL and LP-VERTICAL as the stabbing lines of the unweightedRSRC problem.

In summary, our approximation algorithm for the unweighted RSRC problem con-sists of the following main steps:

1. Formulate a linear program LP-URSRC by using the set of rectangles R and therejecting costs P . Solve the above linear program to obtain an optimal fractionalsolution (x∗, y∗, z∗).

2. Partition R into three subsets RH, RV, and Rrej according to (x∗, y∗, z∗) and thedefinitions of RH, RV, and Rrej.

3. Set zk = 1 for all rk ∈ Rrej and zk = 0 for rk �∈ Rrej.4. Formulate the linear programs LP-HORIZONTAL and LP-VERTICAL. Solve

the two linear programs to obtain the integral solutions x and y.5. Return (x, y, z) as the solution to the unweighted RSRC problem.

Theorem 3. There is a polynomial time 3-approximation algorithm for the unweightedrectangle stabbing problem with rejecting costs.

3.2. Algorithm for the RSRC Problem

To solve the RSRC problem, our main idea is to combine the techniques for the WRSproblem in Section 2 and the techniques for the unweighted RSRC problem to design a3e-approximation algorithm. Below are the main steps of our algorithm:

1. Uniformly randomly choose y from [0, 1).2. Round the weight wi of each ri ∈ R to w′i = αy+kwi , where kwi is the smallest

integer such that wi ≤ αy+kwi .


3. Let Rt = {ri ∈ R|kwi = t} be the set of rectangles whose weights are roundedto αy+t .• For each ri ∈ Rt , modify its rejecting cost pi to p′i = pi/α

y+t .• Apply the 3-approximation algorithm for the unweighted RSRC problem for

the rectangles in Rt with the modified rejecting costs.• Let Lt be the set of lines returned by the 3-approximation algorithm. Reject

the rectangles in Rt which are not stabbed by Lt , associate each line l ∈ Lt

with a cost αy+t , and add it to the solution.

Theorem 4. The above algorithm is a polynomial time 3e-approximation of the RSRCproblem.

Proof. Let OPT(R,W, P) be an optimal solution of the RSRC problem, let LRSRCOPT

(R,W, P) = {l1, . . . , lm} be the set of stabbing lines in OPT(R,W, P), and let CRSRCOPT

(R,W, P) = {c1, . . . , cm} be the corresponding costs of the lines in LRSRCOPT (R,W, P).

Also let Rrej be the set of rectangles rejected by OPT(R,W, P).

Rewrite ci ∈ CRSRCOPT (R,W, P) as α fi+�log

ciα �. Let ki be the smallest integer such that

y + ki ≥ fi + �logciα �, and let W ′ be the set of rounded weights of all rectangles in

R\Rrej. It is easy to see that line set L = {l1, . . . , lm} and the corresponding cost setC = {αy+k1 , . . . , αy+km } form a feasible solution to the WRS problem of R\Rrej and W ′.For each line li ∈ L with cost αy+ki , we make ki additional copies and associate themwith costs αy , αy+1, . . . , αy+ki−1, respectively. Let LC =⋃l∈L{(li , α

y), (li , α1+y), . . . ,

(li , αy+ki )} be the set of line–cost pairs of all lines in L and their corresponding copies,

and let LCt = {(l, c) ∈ LC |c = αy+t } be a subset of LC . It is obvious that each rectanglein Rt\Rrej is stabbed by at least one line of LCt . Therefore, LCt and Rrej ∩ Rt form afeasible solution to the unweighted RSRC problem of Rt (with weight αy+t ). Thus,

∑(l,c)∈LCt

c +∑

rj∈Rrej∩Rt

pj ≥ αy+t OPTURSRC

(Rt ,

1

αy+tP

),

where OPTURSRC(Rt , (1/αy+t )P) is the optimal solution of the unweighted RSRC prob-lem of Rt (with the modified rejecting costs). Since our algorithm returns a solution ofcost no more than 3OPTURSRC(Rt , (1/αy+t )P) for the unweighted RSRC problem of Rt

(with the modified rejecting costs (1/αy+t )P), the total cost of our solution is no morethan

3∑

t

∑(l,c)∈LCt

c + 3∑

rj∈Rrej

pj = 3∑

(l,c)∈LC

c + 3∑

rj∈Rrej

pj .

By Lemma 3, we know that

E

( ∑(l,c)∈LC

c

)≤ e

i=m∑i=1

ci .

Thus, the theorem follows.

200 G. Xu and J. Xu

4. Generalization and Related Problems

In this section we show that the techniques for the WRS and RSRC problems can beeasily extended to several related problems.

4.1. The Stochastic Rectangle Stabbing Problem

In the stochastic rectangle stabbing (SRS) problem, we are given a set R = {r1, r2, . . . ,

rn} of n axis-aligned rectangles and a probability distribution p on F ⊆ 2R , where|F | ≤ O(nk) for some positive integer k. Unlike other rectangle stabbing problems inwhich all rectangles are required to be either stabbed or rejected, the SRS problem onlyrequires stabbing a subset of rectangles. The subset which is needed to be stabbed is,however, unknown in advance. Instead, only a probability distribution p on those likelyto-be-stabbed subsets F is given. The cost for stabbing a rectangle in advance (i.e.,before knowing which subset is the one needed to be stabbed) is 1, but it increases toλ ≥ 1 once the subset to be stabbed is specified. We call a stabbing line determinedin advance a pre-stabbing line and a line determined afterward a post-stabbing line.The objective of the SRS problem is to determine a set of horizontal and/or verticalpre-stabbing lines so that the expected total cost (including the costs of stabbing both inadvance and afterward) is minimized.

The stochastic rectangle (or polygon) stabbing problem also arises in radiationtherapy and other applications. Quite often in radiation therapy a questionable region(or a set of regions) might turn itself into a tumor (or tumors) with a certain probability.To conduct early stage treatment, it is necessary to use radiation beams or needles tostab those regions based on their probabilities. A nonstabbed questionable region mighteventually grow into a (larger) tumor and therefore needs to pay a higher price to stabit. In such a scenario the objective of the treatment planning problem is to determine aset of stabbing lines in advance so that the total expected stabbing cost is minimized.

To solve the SRS problem, we first use a procedure similar to the one for theunweighted RSRC problem to determine sets H and V as candidates for the stabbinglines, and obtain Hk and Vk for each rectangle rk ∈ R. We assume that both H and Vare in sorted order based on their coordinates. Then we formulate the problem as thefollowing integer program:

Minimize∑vi∈V

Xi +∑hj∈H

Yj + λ∑

vi∈V,F∈Fp(F)x F

i + λ∑

hj∈H,F∈Fp(F)yF

j (13)


Xi +∑

hj∈Hk

Yj +∑vi∈Vk

x Fi +

∑hj∈Hk

yFj ≥ 1, ∀rk ∈ F and ∀F ∈ F, (14)

x Fi ∈ {0, 1}, ∀vi ∈ V and ∀F ∈ F, (15)

yFj ∈ {0, 1}, ∀hj ∈ H and ∀F ∈ F, (16)

Xi ∈ {0, 1}, ∀vi ∈ V and ∀F ∈ F, (17)

Yj ∈ {0, 1}, ∀hj ∈ H and ∀F ∈ F . (18)

To obtain a good solution to the above integer program, we first solve the linear re-laxation of the above integer program by replacing constraints (15), (16), (17), and (18)


with Xi ≥ 0, Yj ≥ 0, x Fi ≥ 0, and yF

j ≥ 0, respectively. Let (X∗, Y ∗, x F∗, yF∗)be the optimal fractional solution and let w∗ be the optimal value of the objectivefunction.

Let RV = {rk ∈ R|∑vi∈VkX∗i ≥ 1

4 } and RH = {rk ∈ R\RV|∑

hj∈HkY ∗j ≥ 1

4 }. Ouralgorithm first determines a set of horizontal pre-stabbing lines in H to stab RH and aset of vertical pre-stabbing lines in V to stab RV. Using a similar approach to that inSection 3, we can obtain the optimal stabbing for both RH and RV. Letw∗H andw∗V be thevalue of the optimal stabbing of RH and RV, respectively. We have the following lemma:

Lemma 7. w∗V + w∗H ≤ 4∑

vi∈V X∗i + 4∑

hj∈H Y ∗j .

Proof. By Lemma 6, we know that the optimal stabbing of RH and RV can be obtained bysolving the linear programs LP-HORIZONTAL and LP-VERTICAL. Since 4X∗ and 4Y ∗

are the feasible solutions of the linear programs LP-VERTICAL and LP-HORIZONTAL,respectively, the costs 4X∗ and 4Y ∗ upper bound the optimal stabbing costs of RH

and RV.

Once F ∈ F is specified with probability p(F), we only need to stab the rectanglesin F\(RH ∪ RV). For such rectangles, we use vertical or horizontal post-stabbing linesat cost λ per line to stab them. Let FV = {rk ∈ F\(RH ∪ RV)|

∑vi∈Vk

x F∗i ≥ 1

4 } andFH = {rk ∈ F\(RH ∪ RV)|

∑hj∈Hk

yF∗j ≥ 1

4 }. The following lemma can be easilyproved.

Lemma 8. F = RH ∪ RV ∪ FH ∪ FV.

Proof. Since X∗, Y ∗, x F∗, and yF∗ satisfy constraint (14), one of the four inequalitiesdefining RH, RV, FH, and FV must hold. Thus, the lemma follows.

The post-stabbing lines for FH and FV can be optimally determined in a similar wayas for RH and RV. Let wF∗

H and wF∗V be the optimal value of the stabbing for FH and

FV, respectively. We immediately have the following lemma whose proof is exactly thesame as that of Lemma 7.

Lemma 9. wF∗H + wF∗

V ≤ 4∑

vi∈V x F∗i + 4

∑hj∈H yF∗

j .

Theorem 5. There is a polynomial time 4-approximation algorithm for the SRSproblem.

Proof. From Lemmas 7, 9, and 8, we know that the expected cost of the SRS problemis

λ∑F∈F

p(F)(wF∗H + wF∗

V )+ w∗H + w∗V,

202 G. Xu and J. Xu

which is bounded by

4∑vi∈V

X∗i +4∑hj∈H

Y ∗j +4λ∑

vi∈V,F∈Fp(F)x F∗

i +4λ∑

hj∈H,F∈Fp(F)yF∗

j ≤4OPT(R),

where OPT(R) is the optimal solution of the SRS problem. Thus, the theoremfollows.

4.2. The Polygon Stabbing Problem from k Fixed Directions

For the polygon stabbing problem, we consider both the unweighted and weighted cases.In the unweighted (or cardinality) polygon stabbing problem from k fixed directions, aset O = {o1, o2, . . . , oP} of polygons and k fixed directions, α1, . . . , αk , are given, anda minimum set L of lines from the k fixed directions is sought to stab each polygon in O .

To solve this stabbing problem, for each direction αj , 1 ≤ j ≤ k, we can first projecteach op ∈ O to the direction orthogonal to αj , and then generate the set of representativelines L j = {l j

1 , lj2 , . . . , l

jmj } in the same way as discussed in Section 3. We assume that

L j are in sorted order. Let L jp be the set of lines from L j which stab polygon op. The

stabbing problem can be formulated as the following integer program, where X ji is the

indicator variable for line l ji :

Minimizej=k∑j=1

i=mj∑i=1

X ji (19)

subject toj=k∑j=1

∑l ji ∈L j

p

X ji ≥ 1, ∀op ∈ O, (20)

X ji ∈ {0, 1}, ∀1 ≤ j ≤ k, ∀1 ≤ i ≤ mj . (21)

First we solve the linear relaxation of the above integer program with constraint (21)replaced with X j

i ≥ 0 and let X∗ be the optimal solution. Then we can round the fractionaloptimal solution in a similar way to that discussed in Section 3 and partition O into ksubsets O1, O2, . . . , Ok , where Oj = {op ∈ O :

∑l ji ∈L j

pX j

i ≥ 1/k}. For each Oj ,we find an optimal stabbing by lines in the direction of αj using the same procedurediscussed in Section 3. Since k X∗ is a feasible solution for each directions, we canobtain the performance ratio k in the same way as in the unweighted RSRC problem.If a rejecting cost is introduced, a similar algorithm as the one for unweighted RSRCwill lead to a (k + 1)-approximation for the polygon stabbing problem. Similarly, ake-approximation can be achieved for the weighted polygon stabbing problem from kfixed directions, and a (k+1)e-approximation can be obtained for the weighted polygonstabbing problem from k fixed directions and with rejecting costs.

Theorem 6. There is a polynomial time k-approximation algorithm for the unweightedpolygon stabbing problem from k fixed directions, a (k + 1)-approximation algorithmfor the unweighted polygon stabbing problem from k fixed directions and with rejectingcosts, a ke-approximation algorithm for the weighted polygon stabbing problem from


k fixed directions, and a (k + 1)e-approximation algorithm for the weighted polygonstabbing problem from k fixed directions and with rejecting costs.

Acknowledgments

The authors thank Professor Tetsuo Asano and two anonymous referees for their kind help and thoughtfulsuggestions for improving the presentation of this paper.

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Received December 28, 2004. Online publication July 13, 2005.

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Constant Approximation Algorithms for Rectangle Stabbing and Related Problems