J Comb Optim (2010) 19: 31–42DOI 10.1007/s10878-008-9158-9

An almost four-approximation algorithm for maximumweight triangulation

Shiyan Hu

Published online: 19 April 2008© Springer Science+Business Media, LLC 2008

Abstract We consider the following planar maximum weight triangulation (MAT)problem: given a set of n points in the plane, find a triangulation such that the totallength of edges in triangulation is maximized. We prove an �(

√n) lower bound on

the approximation factor for several heuristics: maximum greedy triangulation, max-imum greedy spanning tree triangulation and maximum spanning tree triangulation.We then propose the Spoke Triangulation algorithm, which approximates the maxi-mum weight triangulation for points in general position within a factor of almost fourin O(n logn) time. The proof is simpler than the previous work. We also prove thatSpoke Triangulation approximates the maximum weight triangulation of a convexpolygon within a factor of two.

Keywords Triangulation · Maximum weight triangulation · Spoke triangulation ·Approximation algorithm · Approximation ratio

1 Introduction

Let P be a set of n points in the plane. A triangulation T(P ) of P is a maximal setof non-intersecting straight-line segments connecting points in P . The weight |T(P )|of T(P ) is the sum of the Euclidean lengths of edges in T(P ). The minimum weighttriangulation (MWT) is a triangulation of P with minimum weight. The maximumweight triangulation (MAT) is a triangulation of P with maximum weight. Comput-ing the former is a well known problem in computational geometry and has beenshown to be NP-hard recently (Mulzer and Rote 2006).

Preliminary version of this paper (Hu 2003) appeared in CCCG’03.

S. Hu (�)Department of Electrical and Computer Engineering, Michigan Technological University, Houghton,MI 49931, USAe-mail: shiyan@mtu.edu

32 J Comb Optim (2010) 19: 31–42

Related work Recent research attention has been paid to solving the maximumweight triangulation problem as this may deepen our knowledge in studying opti-mal triangulations. The first work on it is in Wang et al. (1999) which presents anO(n2) algorithm to find the maximum weight triangulation of an n-sided polygoninscribed in a disk. Later, the conference version of this paper (Hu 2003) proposesthe spoke triangulation which leads to a 6-approximation for maximum weight trian-gulation of the point set in general position. The approximation ratio is improved to4.238 in Chin et al. (2004) through giving better weight bound on maximum weighttriangulation and the ratio approaches four when n goes to infinity. In this paper,the approximation ratio is also proved to be 4 + ε, where ε = 3.5√

nand the proof is

simpler than Chin et al. (2004). When restricted to a convex polygon, a linear-timeapproximation scheme exists (Qian and Wang 2004).

In contrast to very limited research work in MAT, much more results are knownon MWT. An important research direction on MWT is to compute approximationsfor the optimal triangulations. Levcopoulos and Krznaric (1998a) propose a constantfactor approximation to MWT(P ), and a 1 + ε approximation for MWT of a convexpolygon can be computed in linear time (Levcopoulos and Krznaric 1998b). Veryrecently, Remy and Steger show that a 1+ε approximation for a general point set canbe computed in nO(log8 n) time (Remy and Steger 2006). This is interesting especiallyconsidering the NP-hardness nature of the MWT problem (Mulzer and Rote 2006).Other known heuristic algorithms are described in Plaisted and Hong (1987), Lingas(1987) and Heath and Pemmaraju (1994). For example, Lingas (1987) and Heathand Pemmaraju (1994) introduce so-called minimum spanning tree triangulation andgreedy spanning tree triangulation heuristics. However, Levcopoulos and Krznaric(1996) show that these heuristics can have worst-case approximation ratios of �(n),�(

√n), respectively.

Summary of results In this paper, we prove that for MAT, the worst-case approxima-tion lower bound of maximum greedy triangulation, maximum greedy spanning treetriangulation and maximum spanning tree triangulation heuristics (which are the nat-ural analogues of their MWT counterparts) is �(

√n). We also obtain a tight bound of

�(n) for degenerate cases. We then present the Spoke Triangulation and prove that itapproximates the maximum weight triangulation for points in general position withina factor of almost 4 and can be computed in O(n logn) time. A similar result has beengiven in Chin et al. (2004), however, our proof is simpler compared to theirs. We alsoprove that Spoke Triangulation approximates the maximum weight triangulation of aconvex polygon within a factor of 2.

Definition and notation The greedy triangulation GT(P ) of P is obtained by re-peatedly adding a longest possible edge that does not properly intersect any of thepreviously generated edges. The greedy spanning tree triangulation GSTT(P ) of P

is obtained as follows: start with a maximum spanning tree of the greedy triangula-tion, triangulate optimally each of the simple polygons bounded by this spanning treeand the convex hull of P . The maximum spanning tree triangulation MSTT(P ) of P

is obtained similarly to GSTT(P ) except that it starts with the Euclidean maximumnon-crossing spanning tree of P .

J Comb Optim (2010) 19: 31–42 33

Fig. 1 (a) Degenerate case. (b) Maximum spanning tree for the point set

An algorithm is said to be a factor ρ approximation for a maximization problemif the algorithm is guaranteed to produce a solution whose objective function value isat least (1/ρ) of the optimal solution.

Paper organization The rest of the paper is organized as follows: Sect. 2 shows theinapplicability of common heuristics to MAT. Section 3 describes the constant factorapproximation algorithm to MAT. A summary of work is given in Sect. 4.

2 Lower bounds for common heuristics

This section will show that the worst-case approximation lower bound for MAT bymaximum greedy triangulation (GT), maximum greedy spanning tree triangulation(GSTT) and maximum spanning tree triangulation heuristics (MSTT) is �(

√n). The

approximation bound becomes tight, namely �(n), for degenerate cases.We first construct a point set P to prove the claim about GT. We start by allowing

degeneracies, that is, we permit three or more collinear points in P .

Lemma 1 For any integer n ≥ 0, there exists a set P of n points in degenerate posi-tion such that |MAT(P )|

|GT(P )| = �(n).

Proof Assume that the points in P are distributed as shown in Fig. 1(a). All the pointsare on the sides of the right triangle apb, where p is only a virtual point (p �∈ P )satisfying ∠apb = π/2, |ab| = n, |pb| = |pa| = n/

√2, |pc| = 1, and |ppj | < 1 for

1 ≤ j ≤ n − 3.By the definition of greedy triangulation, ab, bc, bpn−3 (recall that p �∈ P ) will

be the first three edges to be added. After that, there is only one possible way tocomplete the triangulation: connect pj to c for 1 ≤ j ≤ n − 3, connect pj to pj+1

for 1 ≤ j ≤ n − 4 and connect a to c. Note that |ab| + |bc| + |bpn−3| + |ac| < 4n

and |pjc| <√

2,1 ≤ j ≤ n − 3, so |GT| = |ab| + |bc| + |bpn−3| + ∑n−4j=1 |pipi+1| +

∑n−3j=1 |pic| + |ac| = �(n). On the other hand, a larger weight triangulation, denoted

by LWT(P ) (clearly |LWT(P )| ≤ |MAT(P )|), is obtained by first connecting pj to a

for 1 ≤ j ≤ n − 3. Then |LWT| ≥ ∑n−3j=1 |pja| = �(n2), thus |MAT(P )|

|GT(P )| = �(n). Thisbound is tight immediately from Lemma 6. �

34 J Comb Optim (2010) 19: 31–42

Fig. 2 Non-degenerate case

We next consider the non-degenerate case, that is, no three or more can becollinear. In Levcopoulos and Krznaric (1996) for proving similar claims for MWT,the authors start with the degenerate construction and then move the points pj

(1 ≤ j ≤ n − 3) from a line to a curve (parabola) to prove the non-degenerate cases.However, this simple strategy will not work in our case. Our idea is that we distrib-ute the first

√n points in a nice way on a unit half-circle such that the total length

of the first√

n edges in GT(P ) is O(√

n). We then show that the total length of allremaining edges is also O(

√n). However MAT(P ) has the weight of �(n), which

completes the proof.

Lemma 2 For any integer n ≥ 0, there exists a set P of n points in general positionsuch that |MAT(P )|

|GT(P )| = �(√

n).

Proof Refer to Fig. 2. Let ab be the diameter of the unit half-circle C and let d lieslightly above the center of C. For any two points x, y on C, let xy denote the arcconnecting x and y, so |ab| = π . All other points will be placed inside or on C. Wesay that p is below a chord xy, when p lies in, or on the boundary of, the closed regionbounded by arc xy and chord xy. Clearly, |xy| > |xp|, |yp| (note that p lies in theclosed half-disk). We distribute the next k − 2 points, denoted by p1,p2, . . . , pk−2,on C as follows.

Place p1 on C such that |bp1| = π/x1, where 1 < x1 < 2. Remaining points willbe placed below bp1. We want to let bp1 be in GT(P ) and this happens if all the

remaining points lie below the chord p1p2 such that |bp1| > |ap2|. Let x2 = |p1b||p1p2|

> 1 and restrict x2 < 2, then x2 > 12−x1

. p1p2 will be in GT(P ) if all other points

lie below the chord p2p3 such that |p1p2| > |bp3|. Let x3 = |p2p1||p2p3| > 1 and restrict

x3 < 2, then x3 > 12−x2

.We now show how to perform the above operations generally. Suppose that we

have just added the edge pj−1pj , that is, the last placed point is pj+1, and allthe remaining points will lie below pjpj+1. Since x1, x2, . . . , xj+1 are determined

and xj+1 = |pj pj−1||pj pj+1| , we have | pj−1pj+1| = (1 − 1

xj+1)|pjpj−1| and |pjpj+1| =

π/∏j+1

k=1 xk .

J Comb Optim (2010) 19: 31–42 35

To add edge pjpj+1, we need to place the point pj+2 on C such that all theremaining points lie below pj+1pj+2 to guarantee |pjpj+1| > | pj−1pj+2|. Let

xj+2 = |pj+1pj || pj+1pj+2| , then | pj+1pj+2| = |pjpj+1|/xj+2 = π/

∏j+2k=1 xk . We have

|pjpj+1| > | pj−1pj+2| (1)

⇒ |pjpj+1| > | pj−1pj+1| + | pj+1pj+2| (2)

⇒ |pjpj+1| > | pj−1pj+1| + |pjpj+1|xj+2

(3)

⇒(

1 − 1

xj+2

)

|pjpj+1| >(

1 − 1

xj+1

)

|pjpj−1| (4)

⇒(

1 − 1

xj+2

) |pjpj−1|xj+1

>

(

1 − 1

xj+1

)

|pjpj−1| (5)

⇒ xj+2 >1

2 − xj+1. (6)

As one can see from the above process, the key to obtain the intended GT is to con-struct a sequence xj such that xj = 1/(2 − xj−1) + ε (ε > 0) and 1 < xj < 2. Notingthat xj > 1/(2 − xj−1) > xj−1, one can see that x1, x2, . . . , xk−2 is a monotonicincreasing sequence. This is why the points are not placed symmetrically on thecircle C. Lemma 3 below can be used to construct this sequence. Suppose that wehave added the points p1,p2, . . . , pk−2 and thus determined x1, x2, . . . , xk−2. Set-ting x1 = (k + 1)/k and ε = 1/kk , the sequence is

x1 = k + 1

k, xj = 1

2 − xj−1+ 1

kkfor j = 2,3, . . . , k − 2. (7)

Since any chord in C is shorter than its corresponding arc, we can safely use thelength of the arc to bound the length of chord. Note that 1/

∏ij=1 xj < 1/xi

1 (sincex1, x2, . . . , xj is a monotonic increasing sequence), then the total length of the firstk − 2 edges is

|ab| + |bp1| +k−4∑

i=1

|pipi+1| (8)

< π + π

k−4∑

i=1

(

1/ i∏

j=1

xj

)

(9)

< π + π

k−4∑

i=1

(1/xi1) (10)

< π + π

x1 − 1(11)

36 J Comb Optim (2010) 19: 31–42

= O

(1

x1 − 1

)

= O(k). (12)

According to Lemma 3,∏k−3

i=1 xi > k+1k

kk−1 · · · 5

4 = k+14 , then | pk−4pk−3| =

π/∏k−3

i=1 xi = O(1/k). We now place the remaining n − k − 1 points, which arebelow pk−3pk−2. They are further restricted to be (arbitrarily) distributed in an ob-tuse triangle pk−3pk−2pk−1 with pk−3pk−2 being the longest edge. We also re-quire pid for k ≤ i ≤ n − 3 must intersect pk−4pk−3. Since pk−4pk−3 has alreadybeen added to GT(P ), no point below pk−4pk−3 can be visible to any point aboveit. Since any edge we can add is bounded above by | pk−4pk−3| = O(1/k), thelength of any triangulation of the region bounded by pk−4pk−3pk−2pk−1 is boundedabove by O(1/k) × �(n − k) = O(n/k). In order to complete the triangulation,we also need to add the edges on convex hull whose length is bounded by π , then|GT| = O(k) + O(n/k) + π . Choose k = √

n, then |GT| = O(√

n).A larger weight triangulation LWT(P ) is easier to compute. We connect d to

p1,p2, . . . , pk−2 and all other n − k − 1 points (pk−1,pk+2, . . . , pn−3). Each of firstk − 2 edges is longer than 1/2 and we have the following observation on the re-maining n − k − 1 edges. Through d we make a line perpendicular to pk−3pk−2intersecting pk−3pk−2 at f . This is valid because d is very close to the centerof C. Since |pk−3pk−2| = O(1/k), |df | >

√1 − 1/k2. Since each of the n − k − 1

remaining edges is bounded below by√

1 − 1/k2, we have |LWT| > (k − 2)/2 +(n− k − 1)

√1 − 1/k2 = �(n) for k = √

n even if LWT is incomplete. It follows that|MAT|/|GT| = �(

√n). �

Lemma 3 For sufficiently large k and ε = 1kk , if a sequence x1, x2, . . . , xk−2 is de-

fined as x1 = (k + 1)/k, and xj = 1/(2 − xj−1) + ε, for j = 2,3, . . . , k − 2, then

1 <k − (j − 2)

k − (j − 1)< xj <

k − (j − 1)

k − j − εkj+ ε for j = 2,3, . . . , k − 2.

Proof First note that 1 < k − j − εkj for j = 2,3, . . . , k − 2. We prove xj <k−(j−1)

k−j−εkj + ε by induction. For j = 2, it is easy to verify x2 < k−1k−2−εk2 + ε. As-

suming xj <k−(j−1)

k−j−εkj + ε holds for j = i − 1, we show that it also holds for j = i

for sufficiently large k,

xi (13)

= 1

2 − xi−1+ ε (14)

<1

2 − (k−(i−2)

k−(i−1)−εki−1 + ε)+ ε (15)

= k − (i − 1) − εki−1

k − i − 2εki−1 − ε(k − (i − 1) − εki−1)+ ε (16)

<k − (i − 1)

k − i − εki+ ε. (17)

J Comb Optim (2010) 19: 31–42 37

We then prove k−(j−2)k−(j−1)

< xj also by induction. For j = 2, it is easy to verifyk

k−1 < x2. Assuming that k−(j−2)k−(j−1)

< xj holds for j = i − 1, we show that it alsoholds for j = i,

xi (18)

= 1

2 − xi−1+ ε (19)

>1

2 − xi−1(20)

>1

2 − k−(i−3)k−(i−2)

(21)

= k − (i − 2)

k − (i − 1). (22)

�

From Lemmas 1 and 2, we conclude

Theorem 4 For any integer n ≥ 0, there exists a set P of n points in general positionsuch that |MAT(P )|

|GT(P )| = �(√

n).

We next construct examples to show that |MAT(P )||GSTT(P )| = �(

√n) and |MAT(P )|

|MSTT(P )| =�(

√n). When degeneracies are allowed, we reuse the example of Fig. 1(a). The max-

imum spanning tree of GT(P ) and maximum spanning tree of P are the same (referto Fig. 1(b)), so |GSTT| = |GT| and |MSTT| = |GT| even if the optimal algorithm isused for completing the triangulation. In non-degenerate case, we reuse the examplein Fig. 2. It is not hard to see that edges ab, bp1,pjpj+1 for j = 1,2, . . . , k − 4 alsobelong to MSTT(P ) and GSTT(P ), thus |GSTT(P )| and |MSTT(P )| are asymptoti-cally the same as |GT(P )| since no point below pk−4pk−3 can be visible to any pointabove pk−4pk−3. Thus, the approximation lower bound will not change. Hence, wehave

Theorem 5 For any integer n ≥ 0, there exists a set P of n points in general positionsuch that |MAT(P )|

|GSTT(P )| = �(√

n), and |MAT(P )||MSTT(P )| = �(

√n).

3 A constant-factor approximation for MAT

3.1 An almost four-approximation algorithm

The diameter of P is the longest segment with both endpoints in P . For points in gen-eral position, we present a new triangulation algorithm, which approximates MAT(P )

within a small constant factor. We call it Spoke Triangulation algorithm (ST in short).Refer to Fig. 3 and Algorithm 1.

38 J Comb Optim (2010) 19: 31–42

Fig. 3 Spoke triangulation(subset R)

Algorithm 1 Spoke triangulation1: Compute the diameter ab of P . Put D = |ab|. Let R = {pi | pi lies to the left

of the line ab} and S = {qi | qi lies to the right of the line ab}. Let n1 (resp. n2)denote the cardinality of R (resp. S). Then n1 +n2 = n− 2. Since at most one ofR and S can be empty, without loss of generality, we assume that R is not empty.Connect a to b.

2: At least one of∑n1

i=1 |pia| and∑n1

i=1 |pib| is larger than n1D/2 since∑n1i=1 |pia| + ∑n1

i=1 |pib| > n1|ab| by triangle inequality. Without loss of gen-erality, we assume

∑n1i=1 |pia| > n1D/2 and connect all pi to a. Similarly, we

connect all qi to b assuming∑n2

i=1 |qib| > n2D/2.3: We now have a spanning tree (called Spoke Spanning Tree or SST in short) of P

and we complete the triangulation as in Graham’s scan algorithm for computingconvex hull. The algorithm for triangulating the region to the left of ab is givenin lines 4–11. The region to the right of ab is triangulated similarly.

4: Sort p1,p2, . . . , pn1 according to the angle piab (from largest to smallest an-gles), which results in a sequence p′

1,p′2, . . . , p

′n1

. Let p′n+1 denote b.

5: Add p′1 in the list Llef t .

6: for i = 2 to n1 + 1 do7: Append p′

i to Llef t and link p′i to p′

i−1.8: while Llef t contains more than two points and the last three points

(p′j ,p

′k,p

′l , l < k < j ) in Llef t make a left turn (i.e., ∠p′

jp′kp

′l > π in quadri-

lateral p′jp

′kp

′la). do

9: Connect p′j to p′

l and remove p′k from the list.

10: end while11: end for

We are to prove the approximation bound of our algorithm. The first step is togive a bound for any MAT. Since any triangulation has at most 3n − 6 edges, theobvious upper bound for the weight of any triangulation is (3n− 6)D where D is thediameter of point set. Motivated by de Berg (2003), we here include the proof for atighter bound on the weight of maximum weight triangulation.

J Comb Optim (2010) 19: 31–42 39

Fig. 4 Almost tight bound forany triangulation

Lemma 6 For any triangulation T(P ) of P with a diameter D, |T(P )| < 2nD +1.75

√nD − 3.3D.

Proof For every triangle i in the triangulation T(P ), we have π(i ) = hidi/2,where π(·) denotes the area, di denotes the length of the longest edge in i , andhi denotes the corresponding height (with respect to di ) of i . Clearly, hi < di . Let|i | denote the weight (i.e., the perimeter) of i . By triangular inequality, |i | < 2hi

+ 2di . Since∑

i∈T(P ) π(i ) ≤ πD2/4 (Scott and Awyong 2000),∑

i∈T(P ) hi · di

≤ πD2/2. Since hi < di , we have∑

i∈T(P ) h2i < πD2/2. Denote by κ the number

of triangles in T(P ). Applying Claim 7, we have∑

i∈T(P ) hi <√

πκD√2

.

|T(P )| = 12

∑i∈T(P ) |i |+ 1

2Cconv where Cconv denotes the perimeter of convexhull of P . Recall that the perimeter of the convex hull is bounded above by πD (Scottand Awyong 2000). We have

|T(P )| <1

2

∑

i∈T(P )

{2di + 2hi} + πD

2

≤∑

i∈T(P )

{D + hi} + πD

2<

∑

i∈T(P )

{D} +√

πκD√2

+ πD

2. (23)

Since we have at most 2n − 5 triangles in T(P ), it follows that |T(P )| < 2nD +1.75

√nD − 3.3D. �

Note that the above upper bound is almost tight: if we place the interior n − 3points very close to an vertex in an equilateral triangle, then the weight of MAT isalmost 2nD. Refer to Fig. 4. Such an example is also observed in Chin et al. (2004).

Claim 7 Given hi > 0, x > 0, and

h21 + h2

2 + · · · + h2n < x2,

one has

h1 + h2 + · · · + hn < x√

n.

40 J Comb Optim (2010) 19: 31–42

Proof We prove the claim by induction. It is easy to verify the fact when n = 2: since(h1 + h2)

2 = h21 + h2

2 + 2h1h2 ≤ 2(h21 + h2

2) < 2x2, we have h1 + h2 <√

2x.Assuming that the claim holds for n = j − 1, we are to prove it is the case

for n = j . Since∑j−1

1 h2i < x2,

∑j

1 h2i < x2 + h2

j . Thus, we need to show that∑j

1 hi <√

x2 + h2j

√j . Noting that

∑j−11 hi < x

√j − 1 or

∑j

1 hi < x√

j − 1 + hj

by induction, it remains to show that x√

j − 1 + hj ≤√

x2 + h2j

√j , which can be

easily verified. �

Theorem 8 Spoke Triangulation algorithm properly triangulates a set of n pointsP (in general position) in O(n logn) time and |MAT(P )|/|ST(P )| < 4 + ε, whereε = 3.5√

n.

Proof It is obvious that the algorithm properly triangulates P . We now bound therunning time of ST(P ). Step 1 requires O(n logn) time. Step 2 requires O(n)

time. Running time of step 3 (including steps 4–11) is bounded above by sortingp1,p2, . . . , pn1 which takes O(n logn) time. Therefore, the total running time ofST(P ) is O(n logn).

We are to bound the approximation ratio. ST(P ) is composed of diameter, pia, qib

and the edges added in step 3. Suppose that there exists pt ∈ R such that |pta| < D/2which means |ptb| > D/2. An example point is p′

2 in Fig. 3. According to ST, inthe final triangulation there always exists a path from pt to b containing no edge in{pia, qib, ab}. An example path is p′

2p′3b in Fig. 3. The length of the path is obvi-

ously longer than |ptb| which means |ST| > D/2 + D + ∑n1i=1 |pia| + ∑n2

i=1 |qib| >1.5D + (n1 + n2)D/2 = nD/2 + D/2.

Otherwise, there is no such point, i.e., for all pi ∈ R, |pia| > D/2. According toST, there must exist an empty triangle abpk in the final triangulation (for example,abp′

5 in Fig. 3), then |ST| > D + |pka| + |pkb| + ∑pi∈R−{pk} |pia| + ∑n2

i=1 |qib| >

nD/2 + D/2.In either case, by Lemma 6 we have

|MAT(P )||ST(P )| <

2nD + 1.75√

nD − 3.3D

nD/2 + D/2< 4 + ε, (24)

where ε = 3.5√n

and ε approaches zero as n grows. Note that a similar result has beengiven in Chin et al. (2004), but our proof is simpler. �

Observation 9 For a convex polygon P , |MAT(P )| < nD + 0.15D.

Proof We have n edges on the convex hull and n−3 edges in its interior. The perime-ter of the convex hull is bounded above by πD according to Scott and Awyong(2000). Then |MAT(P )| < (n − 3)D + πD < nD + 0.15D. �

Since |MAT(P )|/|ST(P )| < (nD + 0.15D)/(nD/2 + D/2) < 2, ST(P ) approx-imates MAT(P ) of convex polygon P within a factor of two and can be computedin O(n logn) time compared with O(n3) time needed to compute MAT exactly byusing dynamic programming. Thus we have shown

J Comb Optim (2010) 19: 31–42 41

Fig. 5 Lower bound forLemma 10: (a) ST(P )

(b) MAT(P )

Theorem 10 If P is a set of n points in convex position, ST(P ) properly triangulatesP in O(n logn) time and |MAT(P )|/|ST(P )| < 2.

Note that the above bound on approximation ratio is sharp since the exampleshown in Fig. 5 indicates that |MAT(P )|/|ST(P )| can be arbitrarily close to two.In the example, every point lies on a circle and two of them are the endpoints of thediameter. Half of the remaining points are near one endpoint of the diameter, and theother half are near the other endpoint.

4 Conclusion

This paper studies the planar maximum weight triangulation problem. We first provethat in the worst case maximum greedy triangulation, maximum greedy spanningtree triangulation and maximum spanning tree triangulation heuristics do not pro-vide a constant factor approximation for the maximum weight triangulation. We thenpresent the Spoke Triangulation whose length is always within a small constant fac-tor from the maximum. Our proof is simpler compared to the previous work (Chin etal. 2004). The future work is to design a polynomial-time approximation scheme forMAT(P ) of a general planar point set P . It is also interesting to give an NP-hardnessproof for the maximum weight triangulation problem.

References

Chin F, Qian J, Wang C (2004) Progress on maximum weight triangulation of a point set. In Proceedingsof the annual international conference on computing and combinatorics (COCOON), pp 53–61

de Berg M (2003) Personal communicationHeath L, Pemmaraju S (1994) New results for the minimum weight triangulation problem. Algorithmica

12(6):533–552Hu S (2003) A constant approximation algorithm for maximum weight triangulation. In Proceedings of

Canadian conference on computational geometry (CCCG), pp 150–154Levcopoulos C, Krznaric D (1996) Tight lower bounds for minimum weight-triangulation heuristics. Inf

Proc Lett 57(3):129–135Levcopoulos C, Krznaric D (1998a) Quasi-greedy triangulations approximating the minimum weight tri-

angulation. J Algorithms 27(2):303–338Levcopoulos C, Krznaric D (1998b) A linear-time approximation scheme for minimum weight triangula-

tion of convex polygons. Algorithmica 21(3):285–311Lingas A (1987) A new heuristic for the minimum weight triangulation. SIAM J Algebr Discrete Methods

8:646–658Mulzer W, Rote G (2006) Minimum weight triangulation is NP-hard. In Proceedings of the 22nd annual

ACM symposium on computational geometry (SoCG), pp 1–10Plaisted D, Hong J (1987) A heuristic triangulation algorithm. J Algorithms 8(3):405–437Qian J, Wang C (2004) A linear-time approximation scheme for the maximum weight triangulations of a

convex polygon. Algorithmica 161–172

42 J Comb Optim (2010) 19: 31–42

Remy J, Steger A (2006) A quasi-polynomial time approximation scheme for minimum weight triangula-tion. In Proceedings of the 38th ACM symposium on theory of computing (STOC), pp 316–325

Scott P, Awyong P (2000) Inequalities for convex sets. J Inequal Pure Appl Math 1(1):6Wang C, Chin F, Yang B (1999) Maximum weight triangulation and graph drawing. Inf Proc Lett

70(1):17–22