c 2014 Society for Industrial and Applied Mathematics

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1 SIAM J. COMPUT. Vol. 43, No. 1, pp c 2014 Society for Industril nd Applied Mthemtics ON A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION ARMAN YOUSEFI AND NEAL E. YOUNG Astrct. Minimum-weight tringultion (MWT) is NP-hrd. It hs polynomil-time constnt-fctor pproximtion lgorithm, nd vriety of effective polynomil-time heuristics tht, for mny instnces, cn find the exct MWT. Liner progrms (LPs) for MWT re well-studied, ut previously no connection ws known etween ny LP nd ny pproximtion lgorithm or heuristic for MWT. Here we show the first such connections: For n LP formultion due to Dntzig, Hoffmn, nd Hu [Mth. Progrmming, 31 (1985), pp. 1 14], (i) the integrlity gp is constnt, nd (ii) given ny instnce, if the forementioned heuristics find the MWT, then so does the LP. Key words. integrlity gp liner progrmming, minimum-weight tringultion, pproximtion lgorithm, AMS suject clssifictions. 68W25, 68U05, 90C59, 68Q25, 90C10, 90C35, 68W40, 90C05, 90C27, 90C90, 90C08 DOI / Introduction. In 1979, Grey nd Johnson [16] listed minimum-weight tringultion (MWT) s one of dozen importnt prolems known to e neither in P nor NP-hrd. In 2006 the prolem ws finlly shown to e NP-hrd [29]. The prolem hs suexponentil time exct lgorithm [33], s well s polynomil-time pproximtion scheme (PTAS) for rndom inputs [19]. Currently it is not known whether, for some constnt c>1, finding c-pproximtion is NP-hrd, ut this is unlikely, s qusi polynomil-time pproximtion scheme exists [32]. MWT hs n O(log n)- pproximtion lgorithm [31] nd, most importnt here, n O(1)-pproximtion lgorithm clled QusiGreedy [25]. The constnt in the ig-o upper ound from [25] is stronomiclly lrge. If restricted to simple polygons, MWT hs well-known O(n 3 )-time dynmicprogrmming lgorithm [18, 23]. Polynomil-time lgorithms lso exist for instnces with constnt numer of shells [2] nd for instnces with only constnt numer of vertices in the interior of the region R to e tringulted [17, sect ]; lso see [20, 5, 34, 24]. Liner progrm (LP) of Dntzig, Hoffmn, nd Hu for MWT. Liner-progrmming methods re primry prdigm for the design of pproximtion lgorithms. For mny hrd comintoril optimiztion prolems, especilly so-clled pcking nd covering prolems, the polynomil-time pproximtion lgorithm with the est pproximtion rtio is sed on liner progrmming, either vi rndomized rounding or the priml-dul method. The design of good pproximtion lgorithm is often synonymous with ounding the integrlity gp of n underlying LP. MWT hs severl strightforwrd liner-progrmming relxtions. Studying their Received y the editors August 13, 2012; ccepted for puliction (in revised form) Septemer 25, 2013; pulished electroniclly Jnury 21, An extended strct of this pper ppered in the proceedings of SODA 2012 [39]. Computer Science Deprtment, University of Cliforni, Los Angeles, Los Angeles, CA (yousefi@cs.ucr.edu). The reserch of this uthor ws prtilly funded y GAANN fellowship. Deprtment of Computer Science nd Engineering, University of Cliforni, Riverside, Riverside, CA (nel.young@ucr.edu). The reserch of this uthor ws prtilly funded y NSF grnts nd

2 26 ARMAN YOUSEFI AND NEAL E. YOUNG integrlity gps my led to etter pproximtion lgorithms, or my widen our understnding of generl methods nd their limittions (s stndrd rndomized rounding nd priml-dul pproches my e insufficient for MWT). Dntzig, Hoffmn, nd Hu [8] introduce the following LP (presented here s reformulted y [10]). Below, denotes the set of empty tringles. 1 R denotes the region to e tringulted, minus the sides of tringles in. The LP sks to ssign nonnegtive weight X t to ech tringle t so tht, for ech point p in the region, the tringles contining it re ssigned totl weight 1: (1.1) minimize c(x) = t c(t)x t suject to X R 0 nd ( p R) t p X t =1. Aove, R 0 is the set of vectors of nonnegtive rels, indexed y the tringles in. The cost c(t) of tringle t is the sum over the edges e in t of the cost c(e) oftheedge, defined to e e /2 (the length of e), unless e is on the oundry of R, inwhichcse the cost is e. (Internl edges re discounted y 1/2 since ny internl edge occurs in either zero or two tringles in ny tringultion.) R s specified is infinite, ut cn esily e restricted to polynomil-size set of points without wekening the LP. (E.g., let R contin, for ech possile edge e, twopointsp nd q, echononesideof e nd very ner e.) For the simple-polygon cse, the ove LP finds the exct MWT (every extreme point hs 0/1 coordintes, nd so corresponds to tringultion). This ws shown y Dntzig, Hoffmn, nd Hu [8, Thm. 7], then (pprently independently) y De Loer et l. [10, Thm. 4.1(i)] nd Kirsnov [22, Cor ]. For summries of these results, see [11, Chp. 8] nd [36]. Kirsnov descries n instnce ( 13-gon with point t the center) for which this LP hs integrlity gp just ove 1, s well s instnces (50 rndom points equidistnt from center point) tht re solved y the LP ut not y the heuristics descried lter. Other uthors hve considered edge-sed LPs, minly for use in rnch-ndound [26, 27, 30, 35, 3]. These edge-sed LPs hve unounded integrlity gps. LPs for mximl independent sets, which re well-studied, re closely relted to ll of the ove LPs, s tringultions cn e defined s mximl independent sets of tringles (or of edges). The ove LPs enforce some, ut not ll, well-studied inequlities for mximl independent sets. It is known to e NP-hrd to determine whether there exists tringultion whose edge set is suset of given set S [28]. For given set S, if we chnge the cost function in the ove LP to c(x) = e/ S t e X t, the LP will hve zero-cost integer solution iff there is such tringultion. Unless P = NP, this implies tht the LP with tht cost function hs unounded integrlity gp. 2 Thus, ny ound on the integrlity gp of the LP with the MWT cost function must rely intrinsiclly on tht cost function. Similrly, given n ritrry frctionl solution X, itisnp-hrd 1 Tht is, tringles lying in the region to e tringulted, whose vertices re in the given set of points, ut otherwise contin none of the given points. 2 If the LP hs ounded integrlity gp, it hs zero-cost frctionl solution iff it hs zero-cost integer solution.

3 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 27 to determine whether there is n integer solution in the support of X. 3 These re ostcles to stndrd rndomized-rounding methods. First new result: Integrlity gp is constnt. We show tht LP (1.1) hs constnt integrlity gp. This is the first nontrivil upper ound on the integrlity gp of ny MWT LP. To show it, we revisit the nlysis of QusiGreedy [25], which shows tht QusiGreedy produces tringultion of cost O( mwt(g) ), where mwt(g) is the length of the MWT of the given instnce G (nd lso the cost of the optiml integer solution to the LP). We generlize their rguments to show tht there exists tringultion of cost O(c(X )), where c(x ) is the cost of the optiml frctionl solution to the LP. Our nlysis lso reduces the pproximtion rtio in their nlysis y n order of mgnitude, ut the pproximtion rtio remins lrge constnt. MWT heuristics. Much of the MWT literture concerns polynomil-time heuristics tht, given n instnce, find edges tht must e in (or excluded from) ny MWT. Here is summry. Gilert [18] oserves tht the shortest potentil edge is in every MWT. Yng, Xu, nd You [38] extend this result y proving tht n edge xy is in every MWT if, for ny edge pq tht intersects xy, xy min{ px, py, qx, qy }. (We refer to the edges stisfying this property s the YXY-sugrph.) This sugrph includes every edge connecting two mutul nerest neighors. Keil [21] defines nother heuristic clled β-skeleton s follows. An edge pq is in the β-skeleton iff there does not exist point z in the point set such tht pzq rcsin(1/β). Thus, n edge pq is in the β-skeleton iff the interior of the two circles of dimeter β pq pssing through p nd q do not contin ny points. Keil [21] then shows tht for β 2, n edge tht is in β-skeleton is in every MWT. Cheng nd Xu [7] strengthen this to β 1/ sin k, where k π/3.1. Ds nd Joseph [9] show tht n edge e cnnot e in ny MWT if oth of the two tringles with se e nd se ngle π/8 contin other vertices. Drysdle, McElfresh, nd Snoeyink [15] strengthen this to ngle π/4.6. This property of e is clled the dimond property. Dickerson, Keil, nd Montgue [12] descrie simple locl-minimlity property such tht, if n edge e lcks the property, the edge cnnot e in ny MWT. Using this, they show tht the so-clled LMT-skeleton must e in the MWT. A primry use of the heuristics is to solve some instnces of MWT exctly in polynomil time s follows: Given n instnce, use the heuristics to identify edges tht re in the MWT. If the regions left untringulted y these edges re simple polygons (equivlently, if the edges spn the given points), then find the MWT of ech region independently using the stndrd dynmic-progrmming lgorithm. (The MWT will e the union of the MWT s of the regions.) According to [12], most rndom instnces with 40,000 points re solvle in this wy. Second new result: The LP generlizes heuristics. We show tht LP (1.1) generlizes these heuristics in tht if the heuristics solve given instnce s descried ove, 3 Given n ritrry suset S of the edges, the prolem of determining whether S contins tringultion reduces to the prolem of determining whether there is n integer solution in the support of given frctionl solution to the LP, s follows. Let set S consist of the empty tringles whose edges re in S, sos contins tringultion (y edges) iff S contins tringultion (y tringles). For ech tringle t S, solve the LP with the cost function tht gives t cost zero, every other tringle in S cost one, nd ll tringles not in S cost infinity. If (for ny t S )the LP for t hs no finite-cost fesile solution, then S contins no tringultion. Otherwise, for ech t S,letX t denote n optiml frctionl solution to the LP for t. Let X = t S Xt / S e the verge of these frctionl solutions. Becuse of the choice of the cost function, if given X t does not give positive weight to t, then no (integer) tringultion in S contins t. Thus, S contins tringultion iff there is tringultion in the support of X.

4 28 ARMAN YOUSEFI AND NEAL E. YOUNG then so does the LP (tht is, the extreme points of the LP re integer solutions incidence vectors of optiml tringultions). In this sense, the LP, whose formultion requires little explicit geometry, generlizes ll of these vried nd generlly incomprle heuristics. (In fct, the LP ppers to e stronger thn the heuristics in tht some nturl instnces re solved y the LP, ut not y the heuristics [22, sect. 3.5]. 4 ) This is the first connection we know of etween the heuristics nd ny MWT LP. Roughly speking, the heuristics re sed on comintion of (i) loclimprovement rguments out the MWT, nd (ii) logicl closure (once the heuristic determines the sttus of one edge with respect to the MWT, this in turn determines the sttus of other edges, nd so on). We extend these rguments to pply to the optiml frctionl tringultion X. This is possile ecuse (i) X looks loclly like n MWT, nd (ii) the LP enforces logicl closure of liner constrints on X. After we finished the ody of this work, we ecme wre of nd exmined dditionl heuristics y Wng, Chin, nd Xu [37] nd Aichholzer et l. [1]. We conjecture tht the LP generlizes them s well. An equivlent formultion of the LP. The following constrints re equivlent to the lst constrints in LP (1.1) (see, e.g., [10, Thm. 1.1(i), Prop. 2.5], [36], or [22, Thm ]) nd re useful for resoning out frctionl tringultions. For ny frctionl tringultion X nd edge e, (1.2) = [e oundry(r)]. t left(e) X t t right(e) X t Here left(e) contins the tringles tht contin e nd lie on one side of e, while right(e) contins the tringles tht contin e nd lie on the other side of e. (If e is on the oundry, tke right(e) =.) The nottion [x S] denotes 1 if x S nd 0 otherwise. Prcticl considertions. Using the O(n 2 ) constrints (1.2) insted of the constrints in (1.1) gives n equivlent LP with totl size (i.e., nonzeros in the constrint mtrix) proportionl to the numer of empty tringles. The empty tringles cn e identified, nd the LP constructed, in time proportionl to their numer [14]. Their numer is lwys O(n 3 ), ut often smller (e.g., O(n 2 )inexpecttionforrndomly distriuted points). The time to construct nd solve the LP cn e further reduced y preprocessing step sed on the heuristics remove ny vrile X t if the heuristics prove ny edge of t to e excluded from every MWT, nd dd constrint t left(e) X t = t right(e) X t = 1 if they prove tht n interior edge e is in every MWT. For rndomly distriuted points, only O(n) edges (in expecttion) hve the dimond property, forming O(n 2 ) possile empty tringles, from which the modified LMT-skeleton cn e computed in O(n 2 )-time [12, 13]. In our d hoc experiments on typicl instnces with 10 4 to 10 5 points, only smll numer of vriles were left undetermined y the heuristics. This llowed us to use stndrd liner-progrmming solvers to quickly solve the remining LP. (This is in keeping with the experiments of [12], which found tht most rndom instnces on 40,000 points were solvle y heuristics.) Similrly, this preprocessing should help integer liner-progrmming solvers to quickly find the MWT (for instnces for which the optiml solution is frctionl). It is known tht, symptoticlly, for n rndom points, the expected numer of remining vriles is Ω(n), ut the leding constnt is pprently stronomiclly smll [6]. Remrks. The results here suggest tht the LP of [8] cptures much of the structure of MWT. This suggests line of ttck for improving the pproximtion rtio: 4 Where G contins the center of unit circle nd n 1 rndom points on the circle.

5 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 29 use systemtic liner-progrmming methods such s rndomized rounding, the primldul method, nd lift-nd-project [4] to study the integrlity gp of the LP. Success would yield etter pproximtion (conceivly, even PTAS, using lift-nd-project). Filure would increse our understnding of the limittions of these techniques. Implicit in our ound on the integrlity gp is polynomil-time lgorithm with mtching pproximtion rtio. Actully, there re two. Both lgorithms first compute Levcopoulos nd Krznric s [25] convex prtition lk of the point set (see our Lemm 2.15), then extend lk y tringulting ech fce f of lk. The tringultion of ech f cn e done either () using the stndrd dynmic progrm to find n MWT of f, or () s follows: compute the frctionl solution X to the LP, then, for ech fce f of lk, trnspose X into frctionl tringultion X f of f (s descried in section 2.1), nd then use the chepest tringultion of f implicit in X f. Tht the first lgorithm ove is n O(1)-pproximtion lgorithm follows from Levcopoulos nd Krznric s previous work [25]. However, the ound we show here 54(λ + 1), whereλ is lrge constnt per Lemm 2.15 is sustntilly smller thn their previous ound. Roughly speking, we otin etter ound y nlyzing the trnsposl opertion t the level of tringles insted of edges. Open prolems. The integrlity gp is constnt, ut there is still huge gp etween the est lower ound known (rely ove 1.0) nd the upper ound shown here (stronomiclly lrge). The next step in improving our upper ound would e to reduce the vlue of λ in Lemm We suspect tht priml-dul nlysis is implicit in the nlysis here; mking the dul solution explicit might e step in this direction. Mny different cost functions (other thn the totl edge length) for tringultions re studied in the literture. The MWT LP extends nturlly y modifying the cost function or restricting the set of llowed tringles. (For exmple, the integrlity of the extreme points of the LP for the simple-polygon cse implies tht the simple-polygon result generlizes to ny liner cost function.) We conjecture tht results similr to those in this pper cn e otined for other cost functions. If MWT heuristics cn solve given instnce of MWT, then so cn the LP. However, the heuristics re lso useful for instnces tht they do not completely solve: in such instnces, the heuristics cn still identify some edges tht re in (or excluded from) every MWT, even if these do not completely determine the tringultion. Cn some nlogous property e shown for the LP? Tht is, is there some condition (e.g., sed on the optiml priml-dul solution to the LP) such tht, if the condition holds for n edge e, tht edge must e in (or excluded from) every MWT? Definition 1.1. The interior of segment pq is pq {p, q}. The interior of polygon P consists of P minus its oundry. Two sets properly intersect (or overlp, or cross) if the intersection of their interiors is nonempty. The (Eucliden) length of line segment pq is denoted pq. For ny set E of segments, E 2 denotes the totl length of segments in E. A plnr stright-line grph (PSLG) is n undirected grph G =(V,E) long with plnr emedding tht identifies ech vertex with plnr point nd ech edge with the line segment connecting its endpoints, so tht ech edge intersects other edges (nd V ) only t its endpoints. The length of G is the sum of the Eucliden lengths of its edges. G prtitions the plne into polygonl fces. 5 A fce or polygon is empty if its interior contins no vertex. 5 Where two points re in the sme fce if there is pth etween them tht intersects no edge, with the cvet tht the term fce excludes the single such unounded region.

6 30 ARMAN YOUSEFI AND NEAL E. YOUNG A digonl, or potentil edge, ofg is ny segment pq E connecting two vertices of fce, nd contined in tht fce, so tht G =(V,E {pq}) is still PSLG. A prtition of G is PSLG tht extends G y dding (noncrossing) digonls; equivlently, the fces of the prtition refine the fces of G. A convex prtition of G is prtition whose fces re empty nd strictly convex. The minimum-length convex prtition of G is denoted mcp(g). Atringultion of G is prtition whose fces re empty tringles. A frctionl tringultion X is fesile solution to the LP. For ny potentil edge e, theweight of edge e in X, denotedx e,is t e X t if e is on the oundry of the region to e tringulted, nd otherwise hlf this mount. Formlly, n instnce of MWT is specified y plnr point set V,implicitly defining PSLG G =(V,E), where E contins the edges on the oundry of the convex hull of V. A solution is minimum-length tringultion of G. Throughout, we fix n instnce G =(V,E) of MWT specified y given point set V. Unless stted otherwise, every grph considered is prtition of G. Since the vertex set V is the sme for ll such grphs, we identify ech prticulr grph y its edge set. 2. Integrlity gp is constnt. This section proves our first new result. Theorem 2.1. Given ny instnce G = (V,E) of MWT, for ny frctionl tringultion X, there exists n integer solution of vlue O(c(X)). Tht is, LP (1.1) hs constnt integrlity gp. Proof. Fix the MWT instnce G nd n ritrry frctionl tringultion X. Fix convex prtition cp of G. (Lter, we will fix cp to e prticulr convex prtition lk with some prticulr properties.) The ide of the proof is to define rounding procedure tht converts X into the desired integer solution. The procedure frctures X into seprte frctionl tringultion X f for ech fce f of cp (where X f covers exctly f). Then, independently within ech fce f of cp, the procedure replces the frctionl tringultion X f y the optiml integer tringultion of f. The finl rounded solution is then the union of these integer tringultions (one for ech fce f of cp), of totl cost t most f cp c(xf ) (nd, hopefully, O(c(X))). In the second step, since ech f is simple polygon, it follows from known results (see, e.g., [8, Thm. 7]; lso see the introduction) tht the cost of the optiml integer tringultion of f is t most the cost of X f. Thus, the integrlity gp will e O(1) s long s the first step tringultes the fces so tht f c(xf )=O(c(X)). The proof is divided into two prts: (i) descriing correct rounding procedure tht frctures X into frctionl tringultion X f for ech fce f of cp (we cll this trnsposing X into f), nd (ii) ounding the cost f c(xf )yo(c(x)) Prt (i) frcturing X into the fces of CP. Fix ny fce f of cp of the convex prtition cp. Our gol is to convert X into frctionl tringultion X f of f. We strt with the oservtion tht X, restricted to tringles tht cross f, cne seprted into independent lyers, where ech lyer is set of tringles tht uniformly covers f (nd possily some points outside f). We sy such lyer lnkets f. Definition 2.2 (lnket). A set B of empty polygons with endpoints in V lnkets the fce f if the union of the polygons contins f nd no two polygons overlp within f (they my overlp outside f). In this susection, the polygons in lnkets re lwys tringles. The next lemm descries how to decompose X (over f) intolnkets. Lemm 2.3. There exists set B of lnkets (ech contining only tringles)

7 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 31 x 1 x 2 Fig. 1. nd weights ɛ B > 0 for ech B Bsuch tht B B ɛ B =1nd, for every tringle t crossing f, X t = B B [t B] ɛ B. Recll tht [t B] is 1 if t is in B, else 0. Proof. Recll tht, for MWT instnces consisting of simple polygon, the LP gives optiml 0/1 solutions (see, e.g., [8, Thm. 7]). We dpt proof of tht property. Choose ny tringle t tht crosses f nd hs X t > 0. If t completely covers f, then stop nd tke B = {t}. Otherwise,someedgee of tringle t crosses the interior of f. Since e hs positive weight, there must e positive-weight tringle t tht hs e s n edge nd lies on e s opposite side (this is implied y constrint (1.2)). Glue t nd t together to form polygonl region. Continue in this wy, growing the polygonl region y repetedly gluing new tringle to ny oundry edge e tht crosses f. Stop when the region hs no oundry edge tht crosses f. The tringles glued together in this wy form the lnket B. Let ɛ B e the minimum weight of ny tringle in B. This gives the first lnket B nd its weight ɛ B. Sutrct ɛ B from ech X t for t B. This reduces X s coverge of f uniformly y ɛ B. To generte the remining lnkets in B (nd their weights), iterte this process s long s X still covers f with positive (nd necessrily uniform) weight. (The process does terminte, s ech itertion rings some X t to zero.) Fix the set B of lnkets of f from Lemm 2.3 nd the corresponding weights ɛ B. We next descrie how to convert ny single lnket B Bintotrue tringultion B f of f. The finl frctionl tringultion X f will e the convex comintion of these tringultions, where the tringultion of B is given weight ɛ B. Recll tht ny lnket B Bconsists of tringles tht together uniformly cover the convex fce f (nd my extend outside of f). To define the tringultion B f,we strt with edge trnsposls (see, e.g., Lemm 4.2 of [25]). For ny edge e tht crosses f, trnsposing e in f slides e to its trnsposl, denoted e f, digonl of f tht hs minimum length mong four or fewer digonls tht re ner e. Wegivetheforml definition next, nd then extend tht to define trnsposls of tringles nd lnkets B B. Definition 2.4 (trnsposing n edge [25]; see Figure 1). Fix ny tringle edge e = x 1 x 2 tht crosses f (tht is, tht intersects the oundry of f in two points or long n edge). The trnsposl of e in f, denoted e f, is defined y the following opertion: Clip the edge x 1 x 2 to chord x 1 x 2 =(x 1x 2 ) f of f. For ech endpoint x i of x 1x 2, if the endpoint lies in the interior of n edge e of f (s opposed to eing vertex of f), then slide x i long e to one of the endpoints of e, clled the destintion of x i. Otherwise (the endpoint is vertex of f), tke tht vertex s the destintion. Choose the destintions (for those where there is choice) to minimize the length of the digonl tht connects the destintions. (Brek ties consistently.) The resulting digonl is e f.

8 32 ARMAN YOUSEFI AND NEAL E. YOUNG Y Z Y Z Y Z Fig. 2. Next we define wht it mens to trnspose tringles nd lnkets. We give somewht uninformtive forml definition, then descrie the importnt properties. Definition 2.5. For ny tringle t, thetrnsposl of t in f, denotedt f,isthe convex hull of the endpoints of the trnsposls of the edges of t tht cross f. For ny lnket B B,thetrnsposl of B in f, denotedb f, is the set contining, for ech tringle t B, the trnsposl t f of t. Tht is, B f = {t f t B}. Consider lnket B B. By definition, the edges of tringles in B do not cross within f. But, priori, their trnsposls might. We next rgue tht this is not the cse. In fct, we prove more: roughly speking, tht trnsposing preserves the topology of the prtition tht B induces on f. More precisely, consider tht prtition, which comes from clipping the edges of the tringles in B into f sshowninfigure 2. Consider ny edge YZ. Focus on just those chords tht hve n endpoint in the interior of YZ. Order these chords, s shown in the left picture, ccording to the order of their endpoints on YZ going from Y to Z. For chords shring n endpoint on YZ, rek ties in fvor of chords tht len closer to Y. Lemm 2.6 (trnsposing preserves order). In the ove ordering of chords long YZ, ll chords whose endpoints hve trnsposl destintion Y precede ll chords whose endpoints hve destintion Z. (Informlly, when trnsposing the edges, when we slide the endpoints to their destintions, the endpoints tht slide to Y precede the endpoints tht slide to Z, sonocrossingsreintroduced.) Proof. Without loss of generlity, ssume tht Z lies (one vertex) clockwise of Y. Focus on the chords xw, wherex is in the interior of YZ. Let C Y contin those chords whose endpoint x hs destintion Y.LetC Z contin those whose endpoint x hs destintion Z. We show tht, if we leve Y nd trvel counterclockwise round the oundry to Z, we encounter the chords in C Y efore we encounter the chords in C Z. This proves the clim, ecuse, s chords in C do not cross, s we trvel counterclockwise we must encounter chords in the sme order tht we would if trveling clockwise from Y to Z. Consider the perpendiculr isector of YZ. Since f is convex, the isector intersects the oundry of f t single point cross from YZ. Suppose tht this intersection point is in the interior of some edge sshowninfigure3. (Ifthe intersection is vertex of f, tke to e tht point nd follow similr resoning.) As we trvel counterclockwise from Y to Z, until we pss the first endpoint of, every chord endpoint w tht we encounter is in some edge e of f tht lies entirely on the Y -side of the isector. Since oth endpoints of e re closer to Y thn to Z, no mtter which endpoint of e is the destintion of w, the destintion of x will definitely e Y. Thus, until we pss the first endpoint of, we encounter only chords in C Y. As we trvel through the interior of edge (or, if is point, through ) for ll chord endpoints w tht we encounter, their chords xw will hve the sme trnsposl (since x is in the interior of YZ nd w is in the interior of, nd trnsposing reks ties consistently). Thus, trveling through, either we encounter only chords in C Y, or we encounter only chords in C Z.

9 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 33 Y x Z Y w Z Fig. 3. c d e f x Y Z c d e f Z c d e f c d e f 1. Y 2. Y 3. Y 4. Y 5. Z Z c d Z Fig. 4. Once we pss the other endpoint of, until we rech Z, everychordthtwe encounter is n edge of f tht lies entirely on the Z-side of the isector, so, resoning s efore, we encounter only chords in C Z. Becuse trnsposing preserves order, the topologicl structure of the trnsposl of ny lnket B is inherited from the prtition tht B induces on f. Seetheexmple in Figure 4. Figure 4 shows five copies of fce f (with gry ckground). Copy 1 shows the fce lnketed y six tringles. In copy 2, the tringle edges re clipped to their chords in the fce, giving the prtition tht B induces on f. In copies 3 through 5, ech chord is shifted to its edge trnsposl y sliding ech endpoint to its destintion. Copy 5 shows the resulting edge trnsposls nd the trnsposl of B in f. Clerly, in the prtition tht B induces on f (copy 2) ech region is of the form t f for some t B. Becuse trnsposing preserves order, moving the edges of tht prtition to their trnsposls preserves the topologicl structure of the prtition: the trnsposl B f of B (copy 5) is convex prtition of f whose edges re the trnsposls of the edges of B, nd whose regions re the trnsposls of the tringles in B. Also, for ech tringle t B, the oundry of its trnsposl t f consists of the trnsposls of the edges of t, together with up to three edges of f. The trnsposl of lnket is convex prtition of the fce, ut not quite tringultion, ecuse ech of its regions my hve up to six sides. To get tringultion, we simply tringulte ech of its regions. Definition 2.7. The tringulted trnsposl of tringle t in f, denoted t f, is the MWT of the trnsposl t f,excepttht,ift f hs no re, then t f is the empty set. The tringulted trnsposl of lnket B in f, denoted B f,istheunionofthe tringulted trnsposls of the tringles in the lnket. The trnsposl B f is convex prtition of f whose regions re the trnsposls of the tringles in B, so the tringulted trnsposl of B indeed tringultes f.

10 34 ARMAN YOUSEFI AND NEAL E. YOUNG Finlly, we define the frctionl tringultion X f of f. We strt with the frctionl tringultion X. We restrict X to tringles crossing f. We decompose this restriction of X into convex comintion of lnkets of f (per Lemm 2.3). Then, in this convex comintion, we replce ech lnket B y its tringulted trnsposl B f, tringultion of f. 6 Here is the forml definition. Definition 2.8. Define the trnsposl of X in f, denotedx f, to e the frctionl tringultion of f formed y the convex comintion of the trnsposls of the lnkets in B so tht X f t = B B[t B f ] ɛ B. We now complete Prt (i) of the proof of Theorem 2.1. Lemm 2.9. Fix ny frctionl tringultion X nd ny convex fce f. The trnsposl X f of X in f defined ove is fesile frctionl tringultion of f. Tht is, it covers the points in f uniformly with weight 1. Proof. As discussed, this holds ecuse X f is convex comintion of tringultions of f. Indeed, it covers ech point p in f with totl weight [p t]x f t = [p t] [t B f ]ɛ B = ɛ B [p t] = ɛ B = 1. t t B B B B t B f B B The first equlity is y definition of X f. The second just exchnges the order of summtion. The third holds ecuse B f tringultes f (so exctly one t B f contins p). The lst follows y Lemm Prt (ii) ounding the cost. Fix the convex prtition cp nd frctionl solution X. By Lemm 2.9, for ech fce f of cp, the trnsposl X f s defined in Prt (i) is frctionl tringultion of f. To complete the proof of Theorem 2.1, we ound the sum of the costs of these frctionl tringultions. We strt y oserving tht we cn view X f s tking the weight of ech tringle t in X nd trnsferring tht weight to (every tringle in) the tringulted trnsposl t f of t in f. Fct X f t = X t. t : t t f Proof. X f t = [t B f ]ɛ B = B B B B [t t f ]ɛ B = t B t : t t f B B [t B]ɛ B = t : t t f X t. The first equlity is the definition of X f. The second holds y definition of Bf (nmely, t B f iff t t f for some t B). The third just exchnges the order of summtion. The lst follows from Lemm 2.3. So fr, we hve considered how lnket of tringles trnsposes into single f. Next we consider how single tringle t trnsposes cross multiple fces. Of course, given tringle t cn cross mny fces, ut in ll ut two its trnsposl will hve no re (nd thus will ply no prt in the tringulted trnsposl of X in f). 6 Alterntively, we could tke X f to e the chepest tringultion B f over ll lnkets B B.

11 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 35 X Y X 1 Y Z 8 Z Fig. 5. Lemm Any given tringle t crossestmosttwofcesf in cp in which its trnsposl t f hs positive re. Thus, for given t, onlytwofcesf hve c( t f ) > 0. Proof. Fix tringle t =ΔXY Z nd consider how the fces of cp cn overlp t. Sy tht fce f is ccommodting if t s trnsposl t f in f hs positive re. In the two exmples in Figure 5, ech dshed edge is n edge trnsposl of n edge of t. Within ech ccommodting fce, the (positive re) trnsposl of t is drk. We clim tht every ccommodting fce touches ll three edges of t. (A fce touches n edge if the intersection of the fce nd the edge, including oundries nd endpoints, is nonempty. For exmple, the ccommodting fce 2 on the left of the figure, nd 2 nd 5 on the right, touch ll three edges of t. Ech other fce is nonccommodting nd, except for 3 nd 4 on the right, touches only two edges of t.) The clim holds ecuse, if fce f touches only two edges of t, the third edge of t lies outside of f, so the two edges cross the interior of single edge of f. Thus, the two edges of t tht touch f must hve identicl trnsposls, forcing t f to hve no re. Now consider the cse tht t hs fce f tht touches the interior of ll three edges of t (s in Figure 5, left). Since the fces re nonoverlpping nd convex, no fce other thn f cn touch ll three edges of t. By the clim, then, only fce f might e ccommodting, so the lemm holds. Assume tht no fce touches the interiors of ll three edges of t. By the clim, ny ccommodting fce f still hs to touch ll three edges of t, ut now there is t lest one edge, sy XY,oftwhose interior f voids. Thus, f must touch XY t n endpoint, sy, Y. (For exmple, consider Figure 5, right. Fces 2, 3, 4, nd 5 touch ll three edges of t, ut not ll three interiors.) Since f touches XY t Y ut does not touch the interior of XY, there must e n edge wy of f tht extends through the interior of t. Sincew is not inside t, wy must cut cross t to the interior of the edge XZ.Thus, in this cse, ny ccommodting fce f must shre some vertex v with t, nd (2.1) n edge of the fce must extend from v cross the interior of t. If there re two ccommodting fces, they must extend n edge cross t from the sme vertex Y, for otherwise the extending edges would cross inside t. Now consider ll edges in cp tht extend from Y cross the interior of t. Let these edges e w 1 Y,w 2 Y,...,w k Y, rotting in order round Y. (In Figure 5, k =3.) cp hs k + 1 corresponding fces f 0,f 1,...,f k, lso in order rotting round Y,where f i 1 nd f i shre edge w i Y. By conclusion (2.1), only these k + 1 fces might e ccommodting. To finish, we oserve tht f i is not ccommodting unless i {0,k} (the first or lst fce). Indeed, for i {0,k} edges w i 1 Y nd w i Y of f i extend from Y cross t

12 36 ARMAN YOUSEFI AND NEAL E. YOUNG s Y Z f e e f ef e f s s Z Fig. 6. to XZ. Since these edges touch t Y, the trnsposl of t in f i is thus just the point Y. Thus, the trnsposl of t in f i hs no re. Our gol is to show tht trnsposing X cross the fces increses the cost of X y t most constnt fctor. For ny tringle t, y the lemm nd Fct 2.9.1, trnsposing X trnsfers the weight X t to the tringulted trnsposls t f of t in t most two fces f. To proceed we ound the cost of ech t f in terms of the cost of t. Recll tht t f is the MWT of its (nontringulted) trnsposl t f, which hs t most six sides (up to three edges of f nd up to three trnsposls of edges of t). We strt y ounding the cost of t f. Our ound depends on the sensitivity of the edges of the convex prtition cp, defined s follows. Definition 2.11 (sensitivity). An edge e is σ-sensitive if, for ny potentil edge e tht crosses e, for ech endpoint x of e, the distnce from x to the closest endpoint of e is t most σ e. In other words, the circle of rdius σ e round ech endpoint of e contins n endpoint of e. For the rest of this section, fix σ such tht ll edges of cp re σ-sensitive. Lemm For ny fce f of cp nd tringle t, the totl length of the edges in t s trnsposl t f tht re not lso edges of cp is t most 2σ times the length of t s edges. Proof. Letf e ny fce of cp nd e e ny edge tht crosses f. We clim tht the length of the edge trnsposl e f of e in f is t most 2σ times the length of e. This clim implies the lemm, ecuse ech edge of the trnsposl of t (ut not of cp) is the edge trnsposl e f of unique edge e of t. To finish, we prove the clim. For n edge e tht crosses f, one of the following three cses holds: (1) e is incident to one vertex of f nd properly intersects one s side of f (s in the left of Figure 6), (2) e properly intersects two sides s nd s of f (s in the right of Figure 6), or (3) e is incident to two vertices of f. In cse (1), let W e the vertex tht f shres with e (nd e f ). Since s is σ- sensitive, nd e crosses s, the endpoint W of e is t most σ e from some endpoint of s. Since e f is the minimum distnce etween W nd ny endpoint of s, this implies e f σ e. In cse (2), let Y e n endpoint of e nd let Z nd Z, respectively, e the closest endpoints of s nd s to Y. Becuse e f is the shortest segment from n endpoint of s to n endpoint of s, e f ZZ. By the tringle inequlity, ZZ YZ + YZ. Becuse s nd s re σ-sensitive, YZ nd YZ re ech t most σ e. In cse (3), the trnsposl e f of e is the sme s e, so the clim holds.

13 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 37 It is strightforwrd to extend the ound to the tringulted trnsposl t f of t. Recll tht the cost of tringle t is the sum of the costs of its edges, where the cost of n edge is hlf its length, unless the edge is on the oundry of the entire region, in which cse the cost of the edge is its length. The cost c( t f ) of tringultion t f is the sum of the costs of the tringles in the tringultion. Lemm For ny fce f nd ny tringle t, thecostc( t f ) of the tringulted trnsposl of t in f is t most three times the cost c(t f ) of the (nontringulted) trnsposl of t in f. Proof. Recll tht t f hs t most six vertices, sy, v 1,v 2,...,v k, ordered clockwise. Tringulte t f y dding up to three interior digonls connecting the odd vertices (e.g., v 1 v 3, v 3 v 5, v 5 v 1 ). The totl length of the dded digonls is t most the totl length of the oundry. Likewise, the sum of costs of the dded digonls is t most the sum of the costs of the edges on the oundry of t f. Ech dded edge occurs in two tringles in this tringultion, wheres ech oundry edge occurs in just one tringle. Thus, dding the digonls gives tringultion of cost t most three times the cost of t f. The lemm follows, s c( t f ) is the minimum cost of ny tringultion of t f. Next we gther the ounds in the previous lemms to ound the totl cost cross ll the fces. We re not finished, s the ound depends on not only the cost of the frctionl tringultion, ut lso the totl length of the edges in the convex prtition cp nd the sensitivity σ of those edges. Lemm The totl cost f c(xf ) is t most 3 cp 2 +12σc(X). Proof. The totl cost is c(x f )= X t c( t f ) y Fct 2.9.1, f,t f 3 [c( t f ) > 0] X t c(t f ) y Lemm 2.13, f,t 3 cp 2 + 6σ t,f 3 cp 2 +12σ t [c( t f ) > 0] X t c(t) y Lemm 2.12, X t c(t) y Lemm 2.10, = 3 cp 2 +12σc(X) y definition of c(x). To proceed further, we need convex prtition whose edges hve constnt sensitivity nd totl length O(c(X)). Levcopoulos nd Krznric hve shown the existence of something close: convex prtition lk whose edges re 4.45-sensitive nd hve totl length O(mcp(G)) (recll tht mcp(g) is the minimum-length convex prtition of G). Lemm 2.15 (see [25]). For some constnt λ>0, for ny MWT instnce G, there exists convex prtition lk of G, whose edges re 4.45-sensitive, hving totl length lk 2 λ mcp(g) 2. Proof. Levcopoulos nd Krznric show tht wht they cll the qusi-greedy convex prtition hs the ove properties: for the first property, see their Lemm 5.4 nd the discussion efore it; for the second property, see their Corollry 5.3 [25]. This convex prtition will work for us: we prove next tht mcp(g) 2 18 c(x). (Note tht mcp(g) 2 is trivilly t most the cost of ny integer tringultion, ut the ound here concerns the frctionl tringultion, so requires proof.)

14 38 ARMAN YOUSEFI AND NEAL E. YOUNG Fig. 7. e v X e e = The proof uses the constrints on X nd leverges previous nlysis of mcp(g) due to Plisted nd Hong [31, Lemm 10]. Lemm mcp(g) 2 18 c(x). Proof. For every vertex v in the interior of the convex hull of the vertex set V, define str t v to e suset of edges incident to v such tht no two successive edges (round v) re seprted y n ngle of 180 degrees or more. For every vertex v on the oundry of the convex hull of V, define the (only) str t v to consist of the two oundry edges incident to v. Let S min (v) denote the minimum cost of ny str t v. Plisted nd Hong show mcp(g) 2 6 v V S min(v) [31, Lemm 10]. We clim S min (v) (3/2) e v X e e, wherex e is t e X t if e is on the oundry of the convex hull; otherwise X e is hlf this mount. As v 2 e X e e =2c(X), the clim implies the lemm. We prove the clim. It is esy to see tht, for ny oundry vertex v, S min (v) e v X e e, sowe restrict our ttention to just n interior vertex v nd its edges. Becuse X stisfies constrint (1.2), rotting clockwise round v, thereisse- quence v 1,v 2,...,v k of distinct vertices such tht for ech i =1,...,k, the tringle vv i v (i+1) mod k hs positive weight in X. (To find the sequence, tke ny positiveweight tringle tht hs v s vertex. Let v 1 nd v 2 e the other vertices, in clockwise order. By constrint (1.2), there is positive-weight tringle tht shres edge vv 2 nd lies clockwise to tht edge. Let v 3 e the other vertex of tht tringle. By constrint (1.2), there is positive-weight tringle tht shres edge vv 3 nd lies clockwise to tht edge. Let v 4 e the other vertex of tht tringle. Continue, stopping when the next vertex v i tht would e dded is v 1 this must hppen y constrint (1.2).) Let e i nd t i denote edge vv i nd tringle vv i v (i+1) mod k, respectively. Note tht ech edge, nd ech tringle, is distinct. Cll the sequence of edges h =(e 1,e 2,...,e k ) helix. Letwrp(h) denote the numer of times h wrps round v. By stndrd construction the X e s cn e expressed s liner comintion of incidence vectors of helices. (Similr to Lemm 2.3 s proof, repetedly find helix h, choose weight ɛ h, nd sutrct ɛ h /wrp(h) from ech tringle X ti in the helix, reducing coverge ner v y ɛ h.) This gives proility distriution ɛ on helices h such tht ech X e = h [e h]ɛ h/wrp(h). Now choose helix h t rndom from the proility distriution ɛ. Prtition h greedily into contiguous susequences of edges such tht ech group g is mximl suject to the constrint tht the totl clockwise ngle round v swept y the group s edges is t most 360. (In Figure 7, white tringles seprte groups.) Ech group contins str, nd (s neighoring groups re seprted y t most 180 )therere t lest 360 wrp(h)/( ) = 2wrp(h)/3 groups. From the rndomly chosen h, choose group g uniformly t rndom from h s

15 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 39 first 2wrp(h)/3 groups. For ny given edge e, the proility tht e is in g is t most h [e h]ɛ h/(2 wrp(h)/3) = (3/2)X e. Thus, y linerity of expecttion, the expected totl length E[ g 2 ]ofedgesing is t most (3/2) e v X e e. On the other hnd, every g contins str, so S min (v) E[ g 2 ]. This proves Lemm For the rounding procedure, fix the (previously ritrry) convex prtition cp to e the prtition lk from Lemm The cost of the finl tringultion is t most c(x f ) ecuse ech fce f is simple, f 3 lk 2 +12σc(X) y Lemm 2.14, 3λ mcp(g) 2 +54c(X) y Lemm 2.15, 3λ 18 c(x)+54c(x) y Lemm 2.16, = 54(λ +1)c(X). Hence, the integrlity gp is t most 54(λ + 1), completing the proof of Theorem The LP generlizes heuristics. This section proves our second new result (the LP generlizes MWT heuristics). Here is summry of heuristics for determining tht given potentil edge e = xy of G is in every MWT of given MWT instnce G =(V,E): β-skeleton: For β =1/ sin k where k π/3.1, there does not exist point z in the point set such tht xzy rcsin(1/β) π/3.1. Equivlently, the two disks of dimeter β e hving e s chord re empty of points. If this condition holds, then e is in every MWT of G [21, 7]. YXY-sugrph: For every potentil edge pq tht crosses e = xy, itssize e is t most min{ px, py, qx, qy }. If this condition holds, then e is in every MWT of G [38, 18]. mximlity: For every potentil edge tht crosses e, tht edge is known to e excluded from every MWT. If so, then e is in every MWT of G (see, e.g., [12]). Here is summry of heuristics for determining tht given potentil edge e of G (not on the oundry of the region to e tringulted) is excluded from every MWT of G: independence: Some potentil edge tht crosses e is known to e in every MWT. If this condition holds, then e is not in ny MWT of G (see e.g., [12]). dimond: Neither of the two tringles with se e nd se ngle π/4.6 isempty. If this condition holds, then e is not in ny MWT of G [9, 15]. LMT-skeleton: For every two tringles t nd t for which e is loclly miniml, oneof the edges of t or t is known to e excluded from every MWT. If this condition holds, then e is not in ny MWT of G [12]. (Edge e is loclly miniml for two tringles t nd t if t t = e nd t nd t together re minimum-length tringultion of the qudrilterl Q = t t tht is, either Q is nonconvex, or e is not longer thn the other digonl of Q.) Let E denote the set of edges tht cn e deduced to e in every MWT y pplying the logicl closure of the ove six rules. (Logicl closure is necessry ecuse the mximlity, independence, nd LMT-skeleton conditions depend on the known sttus of edges other thn e. For exmple, if one of the conditions implies tht some edge e is excluded from every MWT, then the LMT-skeleton condition my then in turn imply tht some new edge e is excluded from every MWT, ecuse e lies on one of two tringles t or t in the pir for which e is loclly miniml.)

16 40 ARMAN YOUSEFI AND NEAL E. YOUNG Idelly, the set E gives prtition of G in which every fce is empty. If this hppens, then the remining edges in the MWT cn e found y tringulting ech remining fce independently using the stndrd dynmic-progrmming lgorithm, nd we sy G is solvle vi the heuristics. According to [12], most rndom instnces with s mny s 40,000 points re solvle vi the heuristics. 7 Here is our second new result. If n instnce is solvle vi the heuristics, then LP (1.1) solves the instnce lso. Theorem 3.1. For ny instnce G of MWT, let E e the prtition of G defined ove. If every fce of E is empty, then every optiml extreme point of the LP (for G) is the incidence vector of minimum-length tringultion. The reminder of this section gives the proof. The first step is to show tht ech condition ove tht ensures tht n edge is in (or excluded from) every MWT lso ensures tht the LP gives the edge weight 1 (or 0) in ny optiml frctionl solution. Sy tht LP (1.1) forces potentil edge e to z (where z {0, 1}) if, for every optiml frctionl tringultion X of G, theweightthtx gives to e is z. Lemm 3.2. If ny of the following conditions hold, the LP forces potentil edge e of G to 1: 1. β-skeleton: The β-skeleton condition ove holds for e. 2. YXY-sugrph: The YXY-sugrph condition ove holds for e. 3. mximlity: The LP forces every potentil edge tht crosses e to 0. Proof ide. Prt 3 is reltively strightforwrd: if X gives weight 0 to every edge tht crosses e, then no tringle t tht crosses e hs positive Xt, so the only wy X cn cover points ner e is with tringles tht hve e s side. The originl β-skeleton nd the YXY-sugrph heuristics re shown to e vlid for MWT y locl-improvement rguments: if the condition holds for n edge e tht is not in the MWT, then polygon P covering e within the MWT cn e retringulted t lesser cost, contrdicting the optimlity of the MWT [21, 7, 38, 18]. Here we extend those rguments to ny optiml frctionl tringultion X : if the condition holds nd X does not give e weight 1, then polygon P covering e whose tringles hve positive weight in X cn e retringulted (lowering the weight of those tringles y ɛ>0 nd rising the weight of other tringles y ɛ), giving frctionl tringultion tht costs less thn X. The originl rguments re intricte geometric cse nlyses, typiclly tking severl pges. The rguments do not extend completely to our setting for the following reson: in the MWT setting, the polygon P identified for retringultion is the union of noncrossing tringles, wheres here, in the frctionl setting, the polygon P is the union of tringles tht cn cross (much s in Lemm 2.3). If the tringles in P do not cross, then the originl rguments pply, ut in generl dditionl nlysis is needed. To illustrte, consider the β-skeleton. Suppose for contrdiction tht the β- skeleton condition holds for n edge e = ut e does not occur in the MWT. Previous works [21, 7] show tht there must e sequence t 1,t 2,...,t m of empty tringles in the MWT whose union P covers e, s shown on the left in Figure 8. Using the β-skeleton condition, [21, 7] show tht this union hs tringultion tht costs less thn does t 1,...,t m, contrdicting the optimlity of the MWT. In the current context, if e hs weight elow 1 in X, then there must (similrly) exist sequence t 1,t 2,...,t m of empty tringles with positive weight in X covering e, 7 Reference [12] defines the modified LMT-skeleton to e the set of edges tht cn e deduced to e in every MWT vi (the logicl closure of) just the dimond, LMT-skeleton, mximlity, nd independence conditions ove. The use of logicl closure is crucil for the effectiveness of the LMT-skeleton.

17 A LINEAR PROGRAM FOR MINIMUM-WEIGHT TRIANGULATION 41 t1 t2 tm t1 tm Fig. 8. ut these tringles cn cross (see the exmple in Figure 8). We extend the rguments of [21, 7] to show tht, even if such crossing occurs, tringultion of lower cost cn still e found. Full proof. Here re the detils of the proofs for prt 1 (β-skeleton) nd prt 2(YXY-sugrph). Prt 3 (mximlity) hs een discussed lredy ove, in the proof ide. Prt 1 (β-skeleton): The originl β-skeleton heuristics re shown to e vlid for MWT y locl-improvement rguments: if n edge e is in the β-skeleton (for β 1/ sin(π/3.1)) ut not in the MWT, then polygon P covering e within the MWT cn e retringulted t lesser cost, contrdicting the optimlity of the MWT [21, 7]. We riefly sketch this rgument nd then extend it to ny optiml frctionl tringultion X. Assume for the reminder of this section tht e goes horizontlly from the point on the left to the point on the right. If is not in the MWT, there exists set of MWT edges tht intersect e. Let e 1,...,e n, e the set of edges indexed in nondecresing order of their length. If the edges re removed from the MWT, n empty polygonl region P results. In [21, 7] it is shown tht P cn e retringulted t lesser cost y set of edges tht contins. The ide is to generte sequence of tringulted polygons P 0,...,P n such tht P 0 is the degenerte polygon, P n is tringultion of P,ndP j 1 P j.tootinp j, P j 1 is expnded to include the endpoints v j nd v j of e j. Assume v j is ove the line through nd v j is elow it. If oth v j nd v j lredy lie on the oundry of P j 1, thenp j = P j 1. Otherwise, t lest one of them will not e on the oundry of P j 1. Assume without loss of generlity tht v j is not on the oundry of P j 1 (if v j is lso outside P j 1, it will e delt with similrly). Since v j is not on the oundry of P j 1, edge e j intersects oundry edge v i v k of P j 1. Consider the sequence δ of vertices on the pth from to on the oundry of P (there re two such pths, ut the one ove the line through is intended). On the sequence δ, vertexv i is the lst vertex efore v j tht elongs to P j 1,ndv k is the first vertex fter v j tht elongs to P j 1. This oservtion llows us to clerly define v i v k in the frctionl setting ecuse in tht setting, polygon P my e self-intersecting nd e j my intersect more thn one oundry edge of P j 1 in the hlf-spce ove. In generl, the tringle v i v j v k contins susequence δ 1 of vertices on δ from v i to v j nd nother susequence δ 2 from v j to v k. The polygon v i δ 1 v j δ 2 v k is then tringulted ritrrily, nd P j is the union of P j 1 nd the tringulted polygon v i δ 1 v j δ 2 v k nd possily nother tringulted polygon to include v j if v j is not on the oundry of P j 1. This construction is shown in Figure 9. The polygon with dshed oundry is P, nd P j 1 is tringultion of the drk gry polygon. The union of P j 1 nd ritrry