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A Polynomial-Time Approximation Scheme For A PolynomialTime Approximation Scheme for Weighted Planar Graph TSP z x y David Karger Philip Klein Michelangelo Grigni Sanjeev Arora Brown U Emory U Princeton U Andrzej Woloszyn Emory U O Abstract in n time where is any constant Then Arora and so on after Mitchell showed that a Given a planar graph on n no des with costs weights PTAS also exists for Euclidean TSP ie the subcase on its edges dene the distance b etween no des i and in which the p oints lie in and distance is measured j as the length of the shortest path b etween i and j using the Euclidean metric This PTAS also achieves Consider this as an instance of metric TSP For any O an approximation ratio in n time More our algorithm nds a salesman tour of total cost 2 O recently Arora improved the running time of his at most times optimal in time n O algorithm to O n log n using randomization We also present a quasip olynomial time algorithm The PTASs for Euclidean TSP and planargraph for the Steiner version of this problem TSP though discovered within a year of each other are quite dierent Interestingly enough Grigni et al had Intro duction conjectured the existence of a PTAS for Euclidean TSP The TSP has b een a testb ed for virtually every algorith and suggested one line of attack to design a PTAS for mic idea in the past few decades Most interesting the case of a shortestpath metric of a weighted planar subcases of the problem are NPhard so attention has graph We note later in the pap er that Euclidean TSP turned to other notions of a go o d solution An early can b e viewed as a Steiner subcase of weighted planar approximation algorithm by Christodes achieves an ap graph TSP proximation ratio on every instance of metric TSP 2 O In this pap er we present an n time PTAS namely one in which the interno de distances form a scheme for weightedplanargraph TSP Our approxima metric This algorithm has not b een improved up on tion scheme starts by extracting a spanner of the input and a general approximation scheme is unlikely since graph the spanner preserves distances b etween vertices metric TSP is MAXSNPhard Nevertheless there but has a lower sum of edgecosts Then like some pre has b een recent progress for certain sp ecial classes of vious results ab out approximation schemes on planar metrics unweighted graphs our approximation scheme Grigni Koutsoupias and Papadimitriou showed employs a technique for obtaining a hierarchical decom that if the metric is the shortestpath metric of an position of the input graph By hierarchical we mean unweighted planar graph that is all edge costs are that the decomp osition is done in steps A step parti one then a polynomial time approximation scheme or tions every current comp onent into two or more pieces PTAS exists It achieves an approximation ratio each of which contains at most a constant fraction of the vertices of its parent Hence the decomp osition tree aroracsprincetonedu Supp orted by NSF CAREER has O log n levels where n is the numb er of vertices Award Alfred Sloan Fellowship and Packard Fellowship The useful asp ect of our decomp osition is that we show y Emory Dept of Math Computer Sci Atlanta GA the existence of a approximate salesman tour Email micmathcsemoryedu z that crosses the b oundary of each region in the decom MIT Lab oratory for Computer Science Cambridge MA p osition O log n times This is reminiscent of the Supp orted by NSF contract CCR and an Alfred P Sloane Foundation Fellowship approximation scheme for the unweighted case In Email kargerlcsmitedu particular we ensure that after contracting the edges URL httptheorylcsmite du karg er that lie on the b oundaries of the regions we obtain a x kleincsbrownedu Supp orted by NSF grant CCR graph where each region is b ounded by O log n ver and a planar embedded graph G with edgecosts vertex tices Dynamic programming can b e used to nd the weights and faceweights nds a Jordan curve C such optimal tour in this contracted graph We show how to that lift this tour to a tour in the uncontracted graph without increasing the cost by much balance condition the interior and exterior of C have In Section we also present a quasip olynomial weight at most of the total weight time algorithm for the Steiner version of this problem faceedge condition C uses at most k face edges that is when we want a nearoptimal tour on some given subset S of the vertices ordinaryedge condition C uses ordinary edges of total cost O k times the total cost of al l the ordinary Denitions and pro cedures edges Spanner Our approach requires that the sum Remark The separating cycle also has the property of the edgecosts of the graph b e within a constant that the ordinary edges comprise at most two paths but factor of the cost of the minimum tour We use the the algorithm does not use this property following spanner result due to Althofer Das Dobkin Joseph and Soares to obtain such a lowcost subgraph Given a planar emb edded graph G and a separator of the input graph that also approximately preserves C satisfying the conditions of Theorem we split distances the graph into two or more derived subgraphs called Theorem For every planar graph G with edge children as follows 0 costs and every there is a subgraph G with the First in G we contract the ordinary edges that are same vertex set such that for every pair of vertices i and in C giving us a contracted planar emb edded graph 0 0 j in V their distance in G is at most times their G the ordinaryedge condition ensures that we do not distance in G Furthermore the sum of costs of edges contract much cost so we do not dramatically p erturb 0 0 appearing in G is O times the cost of a minimum the solution Let C b e the contracted version of the 0 spanning tree in G The subgraph G can be found in separating cycle it has no ordinary edges so the vertices polynomial time it contains if removed separate the graph the numb er of vertices is equal to the numb er of face edges in the Separating the graph Now we give our separa separator so the faceedge condition ensures that there tor theorem which can b e viewed as a generalization of are not to o many We call these vertices the boundary Millers simplecycleseparator theorem The pro of of vertices the theorem app ears in Section We dene the interior piece to b e the interior of 0 It is useful to ensure that our separator while not C together with the b oundary vertices The exterior a cycle in the original graph do es trace out a Jordan piece is similarly dened it to o contains the b oundary curve To capture the resemblance to a cycle we vertices The interior piece and exterior piece each use the notion of face edges Face edges are articial satisfy the following prop erty edges added to the graph while preserving planarity in other words the cycle may cross through the interior of Boundaryvertices prop erty The b ound some faces Our separator will b e a cycle in the graph ary vertices lie on the b oundary of a single face obtained by adding some of these articial edges We We call the single face a hole b ecause it results from call it a cycle separator Such a cycle traverses certain the removal of some of the graph vertices edges and faces of the graph the rest of the The connected comp onents of the interior piece and graph divides into an interior and an exterior the exterior piece are the children of G Because of the The separator is supp osed to achieve some sort of 0 balance condition on the separator the interior of C has balance We sp ecify what is to b e balanced by assigning vertex weight at most twothirds of the vertex weight of weights to vertices and faces the separator then has the G We obtain the following prop erty prop erty that the sum of weights of vertices and faces in the interior is only a constant fraction of the sum of Childweight prop erty Each child has ver all the weights and similarly for the exterior tex weight at most a constant fraction of that We measure the quality of the separator in two of the parent graph ways the numb er of face edges it uses and the sum of costs of the ordinary edges it uses Because all ordinary edges on the b oundary b etween Theorem Separator Theorem There the interior and exterior were contracted we obtain the is an p oly ntime algorithm that given a parameter k following prop erty edgedisjointness prop erty No edge of G obtain the following lemma app ears in more than one child 0 Lemma The edges of G that do not appear in G have total cost at most O P T Moreover the b oundaryvertices prop erty still 0 Clearly the optimal tour in G has length at most holds for each child the b oundary vertices that sep OPT Below we show how to use dynamic programming arate the child from its siblings all lie on the b oundary 0 to compute the optimal tour for G We then extend of a single face a hole this tour back to the original graph as follows We uncontract the contracted paths and use the classical A simpler algorithm doubleMST heuristic on each path to get a tour for To intro duce our techniques we b egin with a simpler those vertices of cost at most path length We then algorithm that runs in quasip olynomial time In the splice the path tour into the global tour Doing this following section we will show how to mo dify it to for all contracted paths raises the tour cost by at most achieve a p olynomial running time OPT so the the cost of the nal tour is at most Our goal is to nd a tour in the input graph that has OPT the other is taken by the spanner cost at most times that of
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