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Approximating the Pathwidth of Outerplanar Graphs 1 Introduction

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Approximating the Pathwidth of Outerplanar Graphs 1 Introduction Approximating the Pathwidth of Outerplanar Graphs y y z Ra jeev Govindan Michael A Langston and Xudong Yan Abstract Pathwidth is a wellknown NP Complete graph metric Only very simple classes of graphs such as trees are known to p ermit practical pathwidth algorithms We present a technique to approximate the pathwidth of outerplanar graphs Our al gorithm works in linear time is genuinely practical and pro duces solutions at most three times the optimum Keywords algorithms pathwidth outerplanar graphs approximation tree de comp osition Introduction Pathwidth was dened by Rob ertson and Seymour in their seminal series of pap ers on Graph Minors Since then this metric has found application in many areas ranging from circuit layout to natural language pro cessing Determining pathwidth is NP Complete Thus it is natural to search for fast approximation algorithms No p olynomialtime relative approxima tion algorithm one whose solution is within a multiplicative constant of the optimum is known for the general problem Moreover no p olynomialtime absolute approximation algorithm one whose solution is within an additive constant of the optimum can exist unless P NP The main result of this pap er is a practical relative approximation algorithm for the path width problem on outerplanar graphs Since outerplanar graphs have treewidth two or less the metho ds in can in principle b e used to compute the pathwidth exactly in p olynomial time This is not a realistic option however b ecause of the high degree of the p olynomial and its enormous multiplicative constant In contrast our algorithm approximates the pathwidth to within a factor of three of the optimum in practical linear time This research is supp orted in part by the Oce of Naval Research under contract NJ y Department of Computer Science University of Tennessee TN z Prism Solutions Inc Hamlin Court Sunnyvale CA Our Approach Tree and Path Decomp ositions We consider only connected graphs without lo ops or multiple edges A tree decomposition of a graph G is a pair T Y where T is a tree and Y fY j i V T g i is a collection of subsets of V G such that i for each edge e E G some Y contains b oth i endp oints of e and ii for all i j k V T if j is on the path b etween i and k in T Y Y Y i k j The width of a tree decomp osition T Y is one less than the size of the largest set in Y The treewidth of G denoted tw G is the smallest width of all its tree decomp ositions A path decomposition of G is a sequence X X of subsets of V G such that i for 1 r each edge e E G some X contains b oth endp oints of e and ii for i j k r i X X X The width of a path decomp osition X X is one less than the size of the i k j 1 r largest set X i r The pathwidth of G denoted pw G is the smallest width of all its i path decomp ositions A Conversion Pro cedure Path decomp ositions can b e derived from tree decomp ositions We employ such a pro cedure tdp d and prove its correctness It requires a routine to construct optimal path decomp osi tions of trees For this we use the lineartime metho d presented in Since the runtime of tdp d is dominated by the time sp ent in this routine tdp d runs in linear time as well Pro cedure tdp d Input A tree decomp osition T Y of a graph G Output A path decomp osition of G b egin pro cedure X X an optimal path decomp osition of T 1 r for i r do [ Y P j i j 2X i output P P 1 r end pro cedure Theorem Let T Y denote a widtht tree decomp osition of a graph G Then tdp dT Y returns a path decomp osition of G with width no more than t pw T Pro of Let X X denote the optimal path decomp osition of T constructed in tdp d 1 r [ and let P P denote the output of tdp d Then for i r jP j j Y j 1 r i j j X i t pw T Thus the width condition is satised and we only need to check that P P is a valid path decomp osition of G 1 r It is easy to see that P P covers all edges in G We prove by contradiction that 1 r P P has the intersection prop erty If the intersection prop erty do es not hold then for 1 r some i j k r there is a vertex v in P P that is not in P Since v P P i k j i k there must exist l X and m X such that v b elongs to Y and Y Consider the subsets i k l m [ [ V and V of V T where V X X and V X X The intersection prop erty 1 2 1 p j 2 p j pj pj of X X implies that V and V are disjoint Moreover there is no edge in T connecting 1 r 1 2 V and V b ecause some X must contain b oth endp oints of such an edge contradicting the 1 2 q disjointness of V and V Thus every path b etween V and V in T contains a vertex from 1 2 1 2 X In particular the path b etween l and m must contain a vertex say h from X By the j j intersection prop erty of T Y v Y Since h X Y P and v P a contradiction h j h j j Path Decomp ositions of Outerplanar Graphs A graph is outerplanar if it has a planar embedding with all vertices lying on a single face Outerplanar graphs have treewidth at most two In this section we develop an algorithm that for an outerplanar graph G constructs an optimal tree decomp osition T Y with pw T pw G By Theorem running tdp d on T Y pro duces a path decomp osition with width at most pw G We say that T Y is simple if T Y has width at most two T is a subgraph of G and v Y for all v V T Since our algorithms use vertex lab els we insist that the lab els of v V T resp ect those of V G Because pathwidth cannot b e increased by taking a subgraph if T Y is simple then pw T pw G Our algorithm constructs T Y by combining tree decomp ositions of Gs subgraphs Let T Y and T Y denote tree decomp ositions of subgraphs G and G resp ectively Supp ose that V T and V T are disjoint and that there are vertices u V T and v V T such that all the vertices in V G V G are in b oth Y and Y Then we may obtain a tree decomp osition of G G by adding the edge uv This u v decomp osition is simple if T Y and T Y are simple and if uv E G E G Biconnected Graphs We concentrate initially on biconnected graphs those without cut p oints Lemma Let G b e biconnected outerplanar and of order at least three Let v denote a vertex in G Then G contains a path P with at least two edges and with endp oints w and x such that the following conditions hold G P fw xg is biconnected outerplanar and contains v w and x are adjacent in G and hence in G P fw xg and every edge in G is either in P or in G P fw xg Pro of If G is a cycle then the lemma is satised by setting P to G fuv g where u is a vertex adjacent to v Otherwise x an outerplanar layout of G Let E denote the set of internal i edges of G those not on the external face Orient the layout so that some edge e is rightmost and some edge e is leftmost in E If v is to the left of e then setting P to the path consisting i of all the external edges to the right of e satises the lemma Otherwise set P to the path containing all the external edges to the left of e If a path P contains at least two edges and has endp oints w and x then it has a widthtwo tree decomp osition T Y such that T P fw xg and for i V T Y fx i j g where j i is the neighbor of i on w s side the sets Y actually form a path decomp osition of P We call i T Y a w extensible tree decomp osition of P Figure shows a path and its w extensible tree decomp osition The sets Y are shown inside the ovals i x 1 2 3 4 5 w {x,1,2} {x,2,3} {x,3,4} {x,4,5} {x,5,w} Figure A path and its w extensible tree decomp osition Note that for every edge ij E P either fi j g Y or fi j g Y We use the notion i j of extensibility to derive b coptd our algorithm to construct simple tree decomp ositions of biconnected outerplanar graphs Pro cedure b coptd Input A biconnected outerplanar graph G of order two or more and a vertex v in G Output A simple tree decomp osition T Y of G with T spanning G fv g b egin pro cedure if jV Gj then b egin u the vertex adjacent to v T fug and Y fY g where Y fu v g u u end else b egin P a path b etween some two vertices w and x that satises Lemma 0 0 T Y b coptdG P fw xg v 0 if fw xg Y w then b egin e the edge incident on w in P 00 00 T Y the w extensible tree decomp osition of P end else b egin e the edge incident on x in P 00 00 T Y the xextensible tree decomp osition of P end 0 00 0 00 T T T feg and Y Y Y end output T Y end pro cedure At this p oint we may as well assume that v is chosen at random A sp ecic choice of v is necessary when G is a biconnected comp onent of a larger graph see Section Lemma Let G b e biconnected and outerplanar and let v denote a vertex in G Let T Y denote the result of the call to b coptdG v Then T Y is a simple tree decomp osition of G and T is a spanning tree of G fv g Pro of We prove using induction on jE Gj a somewhat stronger result We show that T Y is simple that T spans G fv g and that for each edge ij in G either i V T with fi j g Y i or j V T with fi j g Y The lemma holds for the basis case in which G contains just one j edge If jE Gj let P with endp oints w and x denote a path that satises Lemma Let G denote G P fw xg Thus v is in G By the induction hypothesis b coptdG v returns a simple tree decomp osition T Y with T spanning G fv g and with fw xg Y w or fw xg Y Assume without loss of generality that fw xg Y Let T Y denote x w the w extensible tree decomp osition of P Then T P fw xg and fw xg Y where a a is
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