Approximating the Pathwidth of Outerplanar Graphs
y y z
Ra jeev Govindan Michael A Langston and Xudong Yan
Abstract
Pathwidth is a well known 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 de ned 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 olynomial time relative approxima
tion algorithm one whose solution is within a multiplicative constant of the optimum is known
for the general problem Moreover no p olynomial time 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 O ce of Naval Research under contract N J
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
end p 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 end p 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
td p d and prove its correctness It requires a routine to construct optimal path decomp osi
tions of trees For this we use the linear time metho d presented in Since the run time of
td p d is dominated by the time sp ent in this routine td p d runs in linear time as well
Pro cedure td p 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 width t tree decomp osition of a graph G Then td p d T 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 td p d
1 r
[
and let P P denote the output of td p 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 satis ed 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
p j p j
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 end p 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 td p 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 G s 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 satis ed 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 satis es 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 width two
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 c op td our algorithm to construct simple tree decomp ositions of
biconnected outerplanar graphs
Pro cedure b c op td
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 G j
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 satis es Lemma
0 0
T Y b c op td G 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 x extensible 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 eci c 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 c op td G 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 G j 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 G j let P with endp oints w and x denote a path that satis es Lemma Let
G denote G P fw xg Thus v is in G By the induction hypothesis b c op td G 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 w s neighbor in P T is formed by adding an edge b etween vertex w in T and vertex a in
T The only vertices common to G and P are w and x which are contained in b oth Y and
w
Y Therefore T Y is a valid tree decomp osition of G T Y is simple b ecause T Y and
a
T Y are simple with the edge w a existing in G T spans G fv g b ecause T spans G fv g
and T spans P fw xg To complete the induction observe that for each edge ij E G
fi j g Y or fi j g Y b ecause either fi j g fw ag Y or fi j g is contained in one of
i j
a
Y Y Y and Y
i j i j
E ciency
We store the input graph in doubly linked adjacency list format This is space e cient b ecause
outerplanar graphs have a linear number of edges if tw G then jE G j jV G j We
also employ a few additional links To facilitate the removal of an edge ab links are maintained
b etween the copy of b in a s adjacency list and the copy of a in b s adjacency list The only steps
in b c op td that take more than constant time are i nding a path P that satis es Lemma
ii deleting the edges and internal vertices of P from the input graph and iii constructing
an extensible tree decomp osition of P Of these steps ii and iii take at most linear time
over all calls to b c op td Thus the question of e ciency reduces to the implementation of
step i One fast metho d is describ ed b elow
Some prepro cessing is required We rst construct an outerplanar layout of G We scan
the layout in a clo ckwise direction starting at v and number vertices in the order in which
they are encountered Then we rearrange the adjacency list of each vertex a so that neighbors
numbered lower than a o ccur b efore neighbors numbered higher than a Each of these tasks
takes only linear time
Once prepro cessing is completed paths to play the role of P are found during a second
clo ckwise scan It follows from Lemma that until G is reduced to a cycle a pair of vertices
may b e the endp oints of P if and only if they are adjacent by an internal edge and all vertices
with numbers b etween them have degree two Vertices of degree three or more are maintained
on a stack As a new vertex is scanned we check whether it is adjacent by an internal edge to
the vertex on top of the stack If it is then we have found P s endp oints If not we push the
new vertex on the stack and continue the scan
It turns out that no vertex will b e pushed on the stack as long as an internal edge makes
it adjacent it to a lower numbered vertex This in turn implies that G is reduced to a cycle
b efore the scan returns to v Thus the scan terminates after a linear number of steps and
we only need argue that each step can b e accomplished in constant time Let k denote the
vertex b eing scanned and j the vertex on top of the stack We need to check whether j and
k are adjacent by an internal edge Since G is outerplanar and since j cannot b e adjacent to
a lower numbered vertex by an internal edge either j is adjacent by an internal edge only to
k or k is adjacent by an internal edge to no lower numbered vertex other than j In the rst
case j has degree at most three In the second j can only b e one of the rst two elements in
k s adjacency list Therefore we need to scan at most ve elements in the adjacency lists of j
and k Thus step i requires only linear time and so do es b c op td
Tackling Non Biconnected Graphs
We now generalize our algorithm to handle all outerplanar graphs
Pro cedure op td
Input An outerplanar graph G of order two or more and sets B and C of its
biconnected comp onents and cut p oints
Output A simple tree decomp osition T Y of G with T spanning G
b egin pro cedure
if G is biconnected
then b egin
u v any two adjacent vertices in G
0 0
T Y b c op td G v
0 0
T T fv g fuv g and Y Y fY g where Y fv g
v v
end
else b egin
B an element of B that contains exactly one vertex v from C
i
if v is not a cut p oint in G B fv g then C C fv g
i
0 0
T Y b c op td B v
i
00 00
T Y op td G B fv g B fB g C
i i
u an arbitrary neighbor of v in B
i
0 00 0 00
T T T fuv g and Y Y Y
end
output T Y
end pro cedure
Lemma Let G b e outerplanar Let T Y denote the result of the call to op td G Then
T Y is a simple spanning tree decomp osition of G
Pro of The pro of pro ceeds by induction on the number of biconnected comp onents of G The
basis case when G is biconnected follows from Lemma and the mo di cations made to T Y
after the call to b c op td G v So let B v T Y T Y and u b e as de ned in op td
i
Let G denote G B fv g From the pro of of Lemma we know that T Y is simple
i
that it spans B fv g and that fu v g Y there is no Y By the induction hypothesis
i
u v
T Y is a simple spanning tree decomp osition of G Thus by construction T Y is a simple
spanning tree decomp osition of G
Main Result
Biconnected comp onents and cut p oints can b e found using a depth rst search Pro cedure op
td builds an optimal tree decomp osition using b c op td This tree decomp osition is converted
into a path decomp osition using td p d Recalling Theorem and noting that each of the
aforementioned steps requires only linear time we achieve the following result
Theorem If G is outerplanar a path decomp osition of G with width at most pw G
can b e constructed in linear time
Concluding Remarks
We have implemented our algorithm in the C programming language Tests on a SPARC
ULTRA indicate that the implementation is fast in practice taking for instance less than
two seconds to compute the path decomp osition of a graph with ten thousand vertices It is
di cult to gauge the quality of the solutions pro duced b ecause there is no practical way to
obtain optimal path decomp ositions for comparison As a compromise we tested the program
on pseudo random outerplanar graphs of known pathwidth These tests indicate that the
approximate decomp ositions tend to have much smaller width than the worst case guarantee
Our work has exploited the fact that if the width of a tree decomp osition T Y of G
is b ounded and if pw T is within some constant multiple of pw G then we can construct a
path decomp osition of G whose width is at most a constant times pw G Series parallel graphs
also have treewidth at most two Optimal tree decomp ositions for them can b e constructed
quickly We b elieve that for these graphs it is p ossible to ensure pw T pw G yielding a
factor of six relative approximation algorithm
On a more general note we conjecture that any graph G has an optimal tree decomp osition
T Y such that pw T pw G If true a constructive pro of of this would provide a relative
approximation algorithm for any class of graphs whose b ounded width tree decomp ositions can
b e found e ciently Currently this class includes all graphs of treewidth four or less and for
any xed k k chordal graphs k outerplanar graphs and graphs with disk dimension k to name
just a few
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