A Polynomial Time 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 i e the sub case
on its edges de ne 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 quasi p olynomial time algorithm
The PTAS s for Euclidean TSP and planar graph
for the Steiner version of this problem
TSP though discovered within a year of each other are
quite di erent 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 shortest path metric of a weighted planar
sub cases of the problem are NP hard 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 sub case of weighted planar
approximation algorithm by Christo des 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 weighted planar graph 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 MAXSNP hard Nevertheless there
but has a lower sum of edge costs 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 shortest path 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
arora cs princeton edu 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
E mail mic mathcs emory edu
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
E mail kargerlcs mit edu
particular we ensure that after contracting the edges
URL http theory lcs mit e du karg er
that lie on the b oundaries of the regions we obtain a
x
klein cs brown edu 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 edge costs vertex tices Dynamic programming can b e used to nd the
weights and face weights 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 quasi p olynomial
weight at most of the total weight
time algorithm for the Steiner version of this problem
face edge condition C uses at most k face edges that is when we want a near optimal tour on some given
subset S of the vertices
ordinary edge condition C uses ordinary edges of total
cost O k times the total cost of al l the ordinary
De nitions and pro cedures
edges
Spanner Our approach requires that the sum
Remark The separating cycle also has the property
of the edge costs 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 low cost 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 ordinary edge 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 face edge 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
Miller s simple cycle separator theorem The pro of of
vertices
the theorem app ears in Section
We de ne 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 de ned 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 arti cial
satisfy the following prop erty
edges added to the graph while preserving planarity in
other words the cycle may cross through the interior of
Boundary vertices 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 arti cial 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 two thirds 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
Child weight 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 n time algorithm that given a parameter k following prop erty
edge disjointness 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 oundary vertices 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
double MST 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 the optimal tour Let
subgraph approximation
OPT denote the cost of the optimal tour in the input
graph The rst step of our algorithm is to let G b e a
0
Finding the optimal tour in G in quasip oly
spanner of the input graph that preserves distances up
nomial time Now we describ e how to compute the
to a factor of using Theorem The sum of
optimum salesman tour on the contracted graph We
2 edge costs of G is O times the cost of a minimum
O log n log log n
show that the running time is n In
spanning tree in the input graph which is in turn at
the next section we use a more careful analysis to show
most OPT The cost of an optimal tour in G is at most
that p olynomial running time can b e achieved
times OPT Our goal is now to nd a tour in G
The recursive decomp osition of G using the cy
whose cost is at most O P T more than the optimal
0
cle separators induces a recursive decomp osition of G
tour
namely the decomp osition obtained by using the con
Next we x the parameter k c log n where
0
tracted versions C of the separators After the con
n is the numb er of vertices in the given graph and c is
tractions the numb er of vertices in a separator is equal
a constant to b e determined We assign weight one to
to the numb er of face edges in the separator So by
every vertex in G
the face edge condition each separator consists of at
Next we nd a recursive decomp osition of G using
0
most k vertices Consider a subgraph H of G that
the separator algorithm of Theorem we nd a
app ears at level d of the recursive decomp osition It
separator in G determine the children of G nd a
contains at most dk b oundary vertices and they col
separator in each child and determine its children and
0
lectively separate H from the rest of G Note that
so on until each remaining graph has a small numb er
dk O log n b ecause the depth of the decomp osi
of vertices By the child weight prop erty the depth of
0
tion is O log n The optimal tour in G winds in and
the recursion is O log n
out of H via the b oundary vertices The next lemma
Consider the graphs o ccuring at a particular level
guarantees that there is an optimal tour that passes
of the recursive decomp osition By the edge disjointness
through each b oundary vertex at most most twice
prop erty these graphs are edge disjoint Hence the sum
of edge costs over all these graphs is at most the sum
Lemma Patching Lemma Let J be a Jordan
of edge costs in G For each graph the contracted
curve in the plane Let be a closed curve in the plane
0
edges of the separating cycle used to divide the graph
Then we can modify into another closed curve that
have total cost O k times the sum of edge costs in
traverses the same points as but such that at every
0 0
that graph Summing over all these graphs the costs
point p where intersects J crosses J at most twice
of edges app earing in all these separators is O k
Pro of First by eliminating crossings we may assume
times the total cost in G i e O O P T k Summing
that do es not self intersect Then at a p oint of
over all levels of the decomp osition we obtain a b ound
intersection with J the curve alternates going in and
of O OPT log n k on the sum of costs of all edges
out of J We may then reconnect the segments of the
contracted By appropriate choice of the constant c in
curve so that at most two crossings are necessary See
the de nition of k we ensure that this b ound is at most
Figure 2
O P T
0 0
Thus the contracted graph G has a simple struc Let G b e the graph obtained from G by contracting
ture it p ossesses a hierarchical decomp osition such that all the primal separator edges in all the separators We
Jordan Curve
and that these b oundary vertices lie on the b oundaries
of only a constant numb er of holes This last fact
Salesman Tour
together with the planarity of an optimal tour allows
O k
us to prove a Catalan like upp er b ound on the
numb er of subproblems that we must actually consider
at H Putting this together we will have running time
2
O O k O
n n as claimed
The new assignment of weights is actually done dy
namically as we construct the recursive decomp osition
Figure If a planar salesman tour crosses a Jordan
Supp ose we are separating a graph H with total weight
curve more than two times at a b oundary vertex then
w When we nd a separator in H contract its edges
we can clearly reconnect the edges incident to the
and construct its children we create some new b ound
b oundary vertex so that the tour crosses at most twice
ary vertices for the children We assign weight w k to
each new b oundary vertex in a child Since each child
has only k b oundary vertices this adds no more than
each subgraph H arising in the decomp osition is delim
w to a child s total weight We also assign weight
ited by p O log n b oundary vertices and the op
w to the hole in each child on which the b oundary
timum salesman tour enters or leaves H at most twice
vertices lie
via each b oundary vertex
Before additional weight was assigned to b oundary
We use a simple dynamic program to nd the
w The vertices each child s total weight was at most
optimum tour as follows We build a lo okup table for
additional weight increases the child s weight by at most
each subgraph arising in the decomp osition starting
w so the child s weight is now at most w Thus the
from the leaf subgraphs of the decomp osition and
child weight prop erty still holds each child s weight is
working up to larger subgraphs For a subgraph H the
still a constant fraction of its parent s
corresp onding table contains an entry for every p ossible
Now consider the hierarchical decomp osition of the
way the tour can pass through H which can b e sp eci ed
0
graph G obtained by the contractions Let H b e a
by an ordered list of b oundary vertices through which
subgraph arising in the decomp osition First we b ound
the tour enters and exits H Each b oundary vertex
the numb er of b oundary vertices in H The weight of
can app ear at most twice so the numb er of entries is
every b oundary vertex in H is at least k times the
b ounded by the numb er of ways of ordering at most p
O p
weight of H s parent Since H has weight at most ve
items which is p
sixths that of its parent it must contain at most k
The value of an entry is the minimum cost of
b oundary vertices
a set of paths that realize the corresp onding pairing
Next we show that the b oundary vertices lie on a
while visiting all vertices inside H This value can
constant numb er of holes As the recursive decomp osi
b e calculated from similar tables for the children of
0
tion takes place a cycle separator is found for a sub
H Given the contracted cycle separator C used to
graph and its children are constructed each existing
separate H we enumerate the ways the set of paths
0
hole is either preserved if the cycle separator do es not
could cross C For each we lo ok up the minimum cost
pass through the face or destroyed if the cycle passes
realization within each child and then add these costs
through the face In the former case the hole retains
The minimum over all the ways the paths could cross
0
the weight originally assigned to it when it was formed
C is the desired value
and it app ears in one of the children of the subgraph In
To calculate each entry in H s table thus requires
O p O p
the latter case the b oundary of the hole is broken into
p time and there are p entries to ll for a
O p
two pieces and each piece makes up part of the b ound
total of p time We have to build a table for each
ary of the new hole b eing formed in the children Thus
subgraph in the decomp osition The total time and
2
O O p O log n log log n
in this case the old b oundary vertices which lined the
space required is n p n
b oundary of the old hole now lie on the b oundaries of
the new holes app earing in the children one new hole
A p olynomial algorithm
p er child We preserve the prop erty that all b oundary
We now improve the preceding algorithm to achieve a
vertices lie on the b oundaries of holes
p olynomial running time We mo dify the assignment of
We can now use an argument like that used ab ove
weights used in nding the cycle separators in order
in b ounding the numb er of b oundary vertices Let H
to ensure that for every subgraph H arising in the
b e a subgraph arising in the hierarchical decomp osition
decomp osition the numb er of b oundary vertices in
of the contracted graph Each hole has weight at least
H separating it from the rest of the graph is small
vertices and our earlier analysis b ounds the numb er of
O hk
noncrossing matchings among them by In other
words the numb er of choices for the matching M on
the original set of b oundary vertices is at most
O hk
numb er of nonself crossing trees on h p oints
(A) (B)
O h O hk
which is at most h
Figure The order in which a planar tour traverses m
The separator theorem
vertices on the b oundary of a Jordan curve corresp onds
Now we prove Theorem Our pro of uses ideas from
to to balanced arrangements of m pairs of parenthesis
Grigni Koutsoupias and Papadimitriou as well
Each b oundary vertex can contribute two parentheses
as Lipton and Tarjan s planar separator theorem
A shows an invalid pairing and B shows a valid
and Miller s simple cycle separator theorem The
pairing
p erformance criteria in our case are di erent from those
of Lipton and Tarjan and those of Miller so we must
use these techniques di erently
times the weight of H s parent and H s weight is
By rescaling we can assume without loss of gener
at most ve sixths that of its parent so H contains at
ality that all vertex and face weights sum to
most holes
The next lemma is b orrowed with slight mo di ca
Better constants can b e obtained by slightly more
tion from Lipton and Tarjan
complicated ways of breaking up the graph and assign
Lemma Lipton and Tarjan Let T be a spanning
ing weights
tree of a planar triangulated graph G with weights on
Enumerating the pairings To nish the descrip
vertices and faces totaling at most Then there is
tion of the algorithm we need to b ound the numb er
an edge not belonging to T such that the interior and
of ways in which an optimum salesman tour could link
exterior of the simple cycle in T e each contain weight
up the at most k b oundary vertices in a subgraph
no more than
Note that we can assume that the optimal tour do es
Furthermore the same result holds even if not every
not cross itself though it can overlap itself and passes
face is triangular as long as each non triangular face F
through each b oundary vertex at most twice
satis es the fol lowing two properties
If in fact all the b oundary vertices were on a single
face then we could nish with a Catalan b ound That
F has weight less than
is supp ose we have a subgraph with a unique hole with
The boundary of F is the unique simple cycle in
m b oundary vertices The p ortion of the tour on the
T e for some edge e
subgraph that is the exterior of the hole consists
of a non crossing set of paths which matches pairs of
The rst step of the separator algorithm is to
b oundary vertices some more than once but none more
nd the shortest path tree ro oted at some vertex r
O m
than twice The numb er of such matchings is
In the second step describ ed b elow we extract a
as may b e proven by relating this quantity to Catalan
subgraph of the graph and a corresp onding subgraph
numb ers Figure
of the shortest path tree we also add some face edges
However instead of a disk we have the subgraph
Finally we apply the lemma of Lipton and Tarjan to
emb edded on a sphere with h O holes in it The
obtain a simple cycle separator
numb er of b oundary vertices on the b oundary of each
hole is O k Again we want an upp er b ound on the
Extracting a subgraph Note that any p ositive
numb er of top ologically distinct ways to emb ed a non
value x determines a partition of the vertices those
crossing matching on this surface
whose distance from r is less than x and those whose
O h O hk
We claim an upp er b ound of h which
distance is at least x For any subset S of the edges of
O k
is in our case since h is constant This is
the graph let f x denote the numb er of edges in S
S
seen as follows Any emb edded matching M is a
that cross this partition
non crossing multigraph with the holes as its vertices
The multigraph is connected b ecause it represents a Lemma For any subset S of edges and any distances
salesman tour If we cut the surface along any spanning d d if f x k for every d x d then
S
subtree of this multigraph then the h holes merge into d d L k where L is the total cost of the edges
a single hole This single hole has O hk b oundary in S
B
Pro of An edge of length can only contribute to f x
S
an interval of length The lemma follows by simple
for C
averaging 2 C4
C1 C 3
Let S S denote the partition induced by cutting
d d
the shortest path tree at distance d The separator
algorithm will use the top ology of these partitions Note
there are at most n such partitions as we can
that C2
always take d to b e the distance of some vertex from
the ro ot
S is a The set of edges crossing the partition S
d d
co cycle which means that in the planar dual graph the
duals of these edges form a collection of edge disjoint
simple cycles Moreover we can view the emb edding of
Figure C is the deep est dual cycle containing weight
the planar graph in such a way that the interiors of these
more than Note that C may have high edge
dual cycles contain the no des at distance d or greater
cardinality B is C s deep est ancestor having edge
Let C denote this set of dual cycles Essentially we
d
cardinality less than k It contains weight more than
cho ose as our in nite region some face whose b oundary
The dual cycles C C are at the rst level
r
contains the ro ot r
deep er than B such that they have edge cardinality less
As shown in Figure the dual cycles are nested
than k
lies in some cycle of C for d d each cycle of C
d d
1 2
The minimal nontrivial dual cycles each enclose a single
vertex of the primal graph
B the deep est ancestor of C having such a small edge
De ne the depth of a dual cycle in C to b e the
d d
cardinality This contradicts the fact that C s depth
i
maximum d for which the cycle b elongs to C De ne
d
exceeds that of B 2
its edge cardinality to b e the numb er of dual edges
Obtain a graph G from G as follows For each i
comprising it For a collection of such cycles the total
consider the dual cycle C Corresp onding to this dual
edge cardinality is the sum of the edge cardinalities of
i
cycle place a circuit of face edges in G just inside the
the individual cycles i e the total numb er of dual edges
dual cycle where each face edge connects the endp oints
app earing in all of them De ne the weight contained
of a primal edge Then delete all other vertices enclosed
by a dual cycle to b e the sum of weights of vertices and
by this new circuit thus turning the circuit into a face
faces emb edded in the interior of the cycle
Assign the weight of the deleted no des to this face We
Let C denote a deep est dual cycle containing weight
call the new face a leaf face This pro cess is depicted
more than Let B the the deep est ancestor of C
in Figure
whose edge cardinality is less than k See Figure
Note that the circuits are vertex disjoint b ecause
Let d b e the depth of B Note that B contains weight
B
each one connects vertices within a di erent dual cycle
more than De ne d as
C and the interiors of the C s are disjoint Moreover
i i
by Lemma each leaf face has weight at most
d min fd d C has total edge cardinality k g
B d
The dual cycle B is treated similarly add a circuit
of face edges just outside this dual cycle connecting
and let C C b e the cycles in C that are in the
r
d
together those endp oints of the primal edges lying
interior of B
outside the dual cycle Then delete the other vertices in
The next lemma is a consequence of Lemma
the exterior of the dual cycle These deletions cause the
circuit to b e the b oundary of the in nite region which
Lemma d d L k where L is the total cost of
B
we call the root face The weight assigned to the ro ot
edges in the graph 2
face is the total weight of the exterior vertices deleted
Note that since B contained more than weight the
Lemma For each i C contains weight at most
i
weight of the ro ot face is at most
Pro of Supp ose some dual cycle C contained more
i
than half the weight It would then contain in particular Extracting the spanning tree for the sub
the dual cycle C since more than half of the weight is graph Starting with the shortest path tree T we ob
contained in C Thus C is an ancestor of C Since C tain a spanning tree T for the subgraph G we have ex
i i
has edge cardinality less than k C is an ancestor of tracted For each vertex that lies inside B and outside i
Cycles at depth d Cycles at depth d' > d nest
inside the cycles at depth d.
Figure This gure shows how cycles nest as distance increases
6
A leaf dual cycle We then delete the other The primal edges crossing We create a circuit of artificial edges vertices inside, and assign the dual cycle, and the primal by tracing along the dual edges. their weight to the new face.
subgraph inside it.
Figure The construction applied to each dual cycle C to obtain a leaf face
i
every C the vertex s parent edge in T is de ned to b e no edge connects two vertices from di erent leaf faces
i
the vertex s parent edge in T For each but one of the Thus the cycle separator consists of two paths
vertices in the circuit outside B we take its parent edge of original edges joined by at most k face edges
to b e the face edge connecting it to the next vertex in fewer than k from a leaf face fewer than k from
counterclo ckwise order The one exception is to b e the the ro ot face and p ossibly the non tree edge e By
ro ot of T Lemma each of the paths of original edges has length
We similarly treat the dual cycles C For each we at most L k Thus the separator has all the promised
i
determine the interior vertex closest to the ro ot and we prop erties
assign that vertex s parent edge to b e its parent edge
in T For each of the other interior vertices we take The Steiner version of planar TSP
its parent edge to b e the face edge connecting it to the
An obvious next problem to study is a Steiner typ e
next vertex in clo ckwise order
weighted planar graph TSP problem given a weighted
planar graph and a set of terminals which is a subset
Applying Lipton and Tarjan s lemma We
of the set of vertices nd a minimum cost circuit that
have obtained a subgraph G and a spanning tree T
visits all the terminals This problem is an obvious
We add more face edges to G to triangulate every face
generalization of the planar graph TSP and it also
except the ro ot and leaf faces just created and we apply
generalizes the Euclidean TSP To see the latter note
Lemma to obtain a cycle separator in T e where e
that given n p oints in on which we desire a salesman
n
is a non tree edge
tour we can draw all line segments b etween them
We claim that the cycle separator uses face edges
and add a new no de at each intersection This gives a
from at most one leaf face To see this note that the face
weighted planar graph on O n vertices the costs on
edges from each leaf face form a terminal path in T no
the edges are the Euclidean lengths in which it su ces
real edge has such an edge as an ancestor Furthermore
to nd a minimum cost circuit for the original set of n
since the leaf faces corresp ond to edge disjoint co cycles
vertices
Most of our techniques generalize to the Steiner the total cost of the edges in P v v is at most
i j
typ e problem except for the spanner result Theo k times the cost of al l edges in OPT
H
rem Thus the existence of a PTAS would follow
Each part of C v v has at most of the
i j
if the following conjecture is true
vertices of H
Conjecture There exists a function f such that
Now we describ e the approximation scheme Let k
given a weighted planar graph G and a subset
b e c log n where c is a constant to b e determined
0
S of vertices there exists an edge induced subgraph G
and n is the numb er of vertices in the input graph
which approximates al l internode distances in S
The approximation scheme makes use of a recursive
0
and furthermore G has total edge weight at most f
algorithm that takes three inputs a subgraph H of
times the minimum Steiner tree weight for S
the input graph G and an ordered set S of edges and
vertices that connect H to the remainder of G The
Even in the absence of a spanner result we can
output is an approximately min cost way of routing the
give a quasip olynomial time approximation scheme for
tour within H in such a way that it enters and leaves H
the Steiner variant by combining the techniques of this
via the elements of S in the sp eci ed order
pap er with that of Arora s original PTAS for Euclidean
The algorithm rst applies the separator algorithm
TSP In each step the algorithm tries a small
to H which yields O n separators sp eci ed by a way
p oly n size family of separators one of which is
vertex partition C v v and a set P v v of edges
i j i j
guaranteed to work The approximation scheme runs
p oly log n
forming two paths
in n time
For each
The approximation scheme uses a separator algo
such way partition A I nter ior E xter ior B the al
rithm that identi es not one but many separators one
gorithm rst contracts the edges in the corresp onding
of them must b e go o d in a sense to b e describ ed First
0
two paths obtaining two vertices x and y Let H b e
we describ e the algorithm for nding the set of separa
the contracted graph Next the algorithm enumerates
tors for a subgraph H of the input graph
0
orderings S of k subsets of edges within H together
The separator algorithm rst nds a shortest path
with the vertices x and y For each blo ck of the way
tree of H ro oted at an arbitrary vertex For each vertex
partition the algorithm makes a recursive call to nd a
v in H let d v denote the distance of v from the ro ot
0
subtour within the corresp onding subgraph of H that
Let v v v b e the vertices of H in increasing order
n
0
is consistent with the orderings S and S The subtours
of distance from the ro ot For each pair of vertices
0
of H are combined and then lifted to get a subtour of
v v i j de ne H v v to b e the graph obtained
i j i j
H using the same patching scheme as was sketched in
by coalescing v v to a single vertex and deleting
i
the b eginning of Section
v v The coalescing can b e done via edges of the
j n
The b est subtour of H obtained in this way is
shortest path tree so planarity is preserved
returned Lemma guarantees that one of the choices
Apply Lemma to this graph with arti cial edges
0
of way partition and choices of S will lead to a go o d
added temp orarily to triangulate every face The result
subtour b eing returned
is a cycle separator consisting of two paths in the
The error analysis again comes down to b ounding
shortest path tree and a p ossibly arti cial non tree
the total cost of all edges contracted in the recursive
edge Let P v v b e the set of edges in these paths
i j
calls contributing to the b est tour found The choice of
Let C v v denote the vertex partition fv v g
i j i
k including the constant c ensures that the total error
Interior Exterior fv v g where Interior and
j n
is at most OPT where OPT is the cost of the optimal
Exterior denote the set of vertices of H v v in the
i j
tour
interior and exterior of the cycle separator Note we
The depth of our recursion is again O log n and
have O n such partitions
O k
we consider n subproblems for every problem so the
The following lemma can b e proved using Lemma
2
O k log n O log n
total running time is n or n Note
that dynamic programming won t help us much with
Lemma Let OPT be the set of those edges of the
H
this algorithm since the size of S may reach O k log n
optimal tour that lie in H and let k There exists
p erhaps a weighting scheme would help again here
a pair v v of vertices such that
i j
the number of edges in OPT having precisely one
H
Op en problems
endpoint in fv v g is at most k
i
We susp ect that there is a p olynomial time algorithm
the number of edges in OPT having precisely one for the Steiner version of the planar TSP
H
endpoint in fv v g is at most k Another op en problem is to nd a nearly linear time
j n
approximation scheme for the planar graph TSP The
charging argument in strongly uses the geometry
of the plane and do es not seem to apply to the graph
metric case
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