Tehnical Report CS0617 - 1990 - CS Technion
TECHNION - Israel Institute of Technology
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Computer Science Department
ON SEYMOUR'S AND LOMONOSOV'S PLANE INTEGRAL TWO COMMODITY FLOW RESULTS
by
E. Korach and M. Penn Technical Report #617 Technical March 1990
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
ON SEYMOUR'S AND LOMONbsov's PLANE PLANE
INTEGRAL TWO COMMODITY FLOW FLOW
RESULTS
RESULTS
Ephraim Korach Korach
Computer Science Department Department
Technion-Israel Institute of Technology Technology
Haifa, Israel Israel
32000
32000
and
Michal Penn • Faculty of Commerce and Business Administration The University of British Columbia
!
Vancouver, BC, Canada V6T lY8
-II-
This work was done as part of the author's D. Sc. Thesis in the Faculty of Industrial and Management Engineering, Technion-Israel Institute of Technology, Haifa.
•
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
.
Abstract
We consider the maximum integral tWo-Commodity flow problem in augmented planar
I
....
graphs (i.e., with both source-sink edges added) and provide an o(IVI 21ogIVI) algorithm for that problem. Let G = (V, E) be a graph and w : E - t Z+ a weight function. Let T C V be an even subset of the vertices of G. A T-cut is an edge-cutset of the graph which divides T into two odd sets. Lomonosov gave a good characterization of augmented, planar graphs for which the maximum two-commodity flow is integral. We derive Lomonosov's characterization by using results of Seymour on integral packing of T-cuts in the case ITI = 4, and on the correspondence between plane integral multicommodity flow and integral packing of T-cuts.
1
Intro d uction
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
= (V, E)
Z+o be a weight (capacity) function. For a given graph G with q source-sink pairs (Sk' tA:) = fA: the integral multimulti commodity flow problem can be defined as follows. For k = 1,·· . ,q find a flow of value rA: between SA: and tA: presenting the flow of commodity k, if one exists such that the total flow for every edge does not exceed its capacity. Even Itai and Shamir [4] have shown that this problem is NP-complete, even when F, F = {Ii : i = 1,··· q}, consists of two nonadjacent edges. For the event G is an augmented planar graph (i.e., (Le., with both source-sink edges added) In [12] an O(IVjS) algorithm is presented for the half integral multicommodity flow problem, that is where the flows can obtain either integral or half integral values. BaraBara hana showed O(IVI~logIV/) algorithm for that case [1]. For V' S;; V, 6(V') denotes the set of edges with one end in V' and the other in V\V'. Let T S;; V with ITI even then aT-cut is a set of edges of the form 6(8) with 8 c V and 18 n TI odd. Let G
be an undirected graph and let w : E
-+
In this paper we provide a new characterization of certain planar graphs for which
maximum packing of T-cuts is integral (theorem 5). We derived this characterization by using the following: (i) Seymour's theorems on integral packing of T-cuts in the case ITI = 4. (ii) The connection between maximum integral weighted packing of T-cuts and
integral plane multicommodity flow. Theorem 5 enable us to present a simpler proof for Lomonosov's characterization of these planar graphs for which maximum two commodity flow is integral. A main contribution of this paper is an O(lVI 2 loglVI) simple algorithm for the maximum integral two-commodity flow problem in augmented plan¥ graphs that is derived from Theorem 5.
2
Definitions and Notations
Let (G, w) be a pair of an undirected graph G = (V, E) and w : E -+ Z+ a weight function. Let E' S;; E then w(E') = E{w(e} : eEE'}. With9~t loss of generality we may assume that G is loopless and without parallel edges.
2.1
Definitions
2.1.1 A flow with a set of sources X and a set of sinks Y, 1
w~~te _X, Y
C V and X n Y
=¢
is a function I : V x V -+ R+ where I(v, v) = 0 and the following conditions hold: dl(v) = 0 for VEV - (X U Y), diev) ~ 0 for VEX and d,(v) ::; 0 for VEY, where
d,(v) = L{/(v,u) - I(u,v) : (u,v)EE} d,(v) ,
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
The quantity d,(X) = -dl(Y) is denoted by 2.1.2 2.1.2 A Z-flow Z-Jlow is a family of Hows
II I II, where d,(X) = E{dl(v)
: VEX}
{I. : ZEZ} such that:
'" between one end of z to the other end. (i) III III is a flow '" .... (ii) (t~e joint capacity constraints) L{/.(x,y)
+ I. (Y, x) : ZEZ}::; w(x,y) V(x,y)EE.
2.1.3 Let max/(r)(z;w) =maxE{II 2.1.3
I. IIizEZ,{/z} a Z-fiow in
G}
max I(i)(Z; w) = maxE{II I. II: ZEZ, {/z} an integral Z-flow in G}. 2.1.4 Let S ~ V, Z ~ E, O(S) be' a cut then, mino(Z; w) 2.1.4
= min{w(o(S» : S C V, Z C
o(S)} if at least one such Sexists. 2.1.5 (G, w) is called Z-special for a given Z = {Zt, 2.1.5 {ZI' zz} if there exists a 6-partition (RIt , R z, Sh Sz, TI , Tz) of V such that:
(i)
,,
Zi
= (8j,tj) where 8jESj .and tjE1i i
= 1,2.
»
(ii) W(c5(SI (ii) w(c5(St U8 z))= w(c5(TI U Tz)) ::= W(c5(SI URI U Tz = Plin5(Z; w).
»and w(5(Rz)) are odd.
(iii)
both w(c5(R1
.3 Plane Integral Multicommodity Flows and T-Cuts .3 Seym~ur [i3r pointed out on
an interesting correspondence b~tween the problem of
integral.optimum packing of T-cuts and the integral multicom.mod~ty flow problem. ..
.
The integral multicommodity flow prob~em 7an be formuiated as follows: Let G = (V, E) be a graph, F S; E and let w : E -+ Z+. Does there exist a collection of circuits C and a. function ¢J : C -+ Z+ such that: 3.1 3.1
(i) VCj E C : ICi n FI (ii)
VI E F : E{¢J(Ci )
= 1. :
IE Ci } = w(J) and 2
(iii) Ve E E\F: E{4>(Ci ) : e E Cd ~ wee) where F is the set of edges such that Ii E F connects the source is the demand for the itA commodity.
Si
to the sink t i and w(Jd
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
A necessary condition for the existence of such a flow is known as the cut condition,
Le., for every cut D
= 6(X), X
~
V : weD n F)
~
w(D\F);
Even Itai and Shamir [4] have shown that the integral muIticommodity problem is N PP complete, even when F consists of two nonadjacent edges. Few cases have been solved; a partial list in given in [7]. If G = (V, E) is a plane graph and G- = (V·, E-) is its dual, the following dual relations
are well known (see Bondy and Murty [2]): 3.2
(i) Faces of G correspond to vertices of G·. (ii) Edges of G correspond to edges of G-. (iii) Circuits of G correspond to coboundaries of G-.
~
It is easy to see that the multicommodity flow problem in a plane graph G = (V, E) where F C E as expressed in terms of G- is: Let F- ~ E- and w- : E- -to Z+ be given. Does there exist a collection of coboundaries V- in G- and a. function ¢>- : V-
-to
Z+ such
that: 3.3
(i) VD- E V:
ID- nF-' = 1;
(ii) Vr E F- : E{-(D-) : e- E D-}
~
w·(e·).
Let T C V, ITI even T-ioin, F, is a minimal set of edges so that T is exactly the set of all vertices in (V, F) with odd valency. Hence F- is a union of a. T--join and circuits. It is well known that F- is a minimum weight T--join "if and only if for every circuit C- in G- we have:
If w ( e) = w -( e-) for every e E E then the last condition is satisfied if and only if the cut~ondition is
satisfied.
~.
Clearly, every coboundary
n· E O· is a T·-eut and the collection O·
is a packing of
T·-euts in (G.,T·) with value w·(F·). So, 3.4 (Seymour [13]) We can solve the integral multicommodity flow problem in planar 3.4
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
graphs by solving the problem of maximum weight integral packing of T-euts.
4
Lomonosov's and Seymour's Results
Below we present Lomonosov's and Seymour's Theorems. In the sequal based on SeySey mour's results we will derive Lomonosov's theorems. Theorem 4.1 -(Lomonosov -/8/): Let (G, w) be a graph and Z = (Z1I Z2)
c
E. Suppose that
G is planar. Then either 4.1.1
max fCi}(Z;w) = min 5(Z;w), or
4.1.2
max fCi}(Z; fCi}(Zj w) = min 5(Zj 5(Z; w) - l .
Moreover, ~.1.2 holds if and only if (G, w) is Z-special. Z-special.
I
•
Corollary 4.2 (Lomonosov [8]) Let G be a planar graph with Z 4.2.1
= (ZlJ %2) c E, if:
min 5(Zlj 5(ZI; w)+ min 5(~;w) #- min 5(Zj 5(Z; w) then 4.1.1 holds.
=
Remark 4.3:
Let Z (ZI,Z2). (ZI,%2). IT Z2e6(ZI;W), Z2e6(ZljW), where 6(ZI;W) is the cut for which the minimum obtained then, min 6(ZI; w) min 6(Z; 6(Zj w), so the problem can be reduced to
=
maximum flow from
81
to t 1 and obviously 4.1.1 holds.
4.4 Let T ~ V, T = {tl,t2,t 3 ,t-4}. Let Ati be a shortest path from ti to tj 1 Ie W(Al.) + W(A 23 ) ~ Ie w(A12 )
..
<j w(C\F).)
From 5.2.1
a negative cycle. Therefore there is no
negative cycle in (G',w') and from a theorem by Meigu [91 F is an optimal T-join.O Let G* be the planar dual graph of G. It is known that cuts in G correspond to cycles in G*·, vertices in G correspond to faces in G*. Analogously, a dual expression to Z-special graph is what we call F-special graph defined below: 5.3 Definition
Let (G',w') be the extended graph of (G,w) such that G' is a 2-edge
connected planar graph. (G', w') is called F -special if the following holds: 5.3.0 There exists a 6-partition (511 52, TIl T2 , RI, R 2 ) of the faces of G' with boundaries 5.3.0
C(X), X E {5I, 52, T1 , T2 , R 1, R2 }, satisfying: 5.3.1
Ii
E C(5i ) n C(1i)
i
= 1,2
5.3.2 W(C(51)~C(52)) == W(C(Tl)~C(T2)) == W(C(51)~C(T2)~C(Rl))
= 2k = k + w(F)
5.3.3 both w(C(R1)) and w(C(R2)) are odd. l.
Example:
The following graph r* (the dual graph of r) is F special while 6
r
is Z-special.
wee)
=1
w(z;) =
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
w(e·)
0
=1
Ve E E- Z
i
= 1,2
Ve· E E·
z~
t~--------"""tt
graph -
r
Figure 1 The following property which is easy to check is needed in the sequal. Property 5.4: Let G C be a graph, A,B,C mutually disjoint sets of vertices then 5.4.1
8(A)~8(B)~8(C) ,
= 8(A u B U C) 0
Remark 5.5: Let (C, (G, w) be a graph with Z ~ E, and let tV be a weight function on E such that w(e) = wee) Ve E E\Z and w(z) = 0 Vz E Z. Then it is easily implied from the definition of Z-special graphs that (C, (G, w) is Z-special if and only if (C, (G, w) is Z-special. Moreover if (G, (C, w) is Z-special then min 8(Z jw) = min 8(Z, w) now on we will assume that w( Z)
= o.
+ w(Z).
Therefore, from
If (C', (G', w') is the extended graph of (C, (G, w) then
we can assume that W'(f2) > 0, because otherwise there is an integral maximum two-flow from the following reason. If W' (f2) < 0 then min 8(Zj w') < min 8(z;, w'). Hence, there is an integral flow of value min 8(Zj w) from 81 to t h and this flow is a max two commodity flow.
Lemma 5.6 Let (G, (C, w) be a planar J!-edge-connected graph such that Z C E. Let C· G· be the dual graph of G C with F = Z· C E*. Let G = G*\F C*\F and wee) = wee) Ve E E. Let (G·, w*) be an extended graph of (G, w), then (G, (C·, (C, w) is Z-special if and only if (G*, (C*, w*) is F-special. Proof: It is not difficult to see from the dual relations 3.2, Le. edges of G correspond to edges of C* G* and circuits of G C correspond to coboundaries of G*, that 5.6.1 k = min{w·(C*\F) : F cycles of G*, C*, L
..
c
C*, C* E C*}
= min8(Zjw)
where C* is a collection of
Also, since vertices of G C correspond to faces of C* G* we have that (ShSz,RhRz,Tl,Tz)
-
----------
is a 6-partition of V if and only if (Si, S; , Ri, Ri, Ti, Ti) is a 6-partition of the faces of
G·, with (C·(S;), C·(S;), C·(Ri), C·(Ri), C·(Tt), C·(Ti)) as the respective boundaries. Clearly Zi E 8(Sd n 8(Tj ) i
= 1,2 if and only if Ii E C·(St) n C·(Tt) i = 1,2.
From the 6-partition of V, Property 5.4, the definition of w·Ud and 5.6.1 we have
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
that, W(8(Sl U S2)) = w(8(T1 lJ T2 )) if w·(C·(Si)AC·(S;))
= W(8(Sl
T2)) = min8(Zj w) if and only w·(C·(S;)AC·(Ri)AC·(Ti)) = 2k =
U R1 U
= w·(C·(Tt)AC·(T;)) =
k + w(F). Obviously both w(8(Rd) and w(8(R 2 )) are odd if and only if both w·(C·(Ri)) and
w·(C·(R;)) are odd. Hence (G, w) is Z-special if and only if (G·, w·) is F-special.O Procedure 5.7 and Lemma 5.8 are needed for the proof of Theorem 5.9; hence these two results are presented below. Let (G,w) be a graph T S; V,T = {t 1 ,t2,t3 ,t,,}, Ai; a shortest path from ti to t i . Procedure 5.7 provides us with a constructive way for building the collection .A of shortest paths, .A = {At; : 1
:5 i,i :5
4}, such that At;AAtk k
=/: i =/: i,
1 :5 i,;", k :5 4 is a simple
path. Procedure 5.7: Choose a collection of shortest paths Ai;, 1 t
k
< i < j 54. If each Ai;AAtk
=/: i =/: j, 1 :5 i,i, k < 4 is a simple path, we are done.
If not, let us construct the members of the collection .A in the following order: Au, A IS I3 , Au, A z3 zs ,, A 2 ., A 3 ., by the following method. Let Ai; be a shortest path from ti to tj in (G, W(k»), where W(k), defined as follows. 1
w ( e)
= w(e)
"Ie E E and w
() k+l
(e)
={
w(k) (e) W(k) (e)
-
€
if e E At; otherwise
for € < lOjEI' Then Ai; is the unique shortest path from ti to t; in (G, W(k+l»), and .A is the collection of the shortest paths in (G, w(7»). Moreover, since there are only 6 paths and lO~1 we have that
.A is a collection of shortest paths in (G, w) such that AtiAAtk k =/: i =/:j 15 i,j,k < 4 is a simple path. (Note that E(At) = {UE(Ati) :j E {1,2,3,4}-{i}} form a tree, A;, of shortest paths from ti to t i j = 1,2,3,4).0 €
5.9.2.
Let it be a collection of shortest paths that we get as a result of Procedure 5.7. Let us define.
= Au u {II} = Au~{/l} Au~{/l}
C(S2) = A 34 U {f2} = A34~{f2} A34~{f2}
5.9.3 C(SI) 5.9.3
C(R I )
= AI4AA24AA12 AI4AA24AA12
C(R2) = A24AA23AA34 A24AA23AA34
C(Td C(T2 )
= A l3 AA32 U {It} = A13 AA32 A{It} A{It}
= Al3 AA l4 U{II} = AI3 AA14A{f2} A{f2}
where, in this subsection we use
~j
to denote the edge-set of
~j'
~j'
From 5.9.3 and procedure 5.7 one can see that these circuits are 6 simple circuits in G. G.
In 5.9.7 it will be shown that these circuits define a 6-partition of the faces of G'. G'.
From 5.9.3 we have that
Ii E C(Si) n C(1i) i = 1,2. I.e., (G', w') satisfies 5.3.l. 5.3.l.
In order to prove that (G' I w') satisfy 5.3.2, 5.3.3 and that there exists a 6-partition of of
the faces of G' as required, the following claims are needed.
It follows from Lemma 5.2 that F
= {/11/2} is an optimal T-join in (G',w'), therefore
there is no negative circuit relative to F, in particularly:
W'(A I4 ) + W'(A 32 ) > w'(A14'6.A w'(A14'~.A32) 32 ) ~ W'(JI)
+ w'(f2) = Ie W'(A 24 ) + W'(A I3 ) > W'(A 24 AAI3 ) > w'(It) + W'(J2) = Ie w'(A12 ) + W'(A34 ) > w'(A12 12 AA34 ) ~ w'(fI) + w'(f2) = Ie Since w'(F) = Ie (the value of the optimal primal solution) then the value of the optimal packing of T-cuts (the value of dual solution) have to be less than or equal to Ie. Using Seymour's results (Theorem 4.5 and Theorem 4.7) and our assumption 5.9.1, we get:
10
5.9.4 w'(A 14 ) + w'(A3:z) 5.9.4
=k
w'(A 2,,) + W'(A I3 ) = kk
w'(A 12 ) + w'(A3,,) = kk
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
5.9.5 w'(A;;) 5.9.5
+ W'(A ik ) + W'(A;k)
is odd for each choice of i,;·, k in T.
5.9.6 The following pairs of paths are each vertex disjoint (A I4 , AS2 ), (A 24 , A IS ) and 5.9.6
(A l2 , As,,). Using 5.9.4, 5.9.5 and 5.9.6 we shall prove that properties 5.3.2 and 5.3.3 hold for (G', Wi). ",
From 5.9.5 we get that w'(A 14 ) + W'(A 24 ) + w'(A 12 ) is odd therefore w'(Au~A2,,~Al2) is odd hence w'(C(R I)) is odd. In a similar way we can show that w(C(R2)) is odd, and 5.3.3 holds for (G', w'). 5.3.3 From 5.9.4 and 5.9.6 we get:
(i) I
C(Sd~C(S2) = Al2~A3,,~{!l}~{f2}
= (A13~A32~{ft}) w'«C(Td~C(T2)) = 2k
(li) C(TI)~C(T:z) (li)
=>
w'(C(Sd~C(S2)) = 2k.
~(A13~A14~{f:z})
= A3:z~Au~{!l}~{f:z}
(iii) C(Sl)~C(T2)~C(Rd = (A12~{fl}) ~(A13~Al,,~{f2}) ~(Al,,~A2,,~A12) (iii) A24~{h}~{f2} => W'(C(Sl)~C(T:z)~C(Rl)) = 2k
=>
= A13~
5.9.7 It remains to be shown that these six circuits define a 6-partition of the plane. 5.9.7 Clearly K" U F which is isomorphic to r* has 6 faces F I , · · · F6 with the circuits Ch ••• Ca surrounding them, respectively. Our aim is to show that Ch · · · , C6 corcor respond to the 6 circuits in 5.9.3 in the obvious way and analogously Fb ••• I F6 correspond to 6 partition of the .faces to regions in G' with boundaries as listed in 5.9.3 (a subset of faces defined what we call a region, which is the union of the faces
in the region). We shall show it by proving the fact that" each region,in E(A) uF has one corresponding region in K" u F. At the first stage, before the contraction, each region is defined by three different but not necessarily disjoint paths from A. From Lemma 5.8 each Ai; path has an edge ei; contained only in that path. Therefore each circuit in A remains a circuit in E(A)/(E(A)\Ea) = K~ with V(K~) = T. Hence j{ defines a 4-partition of the plane and AuF defines the require 6-partition.
11
A.2.
Based on Lomonosov's result we shall prove the sufficiency of the condition Le.
(G' ,w') is F -special => There does not exist an optimal packing of T -cuts which is integral. Let (G"w·) be the dual graph of (G',w') where w·(e·) = w'(e') for each e' E E'\F
= 0 i = 1,2. It follows from Lemma 5.6 that (G"w·) is Z-Special. Z-Spedal. From Theorem 4.1 we have that there does not exist an optimal flow which is integral (Le. maxf(i)(Z;w·) < minc(Z;w·)). We shall now show that max l{i)(Z;w·) f(i)(Z;w·) < minc(Z;w·) max/{i)(Z;w·)
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
and w·(lt)
= W·(Zi)
implies that there does not exist an optimal packing of T -cuts in (G', w') which is integral. Assume that there exists an optimal packing of T-cuts in (G', V') which is integral. From 5.6.1 and Lemma 5.2 we have that F is an optimal T-join with w(F)
=
k
=
minc(Z;w·). Using Seymour's relations to integral multicommodity flow, 3.4, we have that there exists an integral flow in (G· , w·) of value min c(Z; w·). Hence max I{ f( i) ( Z j w·) ~ minc(Z;w·). A contradiction and we have proved that 5.9.2 => 5.9.1. Assume (G', w') is not 2-edge connected graph. Let
B.
E=
(elt···, el:) be the set of its cut-edges (for G, a connected graph, e E E is a cut-edge if G\e is not connected). E might be empty. Let GI ,··· ~ Gt. be the components of G\E. Let V(Gi ) denote the set of vertices of Gi . If IT n V(Gi)1 = 4 for some 1 < i < i, then by using part A the theorem holds. If not, then (G', w') is not F-special. It remains to show that in that case either there is no finite optimal packing of T-cuts or 5.9.1 does not hold. Assume (G,w) is not connected then either ITnV(Gi )/ = 1 or ITnV(Gi)1 = 2 for some
< i.
If IT n V(Gi)1 = 1, then no T-join exists and hence there is no finite optimal packing of T-cuts. If ITnV(Gi)1 = 2, then it is not difficult to see that 5.9.1 does not hold. 1 ::; i
Now, assume (G', w') is connected and let ei be a cut-edge such that ITn V(Gi)1 :F 0 where
Gi is one of the two components of G\ei. Then either ITnV(Gi)1 is odd or ITnV(Gi)1
= 2,
in both cases it is easy to see that 5.9.1 does not hold. 0
Lemma 5.10
Let (G,w) be a weighted planar graph and (G',w') its e%tended planar
graph. If 5.10.1 fl E C,C E C} +min{w(C): 1% f% E C,C E C}:F min{w(C): (l1,f%) 5.10.1 min{w(C): II (11,1%) C
C, C E C} = 2k then there e:cists an optimal packing of T -cuts which is integral.
Proof:
From Lemma 5.2 F is an optimal T -join; hence there are no negative circuits
relative to F, especially
12
w(A 13 w(A 24 2 ,)) ~ k 13 )) + W(A
..
w(A u )
+ W(A 23 ) > k
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Since min{ w(C) : fl E C} + min{w(C) : f2 E C} i: 2k we have that w(A 12 ) + w(A H ) > k. In addition we have that k == w(F) ~ (optimal integral packing of T~uts). Using Theorem 4.5 we have that there exists an optimal packing of T~uts which is integral.O integral.o
6
Alternative Proofs of Lomonosov's Results.
In previous sections we saw the correspondence between maximum two--commodity flow problem and packing of T ~uts. Also we saw some integrality results for packing of T~uts problem (Theorem 5.9 and Lemma 5.10) which are analogous to Lomonosov's results. Based on these results and Lemma 5.6, we shall present alternative simpler proofs for Theorem 4.1 and Corollary 4.2. Alternative Proof of Theorem 4.1: From Remark 5.5 we may assume w.l.o.g. that w(Zj) == 0, i == 1,2. Let G· be a planar dual graph of G such that F = Z· c E·. Let 6.1
.
G = G·\F and w(e) == w(e), e is the dual edge of e, 'tie E E and T == {fl,f2'~,f,} {tl,f2,~,f4} C V where fl fl == (t (fll ,f2) and /2 == (t (f33 ,f,,). Let (G" w·) be the extended graph of (G, w) . From 5.6.1 and Lemma 5.2, k == w·(F) == min6(Z; w) and F is an optimal T-join. From 3.4 there exists a flow of value min6(Z; min6(Zj w) and hence an optimal one. Using 3.4 and Theorem 4.6 we conclude that there exists an integral flow of value min 6 (Z (Z;i w) - 1 in (G,w), hence max/(i)(Zjw) ~ min6(Zjw)-1. min6(Z;w)-1. The second part of the theorem is derived as follows: By 3.4 we have max f(i)(Zj w) == min6(Z; min6(Zj w) - 1 if and only if there is no optimal packing of T~uts which is integral in (G.,w·). By Theorem 5.9 there is no optimal packing of T~uts which is integral in (G"w·) if and only if (G"w·) is F-special. By Lemma 5.6 (G"w·) is F-special if and only if (G,w) is Z-special.O Remark 6.2: Theorem 6.3 below which was proved by us in [7] is an extention of Theorem 4.6 and hence could replace Theorem 4.6 in the alternative proof of Theorem 4.l. Theorem 6.3 Let T C V he an ellen subset of lIertices and nF the number of components in an optimal T -;join, F·. Let lIw (G) he the lIalue of an optimal integral paclcing of T -cuts
then w(F·) -
lIw (G)
< nF -l.
6.4 Alternative Proof of Corollary 4.2: Let (G·, w·) be the extended graph as defined
i~ 6.1. Clearly, based on 3.2 and 5.6.1 we have, min6(zljw) 13
+ min6(z2;w) min6(z2iw) i: min6(Zjw)
if and only if min{w{G);/I E G,G E C}
+ min{w(G) : h.
E G,G E C}
=1=
min{w{G)
(11,/2) C GjG E C} From Lemma 5.10 we get There exists an optimal packing of T-cuts which is integral and 3.4 implies that
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
max/(i)(Z; w) = min6(Z;w).O max/(il(Z;
7
Algorithms As a consequence of the previous results we present three polynomial time algorithms.
Algorithm 1 is a recognition algorithm. Given (G' , w') an augumented planar graph, with Z C E !Z! = 2 find whether there exists an integral two--commodity flow 1 such that
11/112: min6(Z;w), i.e., find whether there exists an optimal two--commodity flow which is integral in (G' , w').
&
Given (G',w') an augumented planar graph with Z C E, putes the maximum integral two-commodity How in (G', w').
IZ!
= 2, Algorithm 2 comcom
Algorithms 1 and 2 are consequences of Seymour's theorems and algorithms and are not based on Lomonosov's results. Given (G,w) an augumented planar graph with Z C E, IZI = 2, (G,w) is a Z-special graph, then Algorithm 3 provides us with a simple method for defining the 6-partition of the vertices of the graph as described in Definition 2.1.5. First we shall present this three algorithms and discuss their complexity in the sequal. Since these three algorithms start with identical four steps we shall present these four steps steps
under the preprocessing algorithm. algorithm.
Preprocessing Algorithm: Algorithm:
Input: planar graph (G, w), Z
c E, IZI = 2,
w:E
~
Z+, w{Z) = O.
Output: planar graph (G·, w·), such that G· is the dual graph of G with w· as weight function and if IV(Z·)I ~ 4 then, A, a collection of six shortest paths between the ends of Z·, as described in Procedure 5.7. (Note that in Algorithms 1 and 2 any collection of six shortest paths is suffices).
14
Step 1: Using max-flow algorithm calculate min O(Z; w). Step 2: Using Booth and Lueker's [3] algorithm find G-, the dual graph of G. Let Z- = F and define w-(e-) = w(e) \:Ie- E G-\F where e- is the dual edge of e, w-(F) will be
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
defined in Step 4. Step 3: If IV- (F) I ~ 3 stop. Otherwise let v- (F)
= T- where V- (11) = (ti, til.
Otherwise
T- = {ti, ti, t;, t:}. Find A = {Aii : 1 ~ i < j ~ 4} a collection of shortest paths in G-\F from ti to t; and their length for each 1 ~ i < j < 4 by using Frederickson's [5} algorithm and Procedure 5.7.
End. Algorithm 1: Input: (G, w), Z
c E, IZ/ = 2, w: E -+ Z+, w(Z) = o. o.
Output: An answer to the question: "Does there exist an optimal two-commodity How How
which is integral?" Step 0: Perform the Preprocessing Algorithm. Step 1: If jV·(F)1
< 3 go to Step 5.
Step 2: If: w-(A~2) + w-(A;. ~ mino(Z;w)
+ w-(A;.) w-(Ai.) + w-(A;s) w-(A~s)
~ mino(Z; w) ~
mino(Z; w)
go to Step 3, otherwise go to Step 6. Step 3: If there is any strict inequality in Step 2 go to Step 5.
Step 5: There exists an optimal two-commodity How which is integral. Stop. Step 6: There does not exist an optimal
2~ommodity How
End.
15
which is'integral.
Algorithm 2: Input: (G,w), ZcE, /ZI=2, w:E-+Z+, w(Z) =0
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Output: Maximum integral two-commodity flow Step 0: Perform the Preprocessing Algorithm. Step 1: IT I(V"(F)/ ~ 3 there is always an integral packing of TO-cuts and a direct way to find the optimal integral flow.
=
Step 2: IT ITO' 4 using Seymour's technique find po - an optimal integral packing of T°4:uts in (Go,WO). Step 3: Calculate the optimal integral two-commodity flow in (G, w) by the following method: "'Ie E E : fee) = m where e = (eo)" and there are m T°4:uts in po that intersect eO.
End. As was mentioned before, Algorithm 3 provides a simple method to define a 6-partition 6-parlition
of the vertices of a Z-special graph. This simplifies the method for finding the 6-partition as is implicitly suggested by Lomonosov's long proof of Theorem 4.1.
16
16
t
.
Algorithm 3: Input:
(G, w)
Output: If the graph is Z-special then the output is a 6-partition of V as in the definition
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
of Z-special graphs. If not, the output is: "The graph is not Z-special."
Step 0: Perform Algorithm 1 - If there is an optimal flow which is integral go to the End with the answer: "The graph is not Z-special." Step 1: Define the following 6-partition (as defined in the proof of Theorem 5.9) in the following way: Let [(. be a subdivision of K. with T = {tl, t2, t3, t 4} as V (K.) , and ~; 1 < i < j < 4 (the shortest path from t, to t; as can be obtained by Procedure 5.7) as the subdivision of e = (t"
til, e E E(K4 ), in K•.
[(4 is the following graph:
....
It = (tl, t2) and f2 = (t 3, t.) to [(4. [(4.
First add It to 114 so that 11 divides F1 into two regions Let us add
such that F. is divided to two regions graph.
graph.
5; and T;.
I
11
5; and T;, then add f2f2
Eventually we have the following following
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Clearly, {Si,S;,R;.,!li,Ti,T2-} is a 6-partition of the plane, with the following circir cuits, accordinly, as their boundaries. (These circuits were defined previously in 5.9.3.)
C- (Si) = Ai2 u {II} C-(S;) = Ai. u {/2} C-(R;) = Ai.L\A;.L\ A i2 C-(!li) = A 2.L\A;sL\As4 C-(Ti} = A~sL\A;2L\{fl} . C-(T;) = A~sL\Ai,L\{/2} Step 2:
Let (Sh 8 2 , Rt, R 2, Th T2 ) be the sets of vertices that correspond to the regions
in (G"w-) via duality. Then (St,S2,Rt,R2,Tt,T2) is the require 6-partition. End.
Lemma '1.1 The time complexity 01 the Preprocessing Algorithm is O(IVI 2 log IVI). Proof: By checking each step s,eperately, we shall show that the complexity of the algoalgo rithm determined by Step 1. Step 1: By using a simple transformation we can calculate mino(Zjw) by calculating max-flow in the following way: Let Z = {Zl = (t h t2),Z2 = (t3,t4)), where (tht3) be the sources and (t2,t,) the sinks. Let us add two vertices, Sb a super source and 82, 18
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
a super sink. Join 81 to the two source tl and t~h and 82 to the two sinks t2 and t •• Let the weight of each of the new added edges be M> Ecec wee).
It is not 9.ifficult to see that the maximum flow from 81 to 82 is equal to min8(Z; w). We can find the maximum flow from 81 to 82 by using Sleator flow algorithm [14] in O(IVI·IEllog IV/). Since in planar graphs O(IVI) = O(IE/) we have that the complexity of the first step is O(IVI 210gIV/). - The original graph is planar, but after adding the four edges the graph might be nonplanar and therefore we can not use flow algorithms for planar graphs which are more efficient. Step 2: By using Booth and Lueker's algorithm [3] we can find G· the dual graph of G in O(lVI).
t;
Step 3: For constructing A, a collection of 6 shortest paths from ti to tj 1
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Computer Science Department
ON SEYMOUR'S AND LOMONOSOV'S PLANE INTEGRAL TWO COMMODITY FLOW RESULTS
by
E. Korach and M. Penn Technical Report #617 Technical March 1990
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
ON SEYMOUR'S AND LOMONbsov's PLANE PLANE
INTEGRAL TWO COMMODITY FLOW FLOW
RESULTS
RESULTS
Ephraim Korach Korach
Computer Science Department Department
Technion-Israel Institute of Technology Technology
Haifa, Israel Israel
32000
32000
and
Michal Penn • Faculty of Commerce and Business Administration The University of British Columbia
!
Vancouver, BC, Canada V6T lY8
-II-
This work was done as part of the author's D. Sc. Thesis in the Faculty of Industrial and Management Engineering, Technion-Israel Institute of Technology, Haifa.
•
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
.
Abstract
We consider the maximum integral tWo-Commodity flow problem in augmented planar
I
....
graphs (i.e., with both source-sink edges added) and provide an o(IVI 21ogIVI) algorithm for that problem. Let G = (V, E) be a graph and w : E - t Z+ a weight function. Let T C V be an even subset of the vertices of G. A T-cut is an edge-cutset of the graph which divides T into two odd sets. Lomonosov gave a good characterization of augmented, planar graphs for which the maximum two-commodity flow is integral. We derive Lomonosov's characterization by using results of Seymour on integral packing of T-cuts in the case ITI = 4, and on the correspondence between plane integral multicommodity flow and integral packing of T-cuts.
1
Intro d uction
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
= (V, E)
Z+o be a weight (capacity) function. For a given graph G with q source-sink pairs (Sk' tA:) = fA: the integral multimulti commodity flow problem can be defined as follows. For k = 1,·· . ,q find a flow of value rA: between SA: and tA: presenting the flow of commodity k, if one exists such that the total flow for every edge does not exceed its capacity. Even Itai and Shamir [4] have shown that this problem is NP-complete, even when F, F = {Ii : i = 1,··· q}, consists of two nonadjacent edges. For the event G is an augmented planar graph (i.e., (Le., with both source-sink edges added) In [12] an O(IVjS) algorithm is presented for the half integral multicommodity flow problem, that is where the flows can obtain either integral or half integral values. BaraBara hana showed O(IVI~logIV/) algorithm for that case [1]. For V' S;; V, 6(V') denotes the set of edges with one end in V' and the other in V\V'. Let T S;; V with ITI even then aT-cut is a set of edges of the form 6(8) with 8 c V and 18 n TI odd. Let G
be an undirected graph and let w : E
-+
In this paper we provide a new characterization of certain planar graphs for which
maximum packing of T-cuts is integral (theorem 5). We derived this characterization by using the following: (i) Seymour's theorems on integral packing of T-cuts in the case ITI = 4. (ii) The connection between maximum integral weighted packing of T-cuts and
integral plane multicommodity flow. Theorem 5 enable us to present a simpler proof for Lomonosov's characterization of these planar graphs for which maximum two commodity flow is integral. A main contribution of this paper is an O(lVI 2 loglVI) simple algorithm for the maximum integral two-commodity flow problem in augmented plan¥ graphs that is derived from Theorem 5.
2
Definitions and Notations
Let (G, w) be a pair of an undirected graph G = (V, E) and w : E -+ Z+ a weight function. Let E' S;; E then w(E') = E{w(e} : eEE'}. With9~t loss of generality we may assume that G is loopless and without parallel edges.
2.1
Definitions
2.1.1 A flow with a set of sources X and a set of sinks Y, 1
w~~te _X, Y
C V and X n Y
=¢
is a function I : V x V -+ R+ where I(v, v) = 0 and the following conditions hold: dl(v) = 0 for VEV - (X U Y), diev) ~ 0 for VEX and d,(v) ::; 0 for VEY, where
d,(v) = L{/(v,u) - I(u,v) : (u,v)EE} d,(v) ,
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
The quantity d,(X) = -dl(Y) is denoted by 2.1.2 2.1.2 A Z-flow Z-Jlow is a family of Hows
II I II, where d,(X) = E{dl(v)
: VEX}
{I. : ZEZ} such that:
'" between one end of z to the other end. (i) III III is a flow '" .... (ii) (t~e joint capacity constraints) L{/.(x,y)
+ I. (Y, x) : ZEZ}::; w(x,y) V(x,y)EE.
2.1.3 Let max/(r)(z;w) =maxE{II 2.1.3
I. IIizEZ,{/z} a Z-fiow in
G}
max I(i)(Z; w) = maxE{II I. II: ZEZ, {/z} an integral Z-flow in G}. 2.1.4 Let S ~ V, Z ~ E, O(S) be' a cut then, mino(Z; w) 2.1.4
= min{w(o(S» : S C V, Z C
o(S)} if at least one such Sexists. 2.1.5 (G, w) is called Z-special for a given Z = {Zt, 2.1.5 {ZI' zz} if there exists a 6-partition (RIt , R z, Sh Sz, TI , Tz) of V such that:
(i)
,,
Zi
= (8j,tj) where 8jESj .and tjE1i i
= 1,2.
»
(ii) W(c5(SI (ii) w(c5(St U8 z))= w(c5(TI U Tz)) ::= W(c5(SI URI U Tz = Plin5(Z; w).
»and w(5(Rz)) are odd.
(iii)
both w(c5(R1
.3 Plane Integral Multicommodity Flows and T-Cuts .3 Seym~ur [i3r pointed out on
an interesting correspondence b~tween the problem of
integral.optimum packing of T-cuts and the integral multicom.mod~ty flow problem. ..
.
The integral multicommodity flow prob~em 7an be formuiated as follows: Let G = (V, E) be a graph, F S; E and let w : E -+ Z+. Does there exist a collection of circuits C and a. function ¢J : C -+ Z+ such that: 3.1 3.1
(i) VCj E C : ICi n FI (ii)
VI E F : E{¢J(Ci )
= 1. :
IE Ci } = w(J) and 2
(iii) Ve E E\F: E{4>(Ci ) : e E Cd ~ wee) where F is the set of edges such that Ii E F connects the source is the demand for the itA commodity.
Si
to the sink t i and w(Jd
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
A necessary condition for the existence of such a flow is known as the cut condition,
Le., for every cut D
= 6(X), X
~
V : weD n F)
~
w(D\F);
Even Itai and Shamir [4] have shown that the integral muIticommodity problem is N PP complete, even when F consists of two nonadjacent edges. Few cases have been solved; a partial list in given in [7]. If G = (V, E) is a plane graph and G- = (V·, E-) is its dual, the following dual relations
are well known (see Bondy and Murty [2]): 3.2
(i) Faces of G correspond to vertices of G·. (ii) Edges of G correspond to edges of G-. (iii) Circuits of G correspond to coboundaries of G-.
~
It is easy to see that the multicommodity flow problem in a plane graph G = (V, E) where F C E as expressed in terms of G- is: Let F- ~ E- and w- : E- -to Z+ be given. Does there exist a collection of coboundaries V- in G- and a. function ¢>- : V-
-to
Z+ such
that: 3.3
(i) VD- E V:
ID- nF-' = 1;
(ii) Vr E F- : E{-(D-) : e- E D-}
~
w·(e·).
Let T C V, ITI even T-ioin, F, is a minimal set of edges so that T is exactly the set of all vertices in (V, F) with odd valency. Hence F- is a union of a. T--join and circuits. It is well known that F- is a minimum weight T--join "if and only if for every circuit C- in G- we have:
If w ( e) = w -( e-) for every e E E then the last condition is satisfied if and only if the cut~ondition is
satisfied.
~.
Clearly, every coboundary
n· E O· is a T·-eut and the collection O·
is a packing of
T·-euts in (G.,T·) with value w·(F·). So, 3.4 (Seymour [13]) We can solve the integral multicommodity flow problem in planar 3.4
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
graphs by solving the problem of maximum weight integral packing of T-euts.
4
Lomonosov's and Seymour's Results
Below we present Lomonosov's and Seymour's Theorems. In the sequal based on SeySey mour's results we will derive Lomonosov's theorems. Theorem 4.1 -(Lomonosov -/8/): Let (G, w) be a graph and Z = (Z1I Z2)
c
E. Suppose that
G is planar. Then either 4.1.1
max fCi}(Z;w) = min 5(Z;w), or
4.1.2
max fCi}(Z; fCi}(Zj w) = min 5(Zj 5(Z; w) - l .
Moreover, ~.1.2 holds if and only if (G, w) is Z-special. Z-special.
I
•
Corollary 4.2 (Lomonosov [8]) Let G be a planar graph with Z 4.2.1
= (ZlJ %2) c E, if:
min 5(Zlj 5(ZI; w)+ min 5(~;w) #- min 5(Zj 5(Z; w) then 4.1.1 holds.
=
Remark 4.3:
Let Z (ZI,Z2). (ZI,%2). IT Z2e6(ZI;W), Z2e6(ZljW), where 6(ZI;W) is the cut for which the minimum obtained then, min 6(ZI; w) min 6(Z; 6(Zj w), so the problem can be reduced to
=
maximum flow from
81
to t 1 and obviously 4.1.1 holds.
4.4 Let T ~ V, T = {tl,t2,t 3 ,t-4}. Let Ati be a shortest path from ti to tj 1 Ie W(Al.) + W(A 23 ) ~ Ie w(A12 )
..
<j w(C\F).)
From 5.2.1
a negative cycle. Therefore there is no
negative cycle in (G',w') and from a theorem by Meigu [91 F is an optimal T-join.O Let G* be the planar dual graph of G. It is known that cuts in G correspond to cycles in G*·, vertices in G correspond to faces in G*. Analogously, a dual expression to Z-special graph is what we call F-special graph defined below: 5.3 Definition
Let (G',w') be the extended graph of (G,w) such that G' is a 2-edge
connected planar graph. (G', w') is called F -special if the following holds: 5.3.0 There exists a 6-partition (511 52, TIl T2 , RI, R 2 ) of the faces of G' with boundaries 5.3.0
C(X), X E {5I, 52, T1 , T2 , R 1, R2 }, satisfying: 5.3.1
Ii
E C(5i ) n C(1i)
i
= 1,2
5.3.2 W(C(51)~C(52)) == W(C(Tl)~C(T2)) == W(C(51)~C(T2)~C(Rl))
= 2k = k + w(F)
5.3.3 both w(C(R1)) and w(C(R2)) are odd. l.
Example:
The following graph r* (the dual graph of r) is F special while 6
r
is Z-special.
wee)
=1
w(z;) =
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
w(e·)
0
=1
Ve E E- Z
i
= 1,2
Ve· E E·
z~
t~--------"""tt
graph -
r
Figure 1 The following property which is easy to check is needed in the sequal. Property 5.4: Let G C be a graph, A,B,C mutually disjoint sets of vertices then 5.4.1
8(A)~8(B)~8(C) ,
= 8(A u B U C) 0
Remark 5.5: Let (C, (G, w) be a graph with Z ~ E, and let tV be a weight function on E such that w(e) = wee) Ve E E\Z and w(z) = 0 Vz E Z. Then it is easily implied from the definition of Z-special graphs that (C, (G, w) is Z-special if and only if (C, (G, w) is Z-special. Moreover if (G, (C, w) is Z-special then min 8(Z jw) = min 8(Z, w) now on we will assume that w( Z)
= o.
+ w(Z).
Therefore, from
If (C', (G', w') is the extended graph of (C, (G, w) then
we can assume that W'(f2) > 0, because otherwise there is an integral maximum two-flow from the following reason. If W' (f2) < 0 then min 8(Zj w') < min 8(z;, w'). Hence, there is an integral flow of value min 8(Zj w) from 81 to t h and this flow is a max two commodity flow.
Lemma 5.6 Let (G, (C, w) be a planar J!-edge-connected graph such that Z C E. Let C· G· be the dual graph of G C with F = Z· C E*. Let G = G*\F C*\F and wee) = wee) Ve E E. Let (G·, w*) be an extended graph of (G, w), then (G, (C·, (C, w) is Z-special if and only if (G*, (C*, w*) is F-special. Proof: It is not difficult to see from the dual relations 3.2, Le. edges of G correspond to edges of C* G* and circuits of G C correspond to coboundaries of G*, that 5.6.1 k = min{w·(C*\F) : F cycles of G*, C*, L
..
c
C*, C* E C*}
= min8(Zjw)
where C* is a collection of
Also, since vertices of G C correspond to faces of C* G* we have that (ShSz,RhRz,Tl,Tz)
-
----------
is a 6-partition of V if and only if (Si, S; , Ri, Ri, Ti, Ti) is a 6-partition of the faces of
G·, with (C·(S;), C·(S;), C·(Ri), C·(Ri), C·(Tt), C·(Ti)) as the respective boundaries. Clearly Zi E 8(Sd n 8(Tj ) i
= 1,2 if and only if Ii E C·(St) n C·(Tt) i = 1,2.
From the 6-partition of V, Property 5.4, the definition of w·Ud and 5.6.1 we have
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
that, W(8(Sl U S2)) = w(8(T1 lJ T2 )) if w·(C·(Si)AC·(S;))
= W(8(Sl
T2)) = min8(Zj w) if and only w·(C·(S;)AC·(Ri)AC·(Ti)) = 2k =
U R1 U
= w·(C·(Tt)AC·(T;)) =
k + w(F). Obviously both w(8(Rd) and w(8(R 2 )) are odd if and only if both w·(C·(Ri)) and
w·(C·(R;)) are odd. Hence (G, w) is Z-special if and only if (G·, w·) is F-special.O Procedure 5.7 and Lemma 5.8 are needed for the proof of Theorem 5.9; hence these two results are presented below. Let (G,w) be a graph T S; V,T = {t 1 ,t2,t3 ,t,,}, Ai; a shortest path from ti to t i . Procedure 5.7 provides us with a constructive way for building the collection .A of shortest paths, .A = {At; : 1
:5 i,i :5
4}, such that At;AAtk k
=/: i =/: i,
1 :5 i,;", k :5 4 is a simple
path. Procedure 5.7: Choose a collection of shortest paths Ai;, 1 t
k
< i < j 54. If each Ai;AAtk
=/: i =/: j, 1 :5 i,i, k < 4 is a simple path, we are done.
If not, let us construct the members of the collection .A in the following order: Au, A IS I3 , Au, A z3 zs ,, A 2 ., A 3 ., by the following method. Let Ai; be a shortest path from ti to tj in (G, W(k»), where W(k), defined as follows. 1
w ( e)
= w(e)
"Ie E E and w
() k+l
(e)
={
w(k) (e) W(k) (e)
-
€
if e E At; otherwise
for € < lOjEI' Then Ai; is the unique shortest path from ti to t; in (G, W(k+l»), and .A is the collection of the shortest paths in (G, w(7»). Moreover, since there are only 6 paths and lO~1 we have that
.A is a collection of shortest paths in (G, w) such that AtiAAtk k =/: i =/:j 15 i,j,k < 4 is a simple path. (Note that E(At) = {UE(Ati) :j E {1,2,3,4}-{i}} form a tree, A;, of shortest paths from ti to t i j = 1,2,3,4).0 €
5.9.2.
Let it be a collection of shortest paths that we get as a result of Procedure 5.7. Let us define.
= Au u {II} = Au~{/l} Au~{/l}
C(S2) = A 34 U {f2} = A34~{f2} A34~{f2}
5.9.3 C(SI) 5.9.3
C(R I )
= AI4AA24AA12 AI4AA24AA12
C(R2) = A24AA23AA34 A24AA23AA34
C(Td C(T2 )
= A l3 AA32 U {It} = A13 AA32 A{It} A{It}
= Al3 AA l4 U{II} = AI3 AA14A{f2} A{f2}
where, in this subsection we use
~j
to denote the edge-set of
~j'
~j'
From 5.9.3 and procedure 5.7 one can see that these circuits are 6 simple circuits in G. G.
In 5.9.7 it will be shown that these circuits define a 6-partition of the faces of G'. G'.
From 5.9.3 we have that
Ii E C(Si) n C(1i) i = 1,2. I.e., (G', w') satisfies 5.3.l. 5.3.l.
In order to prove that (G' I w') satisfy 5.3.2, 5.3.3 and that there exists a 6-partition of of
the faces of G' as required, the following claims are needed.
It follows from Lemma 5.2 that F
= {/11/2} is an optimal T-join in (G',w'), therefore
there is no negative circuit relative to F, in particularly:
W'(A I4 ) + W'(A 32 ) > w'(A14'6.A w'(A14'~.A32) 32 ) ~ W'(JI)
+ w'(f2) = Ie W'(A 24 ) + W'(A I3 ) > W'(A 24 AAI3 ) > w'(It) + W'(J2) = Ie w'(A12 ) + W'(A34 ) > w'(A12 12 AA34 ) ~ w'(fI) + w'(f2) = Ie Since w'(F) = Ie (the value of the optimal primal solution) then the value of the optimal packing of T-cuts (the value of dual solution) have to be less than or equal to Ie. Using Seymour's results (Theorem 4.5 and Theorem 4.7) and our assumption 5.9.1, we get:
10
5.9.4 w'(A 14 ) + w'(A3:z) 5.9.4
=k
w'(A 2,,) + W'(A I3 ) = kk
w'(A 12 ) + w'(A3,,) = kk
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
5.9.5 w'(A;;) 5.9.5
+ W'(A ik ) + W'(A;k)
is odd for each choice of i,;·, k in T.
5.9.6 The following pairs of paths are each vertex disjoint (A I4 , AS2 ), (A 24 , A IS ) and 5.9.6
(A l2 , As,,). Using 5.9.4, 5.9.5 and 5.9.6 we shall prove that properties 5.3.2 and 5.3.3 hold for (G', Wi). ",
From 5.9.5 we get that w'(A 14 ) + W'(A 24 ) + w'(A 12 ) is odd therefore w'(Au~A2,,~Al2) is odd hence w'(C(R I)) is odd. In a similar way we can show that w(C(R2)) is odd, and 5.3.3 holds for (G', w'). 5.3.3 From 5.9.4 and 5.9.6 we get:
(i) I
C(Sd~C(S2) = Al2~A3,,~{!l}~{f2}
= (A13~A32~{ft}) w'«C(Td~C(T2)) = 2k
(li) C(TI)~C(T:z) (li)
=>
w'(C(Sd~C(S2)) = 2k.
~(A13~A14~{f:z})
= A3:z~Au~{!l}~{f:z}
(iii) C(Sl)~C(T2)~C(Rd = (A12~{fl}) ~(A13~Al,,~{f2}) ~(Al,,~A2,,~A12) (iii) A24~{h}~{f2} => W'(C(Sl)~C(T:z)~C(Rl)) = 2k
=>
= A13~
5.9.7 It remains to be shown that these six circuits define a 6-partition of the plane. 5.9.7 Clearly K" U F which is isomorphic to r* has 6 faces F I , · · · F6 with the circuits Ch ••• Ca surrounding them, respectively. Our aim is to show that Ch · · · , C6 corcor respond to the 6 circuits in 5.9.3 in the obvious way and analogously Fb ••• I F6 correspond to 6 partition of the .faces to regions in G' with boundaries as listed in 5.9.3 (a subset of faces defined what we call a region, which is the union of the faces
in the region). We shall show it by proving the fact that" each region,in E(A) uF has one corresponding region in K" u F. At the first stage, before the contraction, each region is defined by three different but not necessarily disjoint paths from A. From Lemma 5.8 each Ai; path has an edge ei; contained only in that path. Therefore each circuit in A remains a circuit in E(A)/(E(A)\Ea) = K~ with V(K~) = T. Hence j{ defines a 4-partition of the plane and AuF defines the require 6-partition.
11
A.2.
Based on Lomonosov's result we shall prove the sufficiency of the condition Le.
(G' ,w') is F -special => There does not exist an optimal packing of T -cuts which is integral. Let (G"w·) be the dual graph of (G',w') where w·(e·) = w'(e') for each e' E E'\F
= 0 i = 1,2. It follows from Lemma 5.6 that (G"w·) is Z-Special. Z-Spedal. From Theorem 4.1 we have that there does not exist an optimal flow which is integral (Le. maxf(i)(Z;w·) < minc(Z;w·)). We shall now show that max l{i)(Z;w·) f(i)(Z;w·) < minc(Z;w·) max/{i)(Z;w·)
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
and w·(lt)
= W·(Zi)
implies that there does not exist an optimal packing of T -cuts in (G', w') which is integral. Assume that there exists an optimal packing of T-cuts in (G', V') which is integral. From 5.6.1 and Lemma 5.2 we have that F is an optimal T-join with w(F)
=
k
=
minc(Z;w·). Using Seymour's relations to integral multicommodity flow, 3.4, we have that there exists an integral flow in (G· , w·) of value min c(Z; w·). Hence max I{ f( i) ( Z j w·) ~ minc(Z;w·). A contradiction and we have proved that 5.9.2 => 5.9.1. Assume (G', w') is not 2-edge connected graph. Let
B.
E=
(elt···, el:) be the set of its cut-edges (for G, a connected graph, e E E is a cut-edge if G\e is not connected). E might be empty. Let GI ,··· ~ Gt. be the components of G\E. Let V(Gi ) denote the set of vertices of Gi . If IT n V(Gi)1 = 4 for some 1 < i < i, then by using part A the theorem holds. If not, then (G', w') is not F-special. It remains to show that in that case either there is no finite optimal packing of T-cuts or 5.9.1 does not hold. Assume (G,w) is not connected then either ITnV(Gi )/ = 1 or ITnV(Gi)1 = 2 for some
< i.
If IT n V(Gi)1 = 1, then no T-join exists and hence there is no finite optimal packing of T-cuts. If ITnV(Gi)1 = 2, then it is not difficult to see that 5.9.1 does not hold. 1 ::; i
Now, assume (G', w') is connected and let ei be a cut-edge such that ITn V(Gi)1 :F 0 where
Gi is one of the two components of G\ei. Then either ITnV(Gi)1 is odd or ITnV(Gi)1
= 2,
in both cases it is easy to see that 5.9.1 does not hold. 0
Lemma 5.10
Let (G,w) be a weighted planar graph and (G',w') its e%tended planar
graph. If 5.10.1 fl E C,C E C} +min{w(C): 1% f% E C,C E C}:F min{w(C): (l1,f%) 5.10.1 min{w(C): II (11,1%) C
C, C E C} = 2k then there e:cists an optimal packing of T -cuts which is integral.
Proof:
From Lemma 5.2 F is an optimal T -join; hence there are no negative circuits
relative to F, especially
12
w(A 13 w(A 24 2 ,)) ~ k 13 )) + W(A
..
w(A u )
+ W(A 23 ) > k
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Since min{ w(C) : fl E C} + min{w(C) : f2 E C} i: 2k we have that w(A 12 ) + w(A H ) > k. In addition we have that k == w(F) ~ (optimal integral packing of T~uts). Using Theorem 4.5 we have that there exists an optimal packing of T~uts which is integral.O integral.o
6
Alternative Proofs of Lomonosov's Results.
In previous sections we saw the correspondence between maximum two--commodity flow problem and packing of T ~uts. Also we saw some integrality results for packing of T~uts problem (Theorem 5.9 and Lemma 5.10) which are analogous to Lomonosov's results. Based on these results and Lemma 5.6, we shall present alternative simpler proofs for Theorem 4.1 and Corollary 4.2. Alternative Proof of Theorem 4.1: From Remark 5.5 we may assume w.l.o.g. that w(Zj) == 0, i == 1,2. Let G· be a planar dual graph of G such that F = Z· c E·. Let 6.1
.
G = G·\F and w(e) == w(e), e is the dual edge of e, 'tie E E and T == {fl,f2'~,f,} {tl,f2,~,f4} C V where fl fl == (t (fll ,f2) and /2 == (t (f33 ,f,,). Let (G" w·) be the extended graph of (G, w) . From 5.6.1 and Lemma 5.2, k == w·(F) == min6(Z; w) and F is an optimal T-join. From 3.4 there exists a flow of value min6(Z; min6(Zj w) and hence an optimal one. Using 3.4 and Theorem 4.6 we conclude that there exists an integral flow of value min 6 (Z (Z;i w) - 1 in (G,w), hence max/(i)(Zjw) ~ min6(Zjw)-1. min6(Z;w)-1. The second part of the theorem is derived as follows: By 3.4 we have max f(i)(Zj w) == min6(Z; min6(Zj w) - 1 if and only if there is no optimal packing of T~uts which is integral in (G.,w·). By Theorem 5.9 there is no optimal packing of T~uts which is integral in (G"w·) if and only if (G"w·) is F-special. By Lemma 5.6 (G"w·) is F-special if and only if (G,w) is Z-special.O Remark 6.2: Theorem 6.3 below which was proved by us in [7] is an extention of Theorem 4.6 and hence could replace Theorem 4.6 in the alternative proof of Theorem 4.l. Theorem 6.3 Let T C V he an ellen subset of lIertices and nF the number of components in an optimal T -;join, F·. Let lIw (G) he the lIalue of an optimal integral paclcing of T -cuts
then w(F·) -
lIw (G)
< nF -l.
6.4 Alternative Proof of Corollary 4.2: Let (G·, w·) be the extended graph as defined
i~ 6.1. Clearly, based on 3.2 and 5.6.1 we have, min6(zljw) 13
+ min6(z2;w) min6(z2iw) i: min6(Zjw)
if and only if min{w{G);/I E G,G E C}
+ min{w(G) : h.
E G,G E C}
=1=
min{w{G)
(11,/2) C GjG E C} From Lemma 5.10 we get There exists an optimal packing of T-cuts which is integral and 3.4 implies that
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
max/(i)(Z; w) = min6(Z;w).O max/(il(Z;
7
Algorithms As a consequence of the previous results we present three polynomial time algorithms.
Algorithm 1 is a recognition algorithm. Given (G' , w') an augumented planar graph, with Z C E !Z! = 2 find whether there exists an integral two--commodity flow 1 such that
11/112: min6(Z;w), i.e., find whether there exists an optimal two--commodity flow which is integral in (G' , w').
&
Given (G',w') an augumented planar graph with Z C E, putes the maximum integral two-commodity How in (G', w').
IZ!
= 2, Algorithm 2 comcom
Algorithms 1 and 2 are consequences of Seymour's theorems and algorithms and are not based on Lomonosov's results. Given (G,w) an augumented planar graph with Z C E, IZI = 2, (G,w) is a Z-special graph, then Algorithm 3 provides us with a simple method for defining the 6-partition of the vertices of the graph as described in Definition 2.1.5. First we shall present this three algorithms and discuss their complexity in the sequal. Since these three algorithms start with identical four steps we shall present these four steps steps
under the preprocessing algorithm. algorithm.
Preprocessing Algorithm: Algorithm:
Input: planar graph (G, w), Z
c E, IZI = 2,
w:E
~
Z+, w{Z) = O.
Output: planar graph (G·, w·), such that G· is the dual graph of G with w· as weight function and if IV(Z·)I ~ 4 then, A, a collection of six shortest paths between the ends of Z·, as described in Procedure 5.7. (Note that in Algorithms 1 and 2 any collection of six shortest paths is suffices).
14
Step 1: Using max-flow algorithm calculate min O(Z; w). Step 2: Using Booth and Lueker's [3] algorithm find G-, the dual graph of G. Let Z- = F and define w-(e-) = w(e) \:Ie- E G-\F where e- is the dual edge of e, w-(F) will be
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
defined in Step 4. Step 3: If IV- (F) I ~ 3 stop. Otherwise let v- (F)
= T- where V- (11) = (ti, til.
Otherwise
T- = {ti, ti, t;, t:}. Find A = {Aii : 1 ~ i < j ~ 4} a collection of shortest paths in G-\F from ti to t; and their length for each 1 ~ i < j < 4 by using Frederickson's [5} algorithm and Procedure 5.7.
End. Algorithm 1: Input: (G, w), Z
c E, IZ/ = 2, w: E -+ Z+, w(Z) = o. o.
Output: An answer to the question: "Does there exist an optimal two-commodity How How
which is integral?" Step 0: Perform the Preprocessing Algorithm. Step 1: If jV·(F)1
< 3 go to Step 5.
Step 2: If: w-(A~2) + w-(A;. ~ mino(Z;w)
+ w-(A;.) w-(Ai.) + w-(A;s) w-(A~s)
~ mino(Z; w) ~
mino(Z; w)
go to Step 3, otherwise go to Step 6. Step 3: If there is any strict inequality in Step 2 go to Step 5.
Step 5: There exists an optimal two-commodity How which is integral. Stop. Step 6: There does not exist an optimal
2~ommodity How
End.
15
which is'integral.
Algorithm 2: Input: (G,w), ZcE, /ZI=2, w:E-+Z+, w(Z) =0
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Output: Maximum integral two-commodity flow Step 0: Perform the Preprocessing Algorithm. Step 1: IT I(V"(F)/ ~ 3 there is always an integral packing of TO-cuts and a direct way to find the optimal integral flow.
=
Step 2: IT ITO' 4 using Seymour's technique find po - an optimal integral packing of T°4:uts in (Go,WO). Step 3: Calculate the optimal integral two-commodity flow in (G, w) by the following method: "'Ie E E : fee) = m where e = (eo)" and there are m T°4:uts in po that intersect eO.
End. As was mentioned before, Algorithm 3 provides a simple method to define a 6-partition 6-parlition
of the vertices of a Z-special graph. This simplifies the method for finding the 6-partition as is implicitly suggested by Lomonosov's long proof of Theorem 4.1.
16
16
t
.
Algorithm 3: Input:
(G, w)
Output: If the graph is Z-special then the output is a 6-partition of V as in the definition
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
of Z-special graphs. If not, the output is: "The graph is not Z-special."
Step 0: Perform Algorithm 1 - If there is an optimal flow which is integral go to the End with the answer: "The graph is not Z-special." Step 1: Define the following 6-partition (as defined in the proof of Theorem 5.9) in the following way: Let [(. be a subdivision of K. with T = {tl, t2, t3, t 4} as V (K.) , and ~; 1 < i < j < 4 (the shortest path from t, to t; as can be obtained by Procedure 5.7) as the subdivision of e = (t"
til, e E E(K4 ), in K•.
[(4 is the following graph:
....
It = (tl, t2) and f2 = (t 3, t.) to [(4. [(4.
First add It to 114 so that 11 divides F1 into two regions Let us add
such that F. is divided to two regions graph.
graph.
5; and T;.
I
11
5; and T;, then add f2f2
Eventually we have the following following
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
Clearly, {Si,S;,R;.,!li,Ti,T2-} is a 6-partition of the plane, with the following circir cuits, accordinly, as their boundaries. (These circuits were defined previously in 5.9.3.)
C- (Si) = Ai2 u {II} C-(S;) = Ai. u {/2} C-(R;) = Ai.L\A;.L\ A i2 C-(!li) = A 2.L\A;sL\As4 C-(Ti} = A~sL\A;2L\{fl} . C-(T;) = A~sL\Ai,L\{/2} Step 2:
Let (Sh 8 2 , Rt, R 2, Th T2 ) be the sets of vertices that correspond to the regions
in (G"w-) via duality. Then (St,S2,Rt,R2,Tt,T2) is the require 6-partition. End.
Lemma '1.1 The time complexity 01 the Preprocessing Algorithm is O(IVI 2 log IVI). Proof: By checking each step s,eperately, we shall show that the complexity of the algoalgo rithm determined by Step 1. Step 1: By using a simple transformation we can calculate mino(Zjw) by calculating max-flow in the following way: Let Z = {Zl = (t h t2),Z2 = (t3,t4)), where (tht3) be the sources and (t2,t,) the sinks. Let us add two vertices, Sb a super source and 82, 18
Technion - Computer Science Department - Tehnical Report CS0617 - 1990
a super sink. Join 81 to the two source tl and t~h and 82 to the two sinks t2 and t •• Let the weight of each of the new added edges be M> Ecec wee).
It is not 9.ifficult to see that the maximum flow from 81 to 82 is equal to min8(Z; w). We can find the maximum flow from 81 to 82 by using Sleator flow algorithm [14] in O(IVI·IEllog IV/). Since in planar graphs O(IVI) = O(IE/) we have that the complexity of the first step is O(IVI 210gIV/). - The original graph is planar, but after adding the four edges the graph might be nonplanar and therefore we can not use flow algorithms for planar graphs which are more efficient. Step 2: By using Booth and Lueker's algorithm [3] we can find G· the dual graph of G in O(lVI).
t;
Step 3: For constructing A, a collection of 6 shortest paths from ti to tj 1
