Minimum multicuts and Steiner forests for Okamura-Seymour graphs

Minimum mul ticuts and Steiner f orests f or Okamura-Seymour graphs Arindam P al Departmen t of Computer Science and Engineering Indian Institute of T echnology , Delhi New Delhi – 110016, India arindamp@cse.iitd.ernet.in F ebruary 27, 2011 Abstract W e study the problem of finding minimum multicuts for an undirected planar gr aph , where all the terminal vertices are on the b oundary of the outer face. This is known as an Okamur a-Seymour instanc e . W e sho w that for suc h an instance, the minimum multicut problem can b e reduced to the minimum-c ost Steiner forest problem on a suitably defined dual gr aph . The minimum-cost Steiner forest problem has a 2-appro ximation algorithm. Hence, the minimum multicut problem has a 2-approximation algorithm for an Ok am ura-Seymour instance. 1 In tro duction Computing the minimum multicut of a graph is an important problem in com- binatorial optimization. The problem is formally defined b elo w. Minimum Mul ticut: Given an undirected graph G = ( V , E ) on n vertices and m edges with edge costs c e and a set T = { ( s 1 , t 1 ) , . . . , ( s k , t k ) } of k termi- nal pairs. Our goal is to find a set of edges F of minim um cost separating all the terminal pairs ( s i , t i ) , 1 ≤ i ≤ k , i.e. , in the subgraph H = ( V , E \ F ), there is no path b et w een s i and t i , for 1 ≤ i ≤ k . The minimum multicut problem is NP-hard and APX-hard , ev en for trees, whic h excludes the p ossibilit y of a PT AS. Garg, V azirani and Y annak akis gav e a 2-approximation algorithm for trees [ Garg et al. , 1997 ] and an O (log k )- appro ximation algorithm for general undirected graphs [ Garg et al. , 1996 ]. When G is planar and all the terminals are on the b oundary of the outer face of G , 1 w e denote suc h an instance of the minimum multicut problem an Okamur a- Seymour instance. The problem of finding edge-disjoint paths on such type of graphs were studied by Ok amura and Seymour [ Ok amura and Seymour , 1981 ]. They show ed the following imp ortan t theorem ( d F ( S ) denotes the num b er of edges in F exactly one of whose endp oin ts is in S ). Theorem 1 ( Okamura-Seymour ) . L et G = ( V , E ) b e an undir e cte d planar gr aph and let R = { ( s i , t i ) : s i , t i ∈ V , 1 ≤ i ≤ k } b e a set of terminal p airs. Supp ose the fol lowing c onditions ar e satisfie d. 1. A l l terminals ar e on the b oundary of the outer fac e of G . 2. The Euler c ondition is satisfie d: ( V , E ∪ R ) is Eulerian. 3. The cut c ondition is satisfie d: d E ( S ) ≥ d R ( S ) , for al l S ⊆ V . Then ther e exist e dge-disjoint p aths b etwe en s i and t i , for 1 ≤ i ≤ k . They also show ed the following corollary ab out the asso ciated multic ommo dity flow problem. Corollary 2 ( Okamura-Seymour ) . L et G = ( V , E ) b e an undir e cte d planar gr aph and let R = { ( s i , t i ) : s i , t i ∈ V , 1 ≤ i ≤ k } b e a set of terminal p airs. Supp ose the fol lowing c onditions ar e satisfie d. 1. A l l terminals ar e on the b oundary of the outer fac e of G . 2. The cut c ondition is satisfie d: d E ( S ) ≥ d R ( S ) , for al l S ⊆ V . Then ther e exists a fe asible multic ommo dity flow b etwe en s i and t i , for 1 ≤ i ≤ k . Mor e over, if c e ∈ N , ∀ e ∈ E and d i ∈ N , ∀ i : 1 ≤ i ≤ k , then ther e exists a half- inte ger multic ommo dity flow. Sc hw arzler prov ed that computing edge-disjoin t paths in such graphs without the Euler c ondition is NP-hard [ Sch w¨ arzler , 2009 ]. W agner and W eihe gav e a linear-time algorithm to compute edge-disjoint paths in suc h graphs [ W agner and W eihe , 1995 ]. The multicommodity flow problem for an Ok am ura-Seymour instance was studied by Matsumoto et al [ Matsumoto et al. , 1985 ]. Their algo- rithm decides whether G has a feasible multicommodity flow, eac h from a source to a sink and of a given demand, and actually finds them if G has one. If G has n v ertices and k source-sink pairs, their algorithm takes O ( kn + n 2 √ log n ) time and O ( kn ) space. How ev er, the dual problem of computing the minimum m ulticut has not been addressed by them. W e tak e a step in that direction. Our main result is the following. Theorem 3. The minimum multicut pr oblem on an Okamur a-Seymour in- stanc e c an b e r e duc e d to the minimum-c ost Steiner for est pr oblem on an ap- pr opriately define d dual gr aph. The minimum-c ost Steiner for est pr oblem has a 2 -appr oximation algorithm. Henc e, the minimum multicut pr oblem has a 2 - appr oximation algorithm for an Okamur a-Seymour instanc e. 2 The minimum-cost Steiner forest problem is formally defined b elo w. Minimum-cost Steiner forest: Given an undirected graph G = ( V , E ) on n v ertices and m edges with edge costs c e and a set T = { ( s 1 , t 1 ) , . . . , ( s k , t k ) } of k terminal pairs. Our goal is to find a set of edges F of minimum cost connecting all the terminal pairs ( s i , t i ) , 1 ≤ i ≤ k , i.e. , in the subgraph H = ( V , F ), there is some path b et w een s i and t i for 1 ≤ i ≤ k . There is a p olynomial-time 2-approximation algorithm for the minimum-cost Steiner forest problem in an y undirected graph, due to Go emans and Williamson [ Go emans and Williamson , 1995 ]. A comprehensiv e surv ey of all these results is given in [ F rank , 1990 ]. F or a more recent survey , see [ Nav es and Seb o , 2009 ]. 2 Reduction of minim um m ulticut to Steiner forest Let OF be the outer face of G and B be the b oundary of O F . W e construct the dual graph G d = ( V d , E d ) of the planar graph G as follows. 1. F or eac h finite face f of G , we asso ciate a dual v ertex v f of G d . 2. F or each edge e = ( u, v ) ∈ E on the boundary of t wo finite faces f and g of G , we asso ciate a dual edge e d = ( v f , v g ) ∈ E d . 3. F or each terminal pair ( s i , t i ), we add t wo dual v ertices u i , v i ∈ V d suc h that u i , v i are on O F in the planar embedding of G . The terminal pair ( s i , t i ) divides B in to t wo parts. Let us denote by [ s i , t i ] the p ortion encoun tered while trav ersing B from s i to t i in the an ti-clo c kwise direction, and b y [ t i , s i ] the other p ortion. W e asso ciate u i with [ s i , t i ] and v i with [ t i , s i ]. 4. F or each edge e ∈ E on B , we add a dual vertex v e on O F and a dual edge e d = ( v e , v f ) ∈ E d , where e is the common boundary of O F and a finite face f . 5. F or each edge e ∈ E on B that we come across during the trav ersal of [ s i , t i ], w e add a dual edge e d = ( u i , v e ) ∈ E d , where e is the common b oundary of O F and a finite face f and v e is a dual vertex on O F added in the last step. Similarly , we add dual edges ( v i , v e ) for the portion [ t i , s i ]. W e assign a cost c e to dual edges of t yp e (2) and (4). F or dual edges of type (5), w e assign a cost N = P e ∈ E c e , i.e. , the sum of all edge costs in G . Note that there is a technical difference b et w een the dual graph defined here and the traditional definition. Here we hav e many dual vertices on the infinite face, whereas traditionally there is only one dual vertex on the infinite face. This is illustrated in Figure 1 . In general, the dual graph G d is not planar. W e say that a set of edges F ⊆ E c orr esp onds to a set of edges F d ⊆ E d , if for every 3 s 1 t 1 s 2 t 2 s 3 t 3 v 1 u 1 Figure 1: A planar graph G and its dual G d . T erminals are in blue, dual vertices and edges are in red. Only u 1 and v 1 are shown for clarity . The other vertices u 2 , v 2 , u 3 , v 3 and the edges inciden t on them can b e drawn similarly . edge f ∈ F , there exists one and only one edge f d ∈ F d suc h that f d is the dual edge of f . Let R = { u 1 , v 1 , . . . , u k , v k } b e the set of dual vertices on O F con taining all the ( u i , v i ) pairs and let S = V d − R b e the set of remaining v ertices of G d . W e call R the set of r e quir e d vertic es and S the set of Steiner vertic es . Our goal is to connect all the pairs ( u i , v i ) ∈ R . Let S F b e a Steiner for est of the dual graph G d connecting all ( u i , v i ) pairs in R using some Steiner vertices in S . W e call a dual edge an internal e dge , if b oth its endp oin ts are inside the outer b oundary B , an external e dge , if b oth its endp oin ts are outside B , and a cr ossing e dge , if one endp oin t is inside and the other is outside B . Note that the external edges ha ve a cost of N and the other edges ha ve a cost of the corresp onding primal edge. F or con venience, we prov e the following technical result. Lemma 4. Supp ose P i is a p ath in G b etwe en the terminals s i and t i . F urther, Q i is a dual p ath in G d b etwe en the dual vertic es u i and v i such that only its first and last e dges ar e external e dges. Then, P i and Q i must cr oss e ach other. Henc e, if ( V ( G ) , E ( G ) \ F ) c ontains no s i - t i -p ath, then the dual e dge set of F plus two external e dges c ontains a u i - v i -p ath. 4 Pr o of. The dual path Q i starts and ends at the outer face O F and passes through the in terior of B . Hence, it separates s i and t i , i.e. s i and t i lie on differen t sides of Q i . So, any path P i b et w een the terminals s i and t i m ust in tersect the dual path Q i . W e claim that the minimum-cost Steiner forest can’t use to o many external edges of cost N . Lemma 5. L et M S F b e a minimum-c ost Steiner for est in G d c onne cting the p airs ( u i , v i ) ∈ R . Then, M S F c an use at most 2 k external e dges, one for e ach r e quir e d vertex w ∈ R . Pr o of. Supp ose this is not the case. If we use more than 2 external edges to connect a pair ( u i , v i ) ∈ R , we need at least 4 suc h edges. This follows from a simple pairing argumen t: if we go outside, w e hav e to come bac k inside using an external edge and if we use only external edges, w e need at least 4 such edges to connect u i and v i . But we need at least 2 external edges to connect any pair, b ecause all the edges incident on any u i , v i are external edges. Hence, cost of M S F is at least (2 k + 2) N , whereas a Steiner forest using 2 k external edges will hav e cost at most 2 k N + P e ∈ E c e < (2 k + 2) N , which is strictly less than the cost of the M S F , contradicting the fact that M S F is a minimum Steiner forest. Next we show the relationship b etw een Steiner forests and m ulticuts. Lemma 6. Every Steiner for est S F in G d c orr esp onds to a multicut M C in G . Pr o of. Consider the dual path Q i in S F b et ween the dual vertices u i and v i . By Lemma 2, an y path P i in G b et ween the terminals s i and t i m ust in tersect Q i . Therefore, primal edges corresp onding to Q i is a s i − t i cut. Since this is true for all terminal pairs, the primal edges corresp onding to S F is a multicut M C in G . W e observe that not every multicut M C in G is a Steiner forest S F in G d . W e say that M C is a minimal multicut if for an y edge e ∈ M C, M C − { e } is not a multicut, i.e. there exists terminals s i and t i suc h that there is a path P i b et w een s i and t i in G − M C ∪ { e } . W e show the relationship b etw een minimal m ulticuts and Steiner forests in the next lemma. Lemma 7. Every minimal multicut M C in G c orr esp onds to a Steiner for est S F in G d . Pr o of. Since M C is a multicut in G , there is no path b etw een s i and t i in G − M C . By Lemma 2, there must b e a dual path betw een u i and v i . Hence all pairs ( u i , v i ) ∈ R are connected. Moreov er, since M C is minimal, there is only one path b etw een u i and v i . Thus, the dual edges corresp onding to M C is a Steiner forest S F in G d . Com bining Lemmas 3, 4 and 5, we arrive at Theorem 3. 5 3 Conclusion and op en problems In this pap er, we studied the minimum multicut problem for an undirected pla- nar graph, where all the terminal vertices are on the boundary of the outer face. W e sho wed its relation to the minimum-cost Steiner forest problem in the dual graph and gav e a 2-appro ximation algorithm. Are there similar relationships b et w een these problems in a general undirected graph? Is there a direct 2- appro ximation algorithm for the minimum multicut problem without reducing it to the Steiner forest problem? References Andras F rank. Pac king paths, circuits, and cuts - a survey . In Bernhard Korte, Laszlo Lov asz, Hans Jurgen Promel, and Alexander Schrijv er, editors, Paths, Flows, and VLSI-L ayout , pages 47–100. Springer-V erlag, 1990. Na veen Garg, Vijay V. V azirani, and Mihalis Y annak akis. Approximate max- flo w min-(multi)cut theorems and their applications. SIAM J. Comput. , 25 (2):235–251, 1996. Na veen Garg, Vijay V. V azirani, and Mihalis Y annak akis. Primal-dual approx- imation algorithms for in tegral flow and m ulticut in trees. Algorithmic a , 18 (1):3–20, 1997. Mic hel X. Go emans and Da vid P . Williamson. A general approximation tech- nique for constrained forest problems. SIAM J. Comput. , 24(2):296–317, 1995. Kazuhik o Matsumoto, T ak ao Nishizeki, and Nobuji Saito. An efficien t algorithm for finding multicommodity flows in planar net works. SIAM J. Comput. , 14 (2):289–302, 1985. Guyslain Nav es and Andras Seb o. Multiflow feasibility: An annotated tableau. In William Co ok, Laszlo Lo v asz, and Jens Vygen, editors, R ese ar ch T r ends in Combinatorial Optimization , pages 261–283. Springer Berlin, 2009. Haruk o Ok amura and P . D. Seymour. Multicommo dit y flows in planar graphs. J. Combin. The ory Ser. B , 31(1):75–81, 1981. W erner Sch w¨ arzler. On the complexit y of the planar edge-disjoin t paths problem with terminals on the outer b oundary . Combinatoric a , 29(1):121–126, 2009. Dorothea W agner and Karsten W eihe. A linear-time algorithm for edge-disjoint paths in planar graphs. Combinatoric a , 15(1):135–150, 1995. 6