Large Isolating Cuts Shrink the Multiway Cut

Large Isolating Cuts Shrink the Multiw a y Cut Igor Razgon ir45@mcs.le.a c.uk Department of Co mp uter Science, Universit y of Leicester Abstract. W e p ropose a preprocessing algorithm for the m u ltiwa y cut problem that establishes its p olynomial kerneli zability when the d iffer- ence b etw een the parameter k and the size of the smalles t isolating cut is at most log ( k ). T o the b est of our kno wledge, th is is the first progress to wards kernelization of t h e multiw ay cut problem. W e p ose tw o op en questions that, if a n sw ered affirmatively , w ould imply , combined with the prop osed result, unconditional p olynomial kerneliza b ilit y of the multiw ay cut problem. 1 In tro duction 1.1. Ov erview of the prop os ed results. Giv en a pair ( G, T ) wher e G is a graph and T is a sp ecified set of vertices, called t erminals , a (vertex) Multiwa y Cut ( mw c ) of ( G, T ) is a set o f non-terminal vertices whose remov al fro m G separates all the terminals. The mwc pro blem asks to compute the sma lle st mw c of G . It is NP-har d for | T | ≥ 3 [5]. In this pape r we concentrate on the para meterized v er sion ( G, T , k ) of the mw c problem where w e a r e given a parameter k and a sked whether there is an mw c of ( G, T ) o f size at most k . The goal we work tow ards is understanding the kernelizability of the mwc problem. In other words, w e wan t to understand, whether ther e is a p olynomial time algorithm that tra nsforms ( G, T , k ) into a n e quivalent instance ( G ′ , T ′ , k ′ ) (equiv alent in the sense that the for mer is the ’YES’ instance iff the latter is) suc h that | V ( G ′ ) | is upper -b ounded by a p oly- nomial of k ′ and k ′ itself is upp er b ounded by a po lynomial of k . Informally sp eaking we w a n t to shrink the instance of the mwc problem to a size p olyno- mially depe ndent on the par ameter. The kernelizabilit y o f the mw c is co ns idered by the pa rameterized complexity communit y as an interesting and challenging question. In this pa per we prop o se a pa rtial res ult and po se tw o op en questions that, if resolved affirmatively , will imply , together with this result, that the mwc pro blem is kernelizable. An in- formal ov erv iew is g iven b elow. Let ( G, T , k ) b e an instance of the mwc problem. An isolating cut [5 ] is a set of non-terminal vertices separating a ter minal t from the rest of terminals . Let r be the smallest size o f an isolating cut. Clearly we can assume r ≤ k o therwise, ( G, T , k ) is a ’NO’ instance . In this paper w e propo se an algorithm transfor ming the initial instance in to an equiv alent one whose size is O (2 k − r rk 2 ). The runtime of this algorithm is O ( A ( n ) + 2 k − r n 3 r 2 k 4 ) where A ( n ) is the runtime of the constant ratio a pproximation algorithm for the vertex mwc pr o blem prop ose d in [6]. Thu s w e demonstrate that for every fixed co nstant c the subclass of mwc problem consisting of instances w ith k − r ≤ c ∗ l og k is p olynomially k er nelizable. T o the b est of our knowledge, this is the first progre s s to wards kernelization of the mwc problem. Two more merits of the prop osed results a re that it might b e a building block in an unconditional kernelization of the mw c problem and that it g ives a new insig h t into the str ucture of imp ortant sep ar ators [8]. T o justify these merits, w e provide b elow a more detailed o verview of the pro po sed res ult. The main ingredient of the prop osed algorithm is computing for ea c h t ∈ T the union U t of all impo rtant isolating cuts of t of size at most k . An almost immediate consequence of Lemma 3.6. of [8] shows that the union of a ll U t contains a solution o f ( G, T , k ) if such exists. Therefore , ’contracting’ the rest of non-terminal vertices results in an instance equiv alent to ( G, T , k ). P rior to computing the sets U t we e ns ure that the size of | T | is at mos t 2 k ( k + 1). This is done in Section 3 b y running the appr oximation algorithm o f [6 ] a nd pr o cessing the o utput in the flavour o f a simple quadr atic kernelization alg o rithm for the V ertex Co ver problem (i.e. noticing tha t the v e r tices of the giv en mw c adjacent to a large num b er of terminal compo nent s must b e pr e s ent in an y solution and, after remov al of these vertices and the alr eady s e parated terminals, the num b er of remaining terminals is sma ll). But what is the size o f U t and what is the time needed fo r its computation? T o understa nd this, we study (in Section 2 ) imp ortant X − Y separato r s o f G [8] where X a nd Y ar e t wo arbitrar y subsets of vertices. As a result, we obtain a combinatorial theo rem saying that if r is the smallest size of an X − Y separator then for an a rbitrary x the size of the union of all X − Y imp or ta n t separator s of size at most r + x is at most 2 x +1 r a nd these vertices can be computed in time O ( n 3 2 x r 2 ( r + x ) 2 ). 1 The exponential part of the runtime follows from the need to enumerate so-called princip al imp ortant s ep ar ators whose union includes all the needed vertices. W e arg ue that the principal impo rtant separ ators cons titute a genera lly small s ubs et of the whole s et of impo r tant separa tors and po se the first op en question asking whether the num b er of principal sepa rators c a n b e bo unded b y a p olynomia l of n . The affirmative ans w er to this question implies the p olynomial runtime of the alg orithm prop osed in this pap er. In this case the algorithm can b e a first step of a kernelization metho d of the mwc . How ever, it canno t be the only step. W e demonstrate the upp er bo und on the n umber of vertices is tight and hence g enerally ca nno t p olynomially depend on k . Ther e- fore, a natura l questio n is whether the output o f this algor ithm can b e further pro cessed to obtain an unconditional kernelization. W e p ose this as our se c ond op en qu estion . 1.2. R elated work. There are ma ny publications related to the topics con- sidered in the pap er. W e overview only thos e that a re of a dir ect relev ance for the prop osed results. 1 The results are obt ained wi t h out an y regard to the mw c problem, hence they might b e of an ind epen dent in terest. The fix ed-parameter tractability of the mwc problem has b e e n established in [8] and the r un time has b een impr ov ed to O ∗ (4 k ) in [4]. A parameteriza tion of the mwc pr o blem ab ov e a guaranteed v alue has be e n recently pro po sed in [9], where we show that the pro blem is in XP under this pa rameterizatio n leaving op en the fixed-par ameter tr actability sta tus. The notion o f imp ortant sep ar ator has bee n int r o duced in [8]. As noticed in [7], the recent algor ithms for a n umber of challenging g raph separa tion problems, including the one o f [4], are bas ed on en umera tion of imp or tant separ ators. F urther on, [7] prov es a n upp er b ound 4 k on the num b er of imp ortant separators of size at mo st k and notices that the algo rithm of [4] in fact implicitly e s tablishes this upper bound. An alter native upper b ound, suitable for the case where the smallest impo rtant se pa rator is large, is established in [9]. Constant ratio a pproximation algor ithms fo r the mwc problem have b een first prop osed in [5] for the edge version and in [6] for the vertex version. The r e s earch on kernelization has b een given its current shap e by the land- mark pap er [1], which allow ed to classify fixed-pa rameter tractable problems into kernelizable ones and those that ar e probably not. Among the ma ny known ker- nelizability and non kernelizabilit y results, let us mention the kernelization meth- o ds for multicut for trees [3] a nd F eedback V ertex Set [10] and non-kernelizability pro of for the Disjoint Cycles pro blem [2 ]. Although far fr o m b eing ana logous to the mw c pro blem, all these problems ar e rela ted to the flow maximization/c ut minimization tasks and hence might b e a source of idea s useful fo r the final settling of the kernelizability of mwc problem. 2 Bounding t he union of imp ort an t separators Let X and Y b e t wo disjoint se ts of vertices of the given gra ph G . A set K ⊆ V ( G ) \ ( X ∪ Y ) is an X − Y sepa rator if in G \ K ther e is no path fro m X to Y . Let A, B b e t wo disjoint subsets of V ( G ). W e denote by N R ( G, A, B ) the set of vertices that are not re a chable from A in G \ B Let K 1 and K 2 be tw o X − Y separato r s. W e say that K 1 ≺ ∗ K 2 if N R ( G, Y , K 1 ) ⊂ N R ( G, Y , K 2 ). A minimal X − Y separato r K is called imp ortant if there is no X − Y separato r K ′ such that K ≺ ∗ K ′ and | K | ≥ | K ′ | . This notion was first int r o duced in [8] in a slightly different although eq uiv a len t way (see P rop osition 3 of [9]) . Let r b e the s ize of a smallest imp orta n t X − Y separa tor and let S be an arbitrary impo rtant separ ator. W e call | S | − r the exc ess of S . Then the following theo rem holds. Theorem 1. L et U b e the u nion of imp ortant X − Y sep ar at ors of exc ess at most x . Then | U | ≤ 2 x +1 r . Mor e over, U c an b e c ompute d in time O ( n 3 2 2 x r 2 ( r + x ) 2 ) . In this section we prov e Theo rem 1 and show the tightness of the upper bound of | U | . The pro o f of T he o rem 1 in divided into tw o stages. On the first stag e we introduce a pa rtially ordere d family o f subsets of the g iven set satisfying a num b er of certain pro pe r ties. W e c all such family of sets an IS-family . W e prov e Theo rem 1 in terms of the IS family . Then w e show that the family of all important s e pa rators with the ≺ ∗ relation is in fact an IS fa mily fr o m where Theorem 1 immediately follows. The adv antage of such ’axiomatic’ wa y of pr o of is the p ossibility to c learly sp ecify the pro per ties of the family o f imp o rtant separ ators (viewed as a partially ordered family of sets) that imply the above upp er bound. An additional poten- tial adv antage is that so me deep alg ebraic techniques might b eco me a pplicable for further inv estig a tion of the kernelization o f m ultiway cut. 2.1 F rom Imp ortan t Separators to P artiall y Ordered F amilies of Sets Let V be a finite set. Let ( F , ≺ ) b e a pair where F is a family of subsets of V and ≺ is an or der relation on the element s o f F . Let S ∈ F and v ∈ V . W e s ay that S c overs v if ther e is S ′ ∈ F suc h that S ′ ≺ S a nd v ∈ S ′ \ S . W e define P red ( S ) to b e the set of all S ′ such that S ′ ≺ S a nd there is no S ′′ ∈ F such that S ′ ≺ S ′′ ≺ S . Symmetrica lly , we define S ucc ( S ) to b e the set of all S ′ ∈ F such that S ≺ S ′ and there is no S ′′ ∈ F s uc h tha t S ≺ S ′′ ≺ S ′ . W e define the visible set of S denoted by V i s ( S ) to b e the s e t of all v ∈ V satisfying the following t wo conditions : – there is S ′ ∈ P red ( S ) such that v ∈ S ′ ; – v is not cov ered by any elemen t o f P red ( S ). ( F , ≺ ) is ca lled an I S -family if the following conditions a r e true. – Smal lest el emen t (SE) condition. Ther e is a unique element o f F denoted by sm ( F ) such that for a n y o ther S ∈ F , sm ( F ) ≺ S . – Strict m o notonicit y (SM) condition. Let S 1 , S 2 ∈ F . If S 1 ≺ S 2 then | S 1 | < | S 2 | . – Si ng le witnes s (SW) condition. Let S ∈ F and let v ∈ S . Let S ′ be a minimal element such that S ≺ S ′ and v ∈ S \ S ′ . W e call S ′ a witness of v w.r.t. S . The condition requires that there is at most one witness o f v w.r.t. S . – T ransi tiv e El imination (TE) conditi on Let S 1 ≺ S 2 ≺ S 3 be three elements of F and let v ∈ S 1 \ S 2 . Then v ∈ S 1 \ S 3 . – Large visible set (L VS) condi tion Let S ∈ F and let S ′ ∈ P red ( S ). Then | S ′ | ≤ | V is ( S ) | . F or the subsequent pro ofs we will use the extende d L VS condition stating that for ea ch S ′′ ≺ S , | S ′′ | ≤ | V is ( S ) | , whic h immedia tely follows from the com bination of L VS and SM conditions. – Dis tinct visi ble s et (D VS) condition F or e a ch S ∈ F such that S 6 = sm ( F ). Then V is ( S ) 6⊂ S . – Effi cien t Computabili t y (EC) condition Let n = | V | . In O ( n 3 ) we can compute sm ( F ) as well as the witnes s of v w.r.t. S for the given S ∈ F and v ∈ S (or re tur n ’NO’ in cas e such witnes s do es not exist). The relation S 1 ≺ S 2 can be tested in O ( | S 1 | ). In the rest of this subsection we assume that ( F , ≺ ) is an IS-family . Our reasoning consists of three stage s . On the fir st stage, we prove 3 prop ositions stating simple prope r ties of an IS family . On the second stage w e prov e Theorem 2, our main counting result. The main bo dy of the pro of is provided in the 3 preceding lemmas. O n the la st stage we prov e an a nalogue of Theorem 1 for IS families: Co rollary 1 pr ov es the upp er b ound on the s iz e of the union of the resp ective sets a nd Theorem 3 establishes an a lgorithm for computing of these sets. F o r S ∈ F let us denote S \ S S ′ ≺ S S ′ by hat ( S ). Prop ositio n 1 . hat ( S ) = S \ V is ( S ) . Pro of. It is clear fro m the definition tha t hat ( S ) ⊆ S \ V is ( S ). Co n versely , consider v ∈ ( S \ V is ( S )) \ hat ( S ). What ca n we s ay ab out such v ? Fir st, that v ∈ S . Then, since v ∈ S S ′ ≺ S S ′ \ V is ( S ), there may b e t wo p ossibilities. According to one of them, v ∈ S ′ ≺ S such that S ′ / ∈ P red ( S ) and v do es not b elong to any S ′′ ∈ P r ed ( S ). It follows that there is S ′′ ∈ P r ed ( S ) such that S ′ ≺ S ′′ and v ∈ S ′ \ S ′′ . Since S ′′ ≺ S , v / ∈ S b y the TE condition in co n tr adiction to our a ssumption. The other p oss ibility may b e that v ∈ S ′ ∈ P r ed ( S ) and v is covered by another S ′′ ∈ P red ( S ). Then ana lo gous reasoning applies. By definition of a covered vertex, there is S ∗ ≺ S ′′ such that v ∈ S ∗ \ S ′′ and again v / ∈ S by the TE condition, yielding an analog ous con tr adiction.  . Prop ositio n 2 . L et S 1 , S 2 , v b e such that S 1 ≺ S 2 and v ∈ S 1 \ S 2 . L et S ∗ b e the witness of v w.r.t. S 1 . Then S ∗  S 2 . Pro of. Let S ′′ be a minimal e le men t of F such that S 1 ≺ S ′′  S 2 and v ∈ S 1 \ S ′′ . Then S ′′ is a witness of v w .r .t. S 1 . By the SW condition, S ′′ = S ∗ .  Prop ositio n 3 . L et S ∈ F and let v ∈ V is ( S ) \ S . Then ther e is S ∗ ≺ S such that v ∈ hat ( S ∗ ) and S is the witness of v w.r.t. S ∗ . Pro of. L e t S ∗ be a minimal element of F preceding S such that v ∈ S ∗ . Then v ∈ hat ( S ∗ ). Indeed, otherwise, ther e is S ′ such that v ∈ S ′ ≺ S ∗ ≺ S in contradiction to the choice of S ∗ . Assume b y contradiction that S is not the witness of v w.r.t. S ∗ and let S ′′ be this witness. Accor ding to Prop osition 2, S ′′ ≺ S . Let S 2 ∈ P red ( S ) b e such that S ′′  S 2 . Clea r ly S ∗ ≺ S 2 . If S ′′ = S 2 then v ∈ S ∗ \ S 2 by definition of S ′′ . Otherwise, v ∈ S ∗ \ S 2 by the TE condition. It follows that S 2 cov ers v . Consequently , v / ∈ V is ( S ), a contradiction pr oving that S is indeed the witness of v w.r.t. S ∗ .  F o r S ∈ F , let’s call | S | − | sm ( F ) | , the ex c ess of S and deno te it ex ( S ). Lemma 1. L et S ∈ F such that S 6 = sm ( F ) . F or v ∈ V is ( S ) \ S ,let S ( v ) b e such that v ∈ hat ( S ( v )) and S is the witness of v w.r.t. S ( v ) (the existenc e of s u ch S ( v ) fol lows fr om Pr op osition 3). Then | hat ( S ) | ≤ P v ∈ V is ( S ) \ S 2 ex ( S ) − ex ( S ( v )) . Pro of. If V is ( S ) = S then by Prop osition 1, h at ( S ) = ∅ and we ar e done. Otherwise, the DVS co ndition a llows us to fix a v ∗ ∈ V is ( S ) \ S . Let us define a function f on V as follows: f ( v ∗ ) = 2 ex ( S ) − ex ( S ( v ∗ )) and for w 6 = v ∗ , f ( w ) = 1. F o r S ⊆ V , the function naturally extends to f ( S ) = P v ∈ S f ( v ). Claim. | hat ( S ) | ≤ f ( V is ( S ) \ S ) Observe that f ( V is ( S )) = | V is ( S ) \ { v ∗ }| + f ( v ∗ ) = | V is ( S ) | + f ( v ∗ ) − 1 . By the extended L VS condition, the rig h tmos t part of the ab ov e equality does not increase if we repla ce V i s ( S ) by S ( v ), i.e. f ( V is ( S )) ≥ | S ( v ) | + f ( v ∗ ) − 1. Since ex ( S ) − ex ( S ( v )) ≥ 1, by the SM condition, f ( v ∗ ) ≥ ex ( S ) − ex ( S ( v )) + 1. That is, f ( V is ( S )) ≥ | S ( v ) | + e x ( S ) − ex ( S ( v )) = | S | . F ur ther more f ( V is ( S )) = f ( V is ( S ) \ S ) + f ( V is ( S ) ∩ S ) = f ( V i s ( S ) \ S ) + | V is ( S ) ∩ S | . On the other hand, | S | = | S \ V i s ( S ) | + | V is ( S ) ∩ S | = | hat ( S ) | + | V is ( S ) ∩ S | , the last equality follows fro m Prop osition 1. Thus the desired claim follows by remov a l | V is ( S ) ∩ S | from the b oth sides of the inequality f ( V is ( S )) ≥ S .  Observe that due to the SM condition, for each v ∈ V i s ( S ) \ S , 2 ex ( S ) − ex ( S ( v )) ≥ f ( v ), hence f ( V is ( S ) \ S ) ≤ P v ∈ V is ( S ) \ S 2 ex ( S ) − ex ( S ( v )) . Ther efore the lemma follows from the abov e claim.  F o r x ≥ 0, let E x be the subset o f F c onsisting of all the elements of exc e s s at most x . Let S ∈ E x . The x -hat of S denoted b y hat x ( S ) is a subset of hat ( S ) consisting of all elements v such that there is no S ′ ∈ E x such that S ≺ S ′ and v ∈ S \ S ′ . Lemma 2. F or any x ≥ 0 P S ∈ E x 2 x − ex ( S )+1 ∗ | hat x ( S ) \ hat x +1 ( S ) | ≥ P S ′ ∈ E x +1 \ E x | hat ( S ′ ) | . Pro of. Denote the elements of E x by S 1 , . . . , S m . Denote { ( v , i ) | 1 ≤ i ≤ m, v ∈ hat x ( S i ) \ hat x +1 ( S i ) } by O S . F or eac h ( v, i ) ∈ O S , let aw ( v , i ) = 2 x − ex ( S i )+1 . F or OS ′ ⊆ OS , let a w ( O S ′ ) = P ( v, i ) ∈ OS ′ aw ( v , i ). It is not hard to see that the left part of the desired inequality is aw ( OS ). Indeed, for the given i , if we sum up aw ( v , i ) for all ( v , i ) ∈ O S then the total amount will b e exactly 2 x − ex ( S i )+1 ∗ | hat x ( S ) | . Consider ( v, i ) ∈ OS . Then, s ince v / ∈ hat x +1 ( S i ), there is S ′ ∈ E x +1 \ E x such that S i ≺ S ′ and v ∈ S i \ S ′ . W e claim that S ′ is in fact the witnes s of v w.r.t. S i . Indeed, o ther wise, ac c ording to Prop osition 2, S ′ succeeds the witness of v w.r.t. v hence, by the SM condition, the size of the latter is at most x . How ever, this contradicts v ∈ hat x ( S i ). By the SW condition, the ab ov e S ′ is unique for ( v , i ). So, we can sa y that ( v , i ) is witnesse d by S ′ . Denote the elements of E x +1 \ E x by S ′ 1 , . . . , S ′ q . Partition O S into OS 1 , . . . , OS q such that the elemen ts of OS q are witnessed by S ′ q . T o confirm the lemma, it remains to prov e that, for the given i , aw ( O S i ) ≥ | hat ( S ′ i ) | . Let v ∈ V is ( S ′ i ). Accor ding to P r op osition 3, there is S ∗ ≺ S ′ i such that v ∈ hat ( S ∗ ) and S ′ i is the witness of v w.r.t. S ∗ . By the SM condition ex ( S ∗ ) ≤ x , that is S ∗ ∈ E x . Observe that in fact v ∈ hat x ( S ∗ ) \ hat x +1 ( S ∗ ). Indeed, otherwise there is an element S ′′ of E x such that S ∗ ≺ S ′′ and v ∈ S ∗ \ S ′′ . But then S ≺ S ′′ by Pr op o sition 2 in contradiction to the SM c o ndition. Let j ( v ) b e such that S ∗ = S j ( v ) . It follows that ( v , j ( v )) ∈ OS i . Cons equent ly , aw ( O S i ) ≥ P v ∈ V is ( S ′ i ) 2 x +1 − ex ( S j ( v ) ) ≥ P v ∈ V is ( S ′ i ) 2 ex ( S ′ i ) − ex ( S j ( v ) ) ≥ | hat ( S ′ i ) | , the last inequality follo ws from Lemma 1.  F o r x ≥ 0 , denote P S ∈ E x 2 x − ex ( S ) | hat x ( S ) | by M ( x ). Then the following statement tak es place. Lemma 3. F or e ach x ≥ 0 , M ( x + 1) ≤ 2 M ( x ) . Pro of. First of a ll, obser ve that for each S ∈ E x +1 \ E x , hat x +1 ( S ) = hat ( S ) just b ecause, by the SM co ndition, there is no S ′ ∈ E x +1 such that S ≺ S ′ . F urthermor e, by definition, the excess of S is x + 1. Therefore | ha t ( S ) | = 2 x +1 − ex ( S ) | hat x +1 ( S ) | . That is, we ca n rewrite the inequality of Lemma 2 as P S ∈ E x 2 x − ex ( S )+1 ∗ | hat x ( S ) \ hat x +1 ( S ) | ≥ P S ′ ∈ E x +1 \ E x 2 x +1 − ex ( S ) | hat x +1 ( S ′ ) | F ur thermore, observe that for each S ∈ E x , hat x +1 ( S ) ⊆ hat x ( S ), therefor e hat x +1 ( S ) = h at x +1 ( S ) ∩ h at x ( S ). Then we ca n safely add P S ∈ E x 2 x − ex ( S )+1 ∗ | hat x ( S ) ∩ hat x +1 ( S ) | to the left par t of the inequa lit y of the previous paragraph and P S ∈ E x 2 x − ex ( S )+1 ∗ | h at x +1 ( S ) | to the right part of this inequality . Then af- ter noticing that for e ach S ∈ E x , | hat x ( S ) ∩ hat x +1 ( S ) | + | hat x ( S ) \ hat x +1 ( S ) | = | hat x ( S ) | a nd tha t the rig ht part in fact explo res | ha t x +1 ( S ′ ) | for all elemen ts S ′ ∈ E x +1 , the resulting inequality is transformed into: P S ∈ E x 2 x − ex ( S )+1 ∗ | h at x ( S ) | ≥ P S ′ ∈ E x +1 2 x − ex ( S ′ )+1 ∗ | h at x +1 ( S ′ ) | . It remains to notice that the left par t of this inequality is 2 M ( x ) and the right part is M ( x + 1).  Now we are ready to state the main counting r esult. Theorem 2. F or e ach x ≥ 0 , M ( x ) ≤ 2 x | sm ( F ) | . Pro of. Applying inductiv ely Le mma 3, it is easy to see that M ( x ) ≤ 2 x M (0). By definition, M (0) = P S ∈ E 0 2 0 − ex ( S ) | hat 0 ( S ) | . Since the only e lemen t of E 0 is sm ( F ) whose excess is 0 and ha t 0 ( sm ( F )) = sm ( F ), the theorem follows.  The follo wing corollary is the first s tatemen t of Theorem 1 in terms of an IS family Corollary 1. | S S ∈ E x S | ≤ 2 x +1 | sm ( F ) | . Pro of. O bserve that S S ∈ E x S = sm ( F ) ∪ S x i =1 S S ′ ∈ E i \ E i − 1 hat i ( S ′ ). Indeed, by definition, the left set is clea rly a sup erset of the rig ht one, s o let v be a vertex o f the left set. If v ∈ sm ( F ) then the containmen t in the r ight se t is clear. Otherwise, let S ∗ ∈ E x be a minimal set con ta ining v and let j > 0 be the excess of S ∗ . Then, by definition of sets E i , S ∗ ∈ E j \ E j − 1 . F ro m the minimalit y of S ∗ sub ject to the containmen t of v , it follows that v ∈ hat ( S ∗ ). F ur ther more, by the SM condition, there is no S ′′ ∈ E j such that S ∗ ≺ S ′′ . This implies that v ∈ hat j ( S ∗ ), confirming the observ ation. It follows from this equality that | S S ∈ E x S | is upper- b ounded by | sm ( F ) | + P x i =1 P S ∈ E i \ E i − 1 | hat i ( S ) | ≤ M 0 + P x i =1 M i ≤ P x i =0 M i . According to Theo- rem 2, the rightmost item of the ab ove inequality is clearly upp erb ounded b y 2 x +1 | sm ( F ) | , hence the cor ollary follo ws .  T o prov e the second statement of Theorem 1, we need to compute S S ∈ E x S . W e o btain the required algo r ithm in four simple steps. First we intro duc e the notion of princip al sets of F , then w e show that the union of principal sets o f excess at mos t x in fact includes all the vertices of S S ∈ E x S . F urthermo re, we show that the num b er o f principal sets ca n b e upper bounded by 2 x +1 | sm ( F ) | . Finally , w e sho w that sub ject to EC condition, these principal sets c a n b e com- puted in time p olynomia l in their b ound a nd in n = | V | . (Recall that V is the universe of for the sets of F ). W e s ay that a set S ∈ F is princip al if hat ( S ) 6 = ∅ . Denote by Pr x the family of all principal sets of excess at most x . B y definition, S S ∈ Pr x ⊆ S S ∈ E x . F or the other direction, let v ∈ S S ∈ E x . Then, ar guing a s in the pro of of Cor ollary 1 , we observe the e x istence of S ∗ of excess at mos t x such that v ∈ hat ( S ∗ ). Clearly S ∗ ∈ Pr x . Th us w e have established the fo llowing prop ositio n. Prop ositio n 4 . S S ∈ Pr x S = S S ∈ E x S Prop ositio n 5 . F or e ach x ≥ 0 , | Pr x | ≤ 2 x +1 | sm ( F ) | . Pro of. By definition, the num b er of elements of | Pr x | is upper - bo unded by the sum of the sizes of their hats, which in turn, is b ounded b y the sum of sizes of hats of a ll elements of E x . T aking into account that for each 1 ≤ i ≤ x and for each S ∈ E i \ E i − 1 , hat ( S ) = h at i ( S ) (argue as in the pro o f of Corollary 1 ), our upper bo und can be r epresented as | sm ( F ) | + P x i> 1 P S ∈ E i \ E i − 1 | hat i ( S ) | . Now, apply the second paragra ph o f the pro of of Corolla ry 1.  Theorem 3. Pr x c an b e c ompute d in time O ( n 3 2 2 x r 2 ( r + x ) 2 ) wher e r = | sm ( F ) | . Pro of sk etch. The algor ithm works iteratively . Fir st it computes Pr 0 . F o r each i > 0, it co mputes Pr i based on Pr i − 1 . Since Pr 0 = { sm ( F ) } , for i = 0, the result directly follows fro m the EC condition. No w co nsider co mputing of Pr i for i > 0 a ssuming that Pr i − 1 hav e been c o mputed. The algor ithm explores all the elements of Pr i − 1 and for each such elemen t S and for each v ∈ S , applies the witness computation a lg orithm of the EC condition. If the witness S ′ of S ha s bee n returned, S ′ joins Pr i if ex ( S ′ ) = i , S ′ has not b een already gener ated a nd the union of elements of Pr i preceding S ′ is not a s uper set of S . In the rest of the pro of, p ostp oned to the appe ndix, we prove corr ectness and the run time o f this algorithm.  2.2 Bac k to Imp ortan t Separators. Lemma 4. The family of al l imp ortant X − Y s ep ar ators of gr aph G p art ial ly or der e d by the ≺ ∗ r elation is an IS-family. Pro of sketc h. The SE co ndition is established by Lemma 3.3. of [8]. The SM condition immediately follows from the definition of an impo rtant separa tor. F o r the SW condition, let K be a n impor tant X − Y separ ator and let v ∈ K . As- sume that a witness of v w.r.t. K exists . Replace N R ( G, Y , K ) by a sing le vertex x and split v in to n + 1 co pies. Let G ∗ be the resulting g r aph. W e prov e that there is a bijection b etw een the witnesses of v w.r .t. K and smalles t imp orta n t x − Y separato r s of G ∗ and apply to G ∗ the SE condition. F o r the TE condition, we ob- serve (e.g. Pro po s ition 1 of [9]), that if K 1 ≺ ∗ K 2 then K 1 \ K 2 ⊆ N R ( G, Y , K 2 ). Thu s if K 2 ≺ ∗ K 3 , K 1 \ K 2 ⊆ N R ( G, Y , K 2 ) ⊆ N R ( G, Y , K 3 ), the last inclusion is obtained by definitio n o f the ≺ ∗ relation. Thus, no vertex of K 1 \ K 2 can b elong to K 3 . F o r the visible set co nditio ns, we first prove that if K is an imp or ta n t X − Y separato r different fro m the smalle s t one then for ea ch K ′ ∈ P re d ( K ), K ∗ = V is ( K ) \ N R ( G, Y , K ′ ) is also an X − Y separator such that K ′  ∗ K ∗ . The L VS and D VS c o nditions will immediately follo w from this claim combined with the definition of an important separa tor. The O ( n 3 ) algo rithm for comput- ing sm ( F ), as required by the EC condition follows from Lemma 1 in [9]. As shown in the pro of of the SW condition, computing of a witness is ess en tia lly equiv alent to computing o f an imp o rtant sepa rator. Finally the fast testing o f K 1 ≺ ∗ K 2 is easy to e stablish by maintaining an imp or tan t sepa r ator in an appropria te da ta structure.  Pro of of Theorem 1 The theorem immediately follows from combination of Coro llary 1, The o rem 3, and Lemma 4.  2.3 Lo wer b ounds and p ossibi lities for further improv eme n t W e s ta rt with sho wing that the obta ined upper b ound on the num b er of vertices inv olved in imp orta nt separators of size at most x is quite tight . Theorem 4. F or e ach x and r t her e is a gr aph H with two sp e cifie d t erminals s and t such that the size of the smal lest s − t sep ar ator is r and the size of t he union of al l imp ortant sep ar ators of exc ess at m ost x is 2 x +1 r − r . Pro of. T a ke r complete r o oted binary trees o f height x with 2 x leav es (of course, replace arcs b y undirected edg es). Add tw o new vertices s and t . Connect s to the ro o ts of the trees and t to all the leav es. This is the res ulting graph H for the given x and r . It is not hard to s e e that any minimal s − t sepa rator of this graph is an impo r tant one. It only r e ma ins to s how that each non-terminal vertex participates in a s − t separ ator of excess at most x . In fact, w e can show that any v e r tex v whose depth in the res pective bina r y tree is i participates in a s e pa rator of exce s s i . W e compute such separa tor by obtaining a sequence S 1 , . . . , S i of separato rs, where S i is the desired s eparator . S 1 is just the se t of neighbours of s . T o obtain S j +1 from S j , we specify the unique u ∈ S j such tha t u is the ancestor of v (the uniqueness easily fo llows by induction) and replace it by its children. The correctness of this co nstruction ca n b e easily establis hed by induction on the constructed sequence of separa tors, we omit the tedious details.  In the previous subsection we introduced the no tio n of a principa l set o f an IS-family . T he cor resp onding notion of a principal importa nt separator K means that K \ S K ′ ≺ ∗ K K ′ 6 = ∅ . Pro po sition 5 along with Lemma 4 implies that the nu mber of pr incipal impor tant X − Y separato r s of excess x is a t most 2 x +1 r where r is the size of the s mallest impo rtant X − Y separator and the class o f graphs consider ed in Theorem 4 s hows that this bound is tight. On the other hand, the num ber of pr inc ipa l imp ortant separ ators in this c la ss of gra phs is line ar in the ov e r all num b er n of v ertices . This leads us to the following question Op en Question 1 Is the numb er of princip al imp ortant X − Y sep ar ators of the given gr aph G b ounde d by a p olynomial of | V ( G ) | ? First of all obs erve that this question is reasonable b ecause the num b er o f principal s eparator s is gener a lly muc h sma lle r than the ov er all num b er of im- po rtant sepa rators. Indeed, in the class of instances considered in Theor em 4 , the ov era ll num b e r o f imp or tant separa tors is exp onential in n (consider the impo rtant sepa rators including leav es of the binary trees). T o see the significanc e of this op en question, suppose that the answer is yes . Then the alg o rithm claimed in Theorem 1 runs in a p olynomia l time. Indeed, its exp onential r un time is caused by the fact that the algo rithm explo res all pairs of principal impor tant s e pa rators , so, r eplacing the upper b ound has an immediate effect on the runtime. Such poly-time algo rithm would mean that it is pos s ible to test in a p olynomial time whether the given vertex b elongs to an imp ort ant sep ar ator , which is itself quite a n in ter esting achiev ement. Moreov er, the whole prepro cessing algorithm for the mwc problem propos ed in this pap er will have a polynomial time. This means tha t the output of this algorithm can be used for the further prepro cessing , p otent ia lly ma k ing easier the unco nditional kernelization o f the mwc problem. 3 Prepro cessing of mu lt iw a y c ut Let ( G, T ) be an instance of the mwc pro blem. An isolating cut of t ∈ T is a t − T \ { t } s e pa rator. If such sepa rator is imp ortant, we call it imp ortant isolating cut of t . W e start from a pr op osition that allows us to harness the machinery o f im- po rtant separ ators for the prepro ces s ing of the mwc pro blem. The prop osition is easily established by iterative applica tion the argument of Lemma 3 .6 of [8]. Lemma 5. L et ( G, T ) b e an instanc e of the mw c pr oblem. Then t her e is a smal lest mw c S of ( G, T ) such that e ach v ∈ S b elongs to an imp ort ant isolating cut of some t ∈ T .  With Lemma 5 in mind, w e can use the alg orithm cla imed in T he o rem 1 for the pr epro cessing. In par ticular, for each t ∈ T , let r t be the size of the s mallest isolating cut. Compute the set of all v ertices participating in the imp or ta n t isolating cuts of t . Let V ∗ be the set o f all the computed vertices together with the ter minals. Let G ∗ be the gr aph obtained from G [ V ∗ ] by making adjacen t all non-adjacent u, v such that G has a u − v path with all intermediate vertices lying outside V ∗ . It is not ha rd to infer from L emma 5 that the s ize of the optimal solution of ( G ∗ , T ) is the same as of ( G, T ). Accor ding to Theorem 1 , the n umber of v ertices of G ∗ is a t mos t | T | (2 k − r r + 1) where r = min t ∈ T r t and 1 is a dded on the accoun t of terminals. This b ound is not go o d in the sense that | T | ma y b e not b ounded b y k at a ll. Therefore prior to computing the union of impo rtant se parators , we reduce the num ber of terminals. This is p ossible due to the following theorem. Theorem 5. Ther e is a p olynomial-time algorithm that t r ansforms t he instanc e ( G, T , k ) of the mwc pr oblem into an e qu ivalent instanc e ( G ′ , T ′ , k ′ ) such t hat k ′ ≤ k and | T ′ | ≤ 2 k ′ ( k ′ + 1 ) . Then runtime of this algorithm is t he same as the runtime of the fixe d-r atio appr oximation algorithm for the mwc pr oblem [6] 2 . Pro of. W e sta rt from observ atio n that if u is a non-terminal vertex such that there a re k + 2 ter mina ls connected to u by paths intersecting only at u then u participates in a n y mwc o f ( G, T ) of size at mos t k . Indeed, r emov al of a set of at most k vertices not containing u would leave at least 2 of these paths undestroy ed and hence the corresp onding termina ls would be co nnected. An immediate co nsequence of this observ ation is that if S is a mwc of G and there is v ∈ S a djacent to a t lea s t k + 2 comp onents of G \ S containing terminals then this vertex participates in any mw c of G o f size at most k . Having the ab ov e in mind, we apply the r atio 2 approximation algorithm for the mwc pr oblem prop osed in [6]. 3 If the resulting mwc is of size greater than 2 k , the algorithm simply returns ’NO’. Otherwise, let S b e the resulting mwc . If | T | > | S | ( k + 1) then, taking in to account that eac h compo ne nt is adjacent to at least one v ertex of S , it follows fro m the pigeonhole principle that at leas t one vertex of S is adjacen t to at least k + 2 components of G \ S con ta ining terminals. Remov e v and remo ve isolated components of G \ { v } (i.e. thos e that con ta in at most one terminal), decr ease the parameter by 1 and recursively apply the same op eration to the new data. E ventually , one of three p ossible situations o ccur. First, after remov al o f k or less vertices, the resulting graph has no terminals. In this case we have just found the desired mw c of ( G, T ) in a p olynomial time. Second, after remov al of k vertices, there are s till termina ls, not separated by the remov ed vertices. In this case, again in a p olyno mial time, we have found that ( G, T ) has no mw c of size at most k . Finally , it ma y ha pp en that after remov al of some S ′ ⊆ S o f size at most k , the num ber of ter minals in the remaining graph is at most | S \ S ′ | ( k − | S ′ | + 1). Then the resulting graph is returned as the output of the pre pr o cessing.  Thu s , Theor em 5 together with Theorem 1 and Lemma 5 lead to the following result. Corollary 2. Ther e is an algorithm t hat for an inst anc e ( G, T , k ) of the mwc pr oblem finds an e quivalent instanc e of O ( k 2 r 2 k − r ) vertic es in time O ( A ( n ) + 2 k − r n 3 r 2 k 4 ) wher e r is the smal lest isolating cut and A ( n ) is the time c omplexity 2 This algorithm is based on sol v ing a linear program. 3 In fact, the approximati on ratio of this algorithm is 2 − 2 / | T | , but ratio 2 is sufficient for our purp ose. of the ap pr oximation algorithm pr op ose d in [6]. In p articular, if k − r = c ∗ log ( k ) for any fixe d c t hen the mwc pr oblem is p olynomial ly kernelizable. The output of the ab ov e algo rithm is muc h richer than just another instance of the mwc problem. Indeed, fo r each terminal, the alg o rithm in fact co mputes all principal impo rtant isolating cuts. This leads to the follows interesting question. Op en Question 2 Is ther e an algorithm t hat gets t he ab ove out put as input and, in t ime p olynomial in n and the numb er of the princip al isola t ing cuts, pr o du c es an e quivalent instanc e of the mwc pr oblem of size p olynomial in k ? Observe that if Op en Ques tio ns 1 and 2 are answered affirmatively then, together with P r op osition 4, Theor em 3, and Lemma 4, they imply a n uncondi- tional po lynomial kernelization of the mwc problem. Moreover, w e b elieve that inv estigation of Open Question 2 w ould give a significa n t insight into the struc- ture o f the mwc problem. Indeed it would reveal whether or no t we can ’filter’ in a r easonable time some principal isolating cuts, which in tur n would requir e pro of of some interesting structural dependencies related to the mwc problem. References 1. Hans L. Bo dlaender, Ro dney G. Do wney , Michael R. F ellows , and Danny H ermelin. On problems without p olynomial kernels (extend ed abstract). In ICALP (1) , pages 563–574 , 2008. 2. Hans L. Bo dlaender, St´ ephan Thomass ´ e, and Anders Y eo. Kernel b ounds for disjoin t cycles and disjoin t paths. I n ESA , p ages 635–646, 2009 . 3. Nicolas Bousqu et, Jean Daligault, St´ ephan Thomass ´ e, and Anders Y eo. A polyno- mial kernel for m ulticut in trees. In ST ACS , pages 183–194 , 2009. 4. Jianer Chen, Y ang Liu, and Song jian Lu . An improve d parameterized algorithm for the minim um no de multiw ay cut problem. Algorithmic a , 55(1):1–13, 200 9. 5. Elia s Dahlhaus, David S. Johnson, Christos H. Papadimitrio u , Paul D. Seymour, and Mihalis Y annak akis. The complexity of m ultiterminal cu ts. SIAM J. Comput. , 23(4):864– 894, 1994. 6. Na veen Garg, Vijay V. V azirani, and Mihal is Y annaka k is. Multiw ay cuts in node w eighted graphs. Journal of Algorithms , 50(1):49–61, 2004. 7. Daniel Lokshtano v and D´ aniel Marx. Clustering with lo cal restrictions. In ICALP 2011, to app e ar , 20 11. 8. D´ aniel Marx. P arameterized graph separation problems. The or. Comput . Sci. , 351(3):394 –406, 2006. 9. Igor R azgon. Computing multiw ay cut within the give n excess o ver th e largest minim u m isolating cut. CoRR , abs/1011 .6267, 2010. 10. St ´ ephan Thomass ´ e. A 4 k 2 kernel for feedback v ertex set. ACM T r ansactions on Algor i thms , 6(2), 2010. A Pro ofs omitted from the main b o dy The rest of pro of of Theorem 3 F or the corre c tnes s, we need to show that the set of new added elements is precisely Pr i \ Pr i − 1 . This is established in the following three paragraphs. Ev ery elemen t of P r i \ P r i − 1 is collected during the gathering stage. Let S ∈ Pr i \ Pr i − 1 . Since hat ( S ) 6 = ∅ , by Prop ositio n 1, S 6 = V is ( S ). It follo ws from the D VS condition that V is ( S ) * S . Let v ∈ V is ( S ) \ S . By P rop osition 3, there is S ∗ such that v ∈ hat ( S ∗ ) and S is the witness of v w.r.t. S ∗ . T aking int o account the SM condition, we conclude that S ∗ ∈ Pr i − 1 , that is S will b e generated by the above algor ithm. An e lement of P r i \ P r i − 1 will not b e filtered out. hat ( S ) co nsists o f elements tha t are not con ta ine d in any element of F preceding S . The alg orithm chec ks S only against the union of a subset o f such element s . An eleme nt that is not in P r i \ P r i − 1 will b e fil te red. Let S ∈ F be such that ex ( S ) = i and hat ( S ) = ∅ . Let v ∈ S . It is sufficient to show that there is S ∗ ∈ Pr i − 1 such that v ∈ S ∗ and S ∗ ≺ S . Cho os e S ∗ to b e a minimal element pr eceding S and containing v . Due to the minimality of S ∗ , v do es no t belo ng to an y elemen t preceding S ∗ hence v ∈ hat ( S ∗ ), i.e. S ∗ is a principal set. Due to the SM condition, ex ( S ∗ ) ≤ i − 1. It follows that S ∗ ∈ Pr i − 1 . Let us compute the runtime. The main cycle g o e s throug h all element s o f Pr i − 1 , the num b er of such elements is at mos t 2 x +1 r ac c ording to Prop ositio n 5. Let us compute the time s pent per element. Since each element of Pr i − 1 is of excess at most x , i.e. of s iz e at mos t r + x , the algorithm explores at most r + x vertices a nd for each vertex either computes the r esp ective witness or concludes its absence. It follows fro m the EC condition that the overall time sp en t for c o mputation of witnesses p er element of Pr i − 1 is O ( n 3 ( r + x )). Let us compute time spent p er witness. Denote the considered witness by S 1 . Then S 1 is compared against all elements of Pr i − 1 where the n umber of such ele men ts is at mos t 2 x +1 r as noticed ab ov e. F or each S 2 ∈ Pr i − 1 , it is check ed whether S 2 ≺ S 1 which can b e done in O ( r + x ) according to the EC condition. If the test returns a p ositive answer then S 2 is added to the unio n o f elements pre c eding S 1 which, using appro priate data structures 4 can be done in O ( | S 2 | ), i.e. again in O ( r + x ). Thus the total r untime of this op er ation is O (2 x r ( r + x )). After finishing the compariso n a gainst Pr i − 1 , the a lgorithm chec k s whether or not all the elements are in the resulting union of pre decessors. This can b e done in O ( | S 1 | ) i.e. in O ( r + x ), clea rly this runtime can b e igno red in the ligh t of the already sp ent O (2 x r ( r + x )). Multiplying the nu mber of considered witness es by the runtime spent per witnes s, the desired runtime of O ( n 3 2 x r 2 ( r + x ) 2 ) follows.  Pro of o f Lem ma 4 W e show that the set of imp ortant se pa rators par tia lly ordered by the ≺ ∗ meets all the conditions of the IS-family . 4 the union can b e stored as a binary vector of size n indexed by the elemen ts of the universe and add ing a set to the union just means ticking the respective entries | S 1 | times SE Condi tion See Lemma 3.3 of [8]. SM Condi tion Immediately follows from the definition o f an imp ortant sepa- rator. SW Conditio n Let K b e a n impor tant sepa r ator. Let G ∗ be the gra ph obtained from G b y contraction o f a ll the vertices of N R ( G, Y , K ) \ X . In other words, to obtain G ∗ from G , remo ve all vertices of N R ( G, Y , K ) and add an edge b etw een each vertex u ∈ X and v ∈ K such that ther e is a u − v path all int er mediate vertices o f whic h b elong to N R ( G, Y , K ). It is not hard to see that the definition of an impor tant separato r implies that K is the sma lle st X − Y s eparator of G ∗ . Let v ∈ K . Assume that v is not adjacent to Y in G (otherwise ther e is no witness o f v w.r .t. K ). Let G ′′ be a graph obtained from G ∗ by splitting v into n + 1 copies . It is not hard to observe that the set of importa n t X − Y separators of G ′′ is the set of imp o rtant X − Y separ ators of G that do not co n ta in v , moreov er the partial o rder relation is prese r ved. Then a witness o f K w.r .t. v in G is a smalles t importa nt X − Y sepa rator of G ′′ . By the SE condition, this separato r is unique. TE conditi o n. O bserve (e.g. Pr op osition 1 of [9]), tha t if K 1 ≺ ∗ K 2 then K 1 \ K 2 ⊆ N R ( G, Y , K 2 ). Thus if K 2 ≺ ∗ K 3 , K 1 \ K 2 ⊆ N R ( G, Y , K 2 ) ⊆ N R ( G, Y , K 3 ), the la s t inclusion is obtained by definition of the ≺ ∗ relation. Thu s , no vertex of K 1 \ K 2 can b elong to K 3 . In or der to establish the v isible set conditions, we prove an intermediate claim. Claim. Let K be an imp orta n t X − Y separator, whic h is not the smallest one and let K ′ ∈ P r ed ( K ). Then K ∗ = V is ( K ) \ N R ( G, Y , K ′ ) is a X − Y separator such that K ′  ∗ K ∗ . Pro of. Let v ∈ N R ( G, Y , K ′ ) and let p be a v − Y path of G . Let u b e the last vertex of p tha t b elongs to S K ′′ ∈ P r ed ( K ) K ′′ . Clearly , u is not co vered by a n y element of P r ed ( K ) b ecause otherwise it would not b e the last vertex of p that belo ngs to an ele ment of P re d ( K ). Hence by definition u ∈ V is ( K ). Clear ly , u cannot b e long to N R ( G, Y , K ′ ) b e cause otherwise it will b e followed in p by an element of K ′ . Consequen tly , u ∈ V i s ( K ) \ N R ( G, Y , K ′ ), co nfirming the claim.  L VS condition. According to the ab ov e cla im | K ′ | ≤ | K ∗ | b ecause otherwise we get a con tra diction to being K ′ an imp orta nt separ ator. T a king into acco un t that K ∗ ⊆ V i s ( K ), the condition follows. D VS condition. K ∗ ⊆ V is ( K ), hence the latter is an X − Y separator . Therefore, if V is ( K ) ⊂ K then K is not a minimal X − Y separator in co ntra- diction to its imp ortance. EC condition The O ( n 3 ) algor ithm for c o mputing a smallest imp ortant separato r follows from Lemma 1 in [9]. This immediately implies existence of such algo rithm for the witness co mputation. Indeed, the single witness condition pro of of Lemma 4 sho ws that witness computation can b e reduced to computing the smalles t impor tant s eparator and such the r eduction can b e c learly p erformed in O ( n 3 ). Fina lly , observe tha t it is p ossible to main tain an impo rtant separator K in a wa y that for each vertex v testing whether v ∈ N R ( G, Y , K ) ca n b e per formed in O (1): asso ciate K with a binary vector of s iz e n indexed b y V ( G ) where 1-s corr esp ond to the elements of N R ( G, Y , K ). In the ligh t of P rop osition 1 in [9], this immediately implies that K 1 ≺ ∗ K 2 can be tested in O ( K 1 ).  Pro of of Lemm a 5 Let S 1 be an arbitrary smalles t mwc of ( G, T ). If all vertices of S 1 belo ng to imp ortant is o lating cuts, we are do ne. Otherwise, let v ∈ S 1 be a vertex that does no t belong to any smallest isolating cut. Due to the minimalit y of S 1 , there is t ∈ T such that v b e longs to a minima l is olating cut S ′ ⊆ S 1 of t . It follows that there is an imp ortant iso lating cut S ′′ ≻ ∗ S ′ of t suc h that | S ′′ | ≤ | S ′ | . The pro of of Lemma 3.6. of [8] shows that S 2 = ( S 1 \ S ′ ) ∪ S ′′ is a mwc of ( G, T ) o f siz e not e x ceeding S 1 . In other words S 2 is an optimal solution o f ( G, T ) and the n umber of vertices of S 2 not involv ed in an y imp ortant isolating cuts is s maller than that of S 1 . Applying such modification iteratively , we even tually o btain a smallest multiw ay cut without suc h ’undesired’ vertices. 