Fixed Parameter Algorithms for one-sided crossing minimization Revisited

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1 Fixed Prmeter Algorithms for one-sided crossing minimiztion Revisited Vid Dujmović 1, Henning Fernu 2,3, nd Michel Kufmnn 2 1 McGill University, School of Computer Science, 3480 University St., Montrel, QC H3A 2A7, Cnd, vid@cs.mcgill.c 2 Universität Tüingen, WSI für Informtik, Snd 13, Tüingen, Germny fernu / mk@informtik.uni-tueingen.de 3 The University of Newcstle, School of Electr. nd Computer Science, University Drive, Cllghn, NSW 2308, Austrli fernu@newcstle.edu.u Astrct. We exhiit smll prolem kernel for the prolem onesided crossing minimiztion which plys n importnt role in grph drwing lgorithms sed on the Sugiym lyering pproch. Moreover, we improve on the serch tree lgorithm developed in [5] nd derive n O( k + kn 2 ) lgorithm for this prolem, where k upperounds the numer of tolerted crossings of stright lines involved in the drwings of n n-vertex grph. Reltions of this grph-drwing prolem to the lgeric prolem of finding weighted liner extension of n ordering similr to [7] re exhiited. 1 Introduction nd Prolem Definition We consider the following prolem k-one-sided crossing minimiztion, clled k-oscm for short in this pper: Given: A iprtite grph G =(V 1,V 2,E) nd liner order < 1 on V 1. Prmeter: k Question: Is there liner order < on V 2 such tht, when the vertices from V 1 re plced on line (lso clled lyer) L 1 in the order induced y < 1 nd the vertices from V 2 re plced on second lyer L 2 (prllel to L 1 ) in the order induced y <, then drwing stright lines for ech edge in E will introduce no more thn k edge crossings. We denote y OSCM the optimiztion version (tht is, non-prmeterized version) of this prolem. (k-)oscm is the key procedure in the well-known lyout frmework for lyered drwings commonly known s the Sugiym lgorithm. After the first phse (the ssignment of the vertices to lyers), the order of the vertices within the lyers hs to e fixed such tht the numer of the corresponding crossings of the edges etween two djcent lyers is minimized. Finlly, the concrete position of the vertices within the lyers is determined ccording the previously computed order. The crossing minimiztion step, lthough most essentil in the Sugiym pproch, is n NP-complete prolem. The most commonly used method is the G. Liott (Ed.): GD 2003, LNCS 2912, pp , c Springer-Verlg Berlin Heidelerg 2004

2 Fixed Prmeter Algorithms 333 lyer-y-lyer sweep where, strting from i = 1, the order for L i is fixed nd the order for L i+1 tht minimizes the numer of crossings mongst the edges etween lyers L i nd L i+1 is determined. After incresing index i to the mximum lyer index, we turn round nd repet this process from the ck with decresing indices. In ech step, n OSCM prolem hs to e solved. Unfortuntely, this seemingly elementry prolem is NP-complete [6], even for sprse grphs [8]. Jünger nd Mutzel [7] trnsformed the OSCM prolem to liner ordering prolem which they solve with the rnch nd cut method. This elementry grph drwing prolem ttrcted severl reserchers from the re of fixed-prmeter trctle (FPT) lgorithms [2]. The first pproches to the more generl vrint of this prolem hve een pulished y Dujmović et l. in [3] nd [4]. The lst one hs een gretly improved y Dujmović nd Whitesides [5] who chieved n O( k n 2 ) lgorithm for this prolem. In this pper, we derive n O( k + kn 2 ) lgorithm, which significntly lowers the constnts involved in the exponentil serch tree lgorithm prt. Moreover, we exhiit smll prolem kernel for this prolem, which hs not een done efore. More specificlly, this mens tht, with the id of some reduction rules, we cn rrive t n instnce of k-oscm with the numer of vertices involved eing ounded y 3k(k ). 2 Bsics We strt with some formlities. The numer of vertices of grph G =(V,E) is denoted y n = V. For ech vertex v V, let N(v) denote the set of vertices djcent to v nd let deg(v) = N(v) denote the degree of v in G. The sugrph of G induced y the set of vertices V V is denoted y G[V ]. A iprtite grph G =(V 1,V 2,E) together with liner orderings on V 1 nd on V 2 is lso clled drwing of G. This formultion implicitly ssumes drwing of G where the vertices of V 1 nd V 2 re drwn on two (virtul) horizontl lines, the line L 1 corresponding to V 1 eing ove the line L 2 corresponding to V 2.Ifu<vfor two vertices on L 1 or on L 2, we will lso sy tht u is to the left of v. A liner order on V 2 tht minimizes the numer of crossings suject to the fixed liner order of V 1 is clled n optiml ordering nd the corresponding drwing of G is clled n optiml drwing. Since the positions of isolted vertices in ny drwing re irrelevnt, we disregrd isolted vertices of the input grph G in wht follows. If V 2 = 2, then there re only two different drwings. This gives us the useful notion of crossing numer. Given iprtite grph G =(V 1,V 2,E) with V 2 > 1, for ny two distinct vertices, V 2, define c to e the numer of crossings in the drwing of G[{, } (N() N())] when <is ssumed. Furthermore, for ny V 2 with deg(v) > 0, let l e the leftmost neighour of on L 1, nd r e the rightmost neighour of. We cll two vertices, V 2 interfering or unsuited if there exists some x N() with l < x < r,or there exists some x N() with l <x<r. Otherwise, they re clled suited.

3 334 V. Dujmović, H. Fernu, nd M. Kufmnn Oserve tht, for {, } suited, c c = 0. Dujmović nd Whitesides hve shown tht, in ny optiml ordering < of the vertices of V 2,wefind<if r l. This mens tht ll suited pirs pper in their nturl ordering. In the next section we will need the following generl nottion. A directed grph (digrph) otined from n undirected grph G y replcing every edge {u, v} y the two rcs (u, v) nd (v, u) is denoted y D(G). The undirected grph otined from digrph G y putting n edge {u, v} whenever there is n rc (u, v) ing is denoted y U(G). Finlly, the grph complement of G is denoted y G c. It is importnt to note tht we only del with simple (di-)grphs, i.e., grphs hving no (self-)loops or multiple edges. The set of rcs of digrph G is denoted y A(G). 3 Getting Smll Kernel In this section, we give drstic reduction of the complexity of k-oscm. Recll tht prtil order is n irreflexive, symmetric nd trnsitive reltion. A prtilly ordered set (or poset), denoted y P =(V,A), is set V tken together with prtil order A on it. For two distinct elements, V, if either < or <in A, then the pir {, } is comprle in P ; else the pir is incomprle. A prtil order is liner if every pir of elements of V is comprle. If A V V is prtil order on V, then we cll ny prtil order L A completion of A. More lgericlly speking, it cn lso e clled (weighted) liner extension. As is well known, every poset P (V,A) cn e represented s directed cyclic digrph with vertex set V nd rc reltion A which is prtil order. For simplicity, we will equte posets nd directed cyclic grphs in the following, so we refer to posets s ordered or liner digrphs. Consider the following prolem k-weighted completion of n ordering (k-wco): Given: An ordered digrph P = (V,A) nd cost function c mpping A(D([U(P )] c )) into the nonnegtive integers; y setting c to zero for rcs in A(D(U(P ))), we cn interpret the domin of c s V (P ) V (P ). Prmeter: k Question: Is there selection A of rcs from A(D([U(P )] c )) such tht the trnsitive closure (A A(P )) + is liner order nd {c() (A A(P )) + } k? It is quite ovious tht OSCM cn e solved with the help of WCO; the liner order which is imed t is the permuttion of the vertices on the second lyer which minimizes the numer of crossings involved, so tht the crossing numer c is the cost of the rc (, ) in the digrph model. Since it is esy to see tht WCO is nondeterministiclly solvle in polynomil time, nd since Edes nd Wormld [6] hve shown tht OSCM is NP-hrd, we cn immeditely deduce: Lemm 1. WCO is NP-complete. Jünger nd Mutzel [7] gve n ILP-formultion of the OSCM prolem s liner ordering prolem which they could solve with rnch nd cut method.

4 Fixed Prmeter Algorithms 335 In contrst, our lgorithm for OSCM (oth the kerneliztion nd the serch tree prt) cn e seen s growing lrger nd lrger prts of the liner order of the vertices of the second lyer we re iming t. When settling the ordering etween nd, we lso sy tht we re committing <or <. Dujmović nd Whitesides hve shown tht, for ech instnce of OSCM, suited pirs pper in their nturl ordering in ny optiml drwing. Tht justifies the following first reduction rule: RR1: For every suited pir of vertices {, } from V 2 with c > 0, commit <. Furthermore, consider set S V 2 such tht ll the vertices in S re djcent to the exct sme set of neighours in V 1. It is simple to oserve tht ritrrily permuting the vertices of S in ny drwing cnnot ffect the totl numer of crossings. This oservtion justifies the following second reduction rule: RR2: For every pir of vertices {, } from V 2 with N() =N(), (ritrrily) commit <. Notice tht if c = c = 0 then (disregrding isolted vertices) deg() = deg() = 1ndN() = N(). Therefore, fter pplying these two reduction rules to n instnce of k-oscm, we otin the ordered digrph P =(V 2,A), where pir of vertices is incomprle in P only if oth of the crossing numers of tht pir re nonzero. Hence, we cn ssume tht c is mpping ech rc of A(D([U(P )] c )) onto positive integer. We cll the corresponding specilized prolem k-positive completion of n ordering (k-pwco). Our resoning up to now shows tht this specil cse is lso NP-hrd. Exhustively pplying the following two reduction rules shows tht k-pwco hs smll liner-size prolem kernel. RRLO1: If v is vertex in ordered digrph P =(V,A) of mximl degree, tht is, the sum of the indegrees nd outdegrees of v equls V 1, then remove v nd consider the instnce (P [V v],k). To see the soundness of this rule, oserve tht oth P nd P [V v] hve trnsitive rc reltions. After hving pplied RRLO1 exhustively, there is, for ech vertex v V, nother vertex v V (P ) such tht the pir {v, v } is incomprle in P. By the positivity ssumption, committing v<v or v <v will cost t lest one unit nd will, furthermore, t est mximize the degree of v nd v. By n inductive rgument, it follows tht the ordered digrph P in RRLO1-reduced instnce of k-pwco cnnot hve more thn 2k vertices. In n ordered digrph P =(V,A), cll two incomprle vertices u, v V independent with respect to w if {u, w} nd {v, w} re comprle in P ; dependent with respect to w if {u, w} or {v, w} re incomprle in P ; independent (in P )if{u, v} re independent with respect to ll w V \ {u, v}; dependent (in P )if{u, v} re dependent with respect to some w V \{u, v}; trnsitive with respect to w if either {u, w} or {v, w} is comprle ut not oth.

5 336 V. Dujmović, H. Fernu, nd M. Kufmnn Being trnsitive with respect to w mens tht committing v<uor u<v commits y trnsitivity v<w, w<v, u<wor w<u. RRLO2: If the vertices u nd v re independent in given ordered digrph P =(V,A) nd if c((u, v)) <c((v, u)), then reduce the prolem instnce to ((V,A {(u, v)}),k c((u, v)). The soundness of RRLO2 is ovious. In fct, this rule is generlizle: RRLOq for ny fixed q>1: For ech connected component C V of [U(P )] c with C q, solve PWCO optimlly on P [C]. The reduced instnce will see the orderings etween ll pirs of vertices from C settled, nd the prmeter will e reduced ccordingly. Note tht this implies tht ll the vertices of C re isolted in the reduced [U(P )] c nd cn thus e deleted y the RRLO1. This new rule yields kernel of size k q+1 q, since fter exhustive ppliction of this rule nd RRLO1, ech connected component of [U(P )] c hs t lest (q+1) vertices nd thus t lest q edges. Correspondingly, t lest q rcs with weight of t lest one per rc hve to e dded per component, therefore there re t most k/q components. In other words, kernel size of k cn e pproximted with ritrry precision. This type of ehviour hs not een found in other prmeterized lgorithms yet. However, this pproximtion comes t price: Solving this prolem on grphs with q vertices cn only e done in exponentil time (ssuming P NP). Still, ritrry constnts q or q = log k re possile choices for polynomil-time kerneliztion lgorithms. It is therefore desirle to find esy optimlity criteri for smll vlues of q (s q = 2). The soundness of the reduction rule RRLOq is redily seen.assume tht C is connected component of [U(P )] c. Consider vertex x C. Since C is connected component in [U(P )] c, x is comprle with ech y C (otherwise, y nd x would e connected in the complement grph). Hence, C cn e portioned into C l = {y C y<x} nd C r = {y C x<y}. Since C l <x<c r, either C l = or C r = ; otherwise, there would e no connection etween vertices of C l nd vertices of C r in the complement grph. Hence, x will never e ordered in-etween two vertices from C. Therefore, we cn conclude: Theorem 1. Fix some 1 <α 1.5. Then, ech instnce of k-pwco dmits prolem kernel of size αk. Considering k-oscm in slightly more generl fshion, nmely, llowing some pirs of V 2 to e lredy committed s prt of the input, we cn conclude: Corollry 1. Fix some 1 <α 1.5. Then, ech instnce of k-oscm cn e reduced to n instnce (P =(V 2,A),k) of k-pwco, with V 2 αk. Proof. Applying RRLO1 nd RRLO2 (nd RRLOq) tok-oscm exhustively results in prtil order on V 2 where the crossing numers of the incomprle pirs re positive. Therefore, this prtil order on V 2, together with the positive crossing numers of incomprle pirs comprises n instnce of k-pwco, which y Theorem 1 hs the kernel of size V 2 αk.

6 Fixed Prmeter Algorithms 337 This hs not yet given us prolem kernel for the originl prolem, since V 1 (nd hence E ) hs not yet een ounded y function of k. With tht im, consider the following simple reduction rule RRlrge: Ifc >kthen do: if c k then commit <else return NO. This is clerly sound rule. Moreover notice, tht if vertex V 2 hs deg() > 2k + 1, then for every vertex V 2, c >kor c >kre true. Therefore, fter performing RRlrge exhustively in comintion with RRLO1, ll the vertices of V 2 of degree lrger thn 2k + 1 will hve een removed from the OSCM-instnce. This yields: Theorem 2. For some 1 <α 1.5, k-oscm dmits prolem kernel G = (V 1,V 2,E) with V 1 (2k +1) V 2 ( 3k(k )), V 2 αk, nd E (2k + 1) V 2. The following lgorithm summrizes how to otin prolem kernel: Algorithm 1 (Kerneliztion for k-oscm). Compute the crossing numers c for ll pirs (, ). If k<, V 2 min(c,c ) then nswer NO. If k, V 2 mx(c,c ) then YES; output n ritrry liner order. Apply reduction rules RR1, RR2 (nd RR3). Apply reduction rules RRLO1, RRLO2, nd RRlrge exhustively. The rule RR3 is derived in Section 6, s it is needed y the serch tree prt of our lgorithm. For now, while considering kerneliztion only, we cn disregrd this reduction rule. Since computing the crossing numers is the predominnt computtionl prt (tking time O(kn 2 ) even when comined with RRlrge ccording to Dujmović nd Whitesides), we conclude tht the kerneliztion preprocessing tkes time O(kn 2 ). As result of this kerneliztion, the vertices of V 2 re prtilly ordered. Our serch tree lgorithm presented in the next section cn cope with such pre-set instnces of k-oscm. Therefore, we will otin running time of the form O(f(k)+kn 2 ) for the k-oscm prolem. 4 A Generl Overview of the Serch Tree Algorithm Our serch tree for the OSCM prolem slightly devites from the one constructed y Dujmović nd Whitesides in their pper. Hence, we give some detils in wht follows. Consider two distinct vertices, V 2 with c = i nd c = j. Then, the pir {, } is lso clled n i/j pttern. As is the stndrd prctice when developing serch tree sed FPT lgorithms, ech node of serch tree is ssocited with prolem instnce. In the cse of k-oscm, we let n instnce consist of n ordered digrph P =(V 2,A) nd n integer k. P contins the rcs corresponding to ll the pirs of vertices of V 2 committed thus fr, nd k gives the remining numer of llowed edge crossings, tht is, k = k (v,u) A c vu. Initilly (in the root node) the rcs

7 338 V. Dujmović, H. Fernu, nd M. Kufmnn of P re determined y the kerneliztion. In ech node of the serch tree, if k is negtive, then the node returns NO, otherwise we look for pir of dependent vertices {, } tht form n i/j pttern such tht either i+j 4, or i =2,j =1 nd {, } is trnsitive. If such pir {, } is found we rnch the prolem instnce into two new prolems instnces. In one we commit <nd in the other we commit >; consequently, in ech prolem instnce we updte the ordered digrph P nd lower the prmeter k ccordingly. (By dding (v, w) to A nd then computing trnsitive closure (A (v, w)) + we get the updted ordered digrph P =(V 2, (A (v, w)) + ).) If no pir s descried ove is found, then we commit ll the remining incomprle pirs in P deterministiclly. The detils nd the correctness of this pproch will e discussed in Section 7. The running time of FPT lgorithms is dominted y the prt exponentil in prmeter k. In our cse, tht prt is ounded y the size of the serch tree. We now nlyse tht size. Firstly, oserve tht ech internl node of our serch tree hs two rnches. If one rnch lowers the prmeter k y 1, nd the other y 2, we denote the corresponding rnching vector y ( 1, 2 ). Since ll i/j ptterns with i =0orj = 0 re lredy committed (y the reduction rules RR1 nd RR2 during the kerneliztion), ech internl node of the serch tree hs 1 > 0 nd 2 > 0. Furthermore, ech node tht rnches on trnsitive 2/1 pttern commits y trnsitivity n extr i/j pttern where i, j > 0. Thus ech internl node hs Therefore, in the worst cse, ech node of our serch tree hs rnching vector (2, 2) or (3, 1). The size s(k) of the serch tree oeying these rnching vectors cn e deduced s follows: The recurrence corresponding to the (2, 2) rnching vector is s(k) = 2s(k 2) + O(1). Solving this recurrence gives s(k) < k. The recurrence corresponding to the (1, 3) rnching vector is s(k) = s(k 1) + s(k 3) + O(1). Solving this recurrence gives s(k) < k. Thus in the worst cse, the size of our serch tree is less thn k. This is in contrst to the serch tree with (2, 1) rnching vector derived y Dujmović nd Whitesides tht gives s(k) < k. In node of our serch tree tht does not rnch (tht is, in lef), ll the incomprle pirs form 1/1 nd 2/1 ptterns. To e le to commit these pirs deterministiclly, we study the structurl properties of such ptterns in the next section. In Section 7, we finlly give detiled lgorithm for the k-oscm prolem nd the proof of its correctness. 5 Some Structurl Properties of 2/1 nd 1/1 Ptterns We strt this section with the following known nd useful equlity [1, Chpter 9]: c + c = deg() deg() N() N(). (1) This implies tht if deg() deg() then (deg() 1) deg() c + c deg() deg(). (2)

8 Fixed Prmeter Algorithms 339 We now study 1/1 nd 2/1 ptterns. As convention, we will lel vertices from the first lyer y (decorted) letters x, y, z nd vertices from the second lyer y letters,, c. For clrity, we will drw neighours of s non-filled circles nd neighours of s filled-in circles. Lemm 2. In the cse of 1/1 or 2/1 ptterns, nd if c =1, we must find the sitution depicted in the figure elow. x y Moreover, ech remining neighour of (if ny) must e to the right of y (or y itself), while ech remining neighour of (if ny) must e to the left of x (or x itself). (Otherwise, c > 1.) Theorem 3. If c = c =1, then there re two siclly different situtions (tht cn e otined y enhncing the sitution sketched in Lemm 2): 1. nd re ech djcent to x nd y only. In other words, deg() = deg() = 2 nd N() =N(), s illustrted in the figure elow. x y 2. Two sucses rise: () If deg() =1, then hs (esides x) nother neighour x to the right of y. In ddition to x nd x, my only e djcent to y. () If deg() =1, then hs (esides y) nother neighour y to the left of x. In ddition to y nd y, my only e djcent to x. Both situtions re illustrted in the figure elow. x y x y x y Proof. By inequlity (2) either deg() = deg() = 2, or one of, hs degree one. Consider first the cse where deg() = deg() =2. Then y equlity (1), N() =N() s otherwise c + c 3. Consider now the cse where deg() = 1. Then hs (esides x) exctly one neighour x strictly to the right of y s otherwise either the pir {, } is suited or c > 1.

9 340 V. Dujmović, H. Fernu, nd M. Kufmnn Vertex cnnot hve neighour strictly to the left of y s otherwise c > 1. Thus in ddition to x nd x, cn only e djcent to y. The remining cse where deg() = 1 is symmetric. Theorem 4. Let the pir {, } form 1/2 pttern, tht is, c =1nd c =2. Then there re two siclly different situtions: 1. In the following figures, neither nor cn hve ny other neighours. x = y x y x y y = x 2. In the following figure (on the left-hnd side), in ddition to x, x nd x, vertex my only e djcent to y, while hs no other neighours. The figure on the right-hnd side cn e symmetriclly interpreted. x y x y x y y x Proof. By inequlity (2) either deg() = deg() = 2, or one of, hs degree one. Consider first the cse where deg() = deg() = 2 nd let x nd y denote the remining neighours of nd, respectively. Hving oth x y nd y x is not possile s otherwise y equlity (1), c + c = 4. Hving oth x = y nd y = x gives the first cse of the previous theorem. Therefore, if x = y then x<y <ys otherwise, {, } is suited or c =1. On the other hnd, if y = x, then x<x <ys otherwise, {, } is suited or c =1. Consider now the cse where deg() = 1. The cse where deg() = 1 is symmetric. In ddition to x, hs exctly one neighour x strictly to the left of y s otherwise, c 2. Vertex hs exctly one neighour x strictly to the right of y s otherwise, either the pir {, } is suited or c > 1. Thus in ddition to x, x nd x, cn only e djcent to y.

10 6 A Reduction Rule for 2/1 Pttern Fixed Prmeter Algorithms 341 The reduction rule RR2 presented in Section 3 resolves the sitution descried in the first cse of Theorem 3. Unfortuntely, the second cse of Theorem 3 does not dmit similr simple resolution. Let us now reconsider the 2/1 ptterns identified in Theorem 4. The second kind of the pttern descried in tht theorem is in sense similr to (ut more complicted thn) the second pttern of Theorem 3 nd hence dmits no deterministic solution. Consider the first cse in Theorem 4 with hving neighour distinct from x nd y. Let I denote ny drwing where <nd let II denote the drwing otined from drwing I y swpping nd. Let the totl numer of crossings in I nd II e denoted y c I nd c II respectively. Consider the following figure nd tles. T I A 1 A 2 A 3 T II A 1 A 2 A 3 X X X X X X X X X X X X X X These tles red s follows: if there re m ij edges connecting vertices from X i with vertices from A j, then there re T x (X i,a j )m ij crossings etween these m ij edges nd the edges shown in the ove sketches for cse x {I,II}. Itis cler tht the columns lelled A 1 nd A 3 re identicl in oth tles: swpping nd cn only ffect the reltive order of vertices in A 2 compred to nd. The oxed entries show the only differences etween the tles. It follows tht c I c II = c c + m 42 +2m 52 + m 62 > 0, since m 42 +2m 52 + m 62 0 nd c c = 1. This implies tht whenever this sitution from Theorem 4 rises, ny optiml drwing shows <. The first cse in Theorem 4 where hs neighour distinct from x nd y is symmetric; the sme nlysis pplies. This justifies the next reduction rule: RR3: In the situtions descried in the first cse of Theorem 4, lwys let <. Since in every optiml drwing ll the pirs of this type pper in their cheper ordering, we pply this reduction rule t the very eginning of our lgorithm, tht is we dd this rule to the kerneliztion s shown in Algor. 1.

11 342 V. Dujmović, H. Fernu, nd M. Kufmnn Hving this reduction rule is instrumentl in ounding the size of the serch tree, s will ecome cler from the two lemms in the next section. 7 The Algorithm for the OSCM Prolem We re now redy to present the prmeterized lgorithm we suggest for OSCM: Algorithm 2 (A prmeterized lgorithm for OSCM). - kernelize In node of serch tree with instnce (P,k ) do 0: if k < 0 return NO. 1: Apply RRLO1, RRLO2 nd RRlrge exhustively. 2: if in P there is n incomprle i/j pttern {,} such tht i + j 4 rnch on < nd <; updte P nd k ccordingly 3: else if in P there is dependent 2/1 pttern {,} rnch on < nd <; updte P nd k ccordingly 4: else //the remining 2/1 ptterns re independent; thus, RRLO2 pplies resolve ll remining 1/1 ptterns ritrrily updte P nd k ccordingly; if k < 0 return NO else YES. The correctness of this lgorithm is cler from the nlysis presented in the previous sections. Wht remins is to nlyse the rnching vectors of the internl nodes in the serch tree ssocited with the lgorithm. As required y the nlysis presented in Section 3, we now show tht the rnching vector ( 1, 2 ) in ech internl node hs nd 1, 2 > 0. Lemm 3 (Min Lemm). Let {, } e pir of dependent vertices forming 2/1 pttern in step 3 of the lgorithm. Then {, } is trnsitive pir. Proof. Since {, } is dependent in P, there must e vertex c such tht {, c} or {, c} re incomprle in P. It suffices to show tht one of these two pirs re comprle in P. In the step 3 of the lgorithm, the only remining incomprle pirs re 1/1 ptterns of the second type in Theorem 3 nd 2/1 ptterns of the second type in Theorem 4. Therefore, let without loss of generlity deg() = 3 (or deg() = 4) nd deg() = 1. If deg(c) 2 then y the inequlity (2), c c + c c 4 nd thus the ordering of {, c} is settled either y RR1 or y the step 2 of the lgorithm. Therefore, deg(c) = 1. In tht cse, the pir {, c} forms 0/i pttern nd its ordering is settled y either RR1 or RR2. A direct consequence of this lemm is the following. Lemm 4. The rnching vector ( 1, 2 ) in ech internl node of the serch tree hs nd 1, 2 > 0.

12 Fixed Prmeter Algorithms 343 Proof. Bsed on the reduction rules RR1 nd RR2, in ech node of serch tree ll the incomprle (nd dependent) ptterns i/j hve i > 0 nd j > 0, thus in ech node 1, 2 > 0. Furthermore, ll the nodes rnched in the step 2, hve i + j 4 nd thus hve The remining nodes re rnched in step 3. Ech such node rnches on dependent 2/1 pttern {, }. By the previous lemm, committing either <or <determines, without loss of generlity, the ordering of pir {, c}. Hving een incomprle t step 3 initilly, the pir {, c} hs c c,c c 1. Therefore, we hve in this cse, too. 8 Conclusions In this pper, we present new serch tree sed prmeterized lgorithm for the k-oscm prolem. Due to possile rekerneliztions [9], we my deduce: Theorem 5. The prolem of k-oscm cn e solved in time O( k + kn 2 ). It remins to e determined whether further progress is possile, especilly in further lowering the se of the exponent in the running time of the serch tree lgorithm. Our present cse nlysis shows t two plces rnching ehviour which mtches the upper ound stted in the previous theorem, nmely when rnching t 3/1 nd t 2/1 ptterns. A detiled nlysis of 3/1 ptterns my e worthwhile, ut will proly e very tedious, requiring considertions of ll the possile comintions. Let us finlly remrk tht even with the possily more difficult prolem k- PWCO, we re le to derive serch tree lgorithm with etter rnching ehviour thn Dujmović nd Whitesides did for k-oscm. Since the result is of certin interest on its own, we will stte it in the following. Theorem 6. k-pwco cn e solved in time O( k + kn 2 ). In fct, the worst cse rnching vector in our lgorithm is (2, 4, 4, 4). Acknowledgments. We re grteful for hints of F. Rosmond, P. Shw nd M. Sudermn on drft versions of this pper. References 1. Di Bttist, G., Edes P., Tmssi R. nd I. G. Tollis, Grph Drwing: Algorithms for the Visuliztion of Grphs, Prentice Hll, R. Downey nd M. Fellows. Prmeterized Complexity, Springer, V. Dujmović, M. Fellows, M. Hllet, M. Kitching, G. Liott, C. McCrtin, N. Nishimur, P. Rgde, F. Rosemnd, M. Sudermn, S. Whitesides,nd D. Wood, An efficient fixed prmeter pproch to two-lyer plnriztion. In P. Mutzel nd M. Jünger, eds., Grph Drwing GD 01, LNCS 2265, pges Springer, V. Dujmović, M. Fellows, M. Hllet, M. Kitching, G. Liott, C. McCrtin, N. Nishimur, P. Rgde, F. Rosemnd, M. Sudermn, S. Whitesides,nd D. Wood, On the prmeterized complexity of lyered grph drwing, In F. Meyer uf der Heide, ed., Europen Symposium on Algorithms ESA 01, LNCS 2161, pges Springer, 2001.

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